diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhyhf" "b/data_all_eng_slimpj/shuffled/split2/finalzzhyhf" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhyhf" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{introduction}\nIn \\cite{SpeyerSturmfels} the authors associated Gr\\\"obner toric\ndegenerations $\\mathrm{Gr}_2(\\mathbb{C}^n)^{\\mathcal{T}}$ of $\\mathrm{Gr}_2(\\mathbb{C}^n)$\nto each\ntrivalent tree $\\mathcal{T}$ with $n$ leaves. These degenerations\ninduce toric\ndegenerations $M_{\\mathbf{r}}^{\\mathcal{T}}$ of $M_{\\mathbf{r}}$, the\nspace of $n$ ordered, weighted (by $\\mathbf{r}$) points on the projective line. \nWe denote the corresponding toric fibers by\n$\\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$ and $(M_{\\mathbf{r}})_0^{\\mathcal{T}}$ respectively.\nOur goal in this paper is to give a\ngeometric (Euclidean polygon) description of the toric fibers as\nstratified symplectic spaces (see \\cite{SjamaarLerman} for this\nnotion) and describe the action of the compact part of the torus\nas ``bendings of polygons.''\n\n\n\\subsection{The Grassmannian and imploded spin-framed polygons}\nWe start by identifying the Grassmannian $\\mathrm{Gr}_2(\\mathbb{C}^n)$\nwith the moduli space of\n``imploded spin-framed'' $n$-gons in $\\mathbb{R}^3$.\nWe define the space of \\emph{imploded framed vectors}, which is\ntopologically the cone $C \\mathrm{SO}(3,\\mathbb{R})$ of $\\mathrm{SO}(3,\\mathbb{R})$, as the space\n$$\\Big\\{(F,e) \\in \\mathrm{SO}(3,\\mathbb{R}) \\times \\mathbb{R}^3 \\mid\n\\text{ $e = tF(\\epsilon_1)$ for some $t \\in \\mathbb{R}_{\\geq 0}$}\\Big\\}$$\nmodulo the equivalence relation\n$(F_1,0) \\sim (F_2,0)$ for all $F_1,F_2 \\in \\mathrm{SO}(3,\\mathbb{R})$,\nwhere $\\epsilon_1 = (1,0,0)$ is the first standard basis vector of\n$\\mathbb{R}^3$.\nWe will see later that this equivalence relation is ``implosion'' in\nthe sense of \\cite{GuilleminJeffreySjamaar}.\nWe call the equivalence class of $(F,e)$ an ``imploded framed vector''. Note that the isotropy $T_{\\mathrm{SO}(3,\\mathbb{R})}$ of $\\epsilon_1$ in $\\mathrm{SO}(3,\\mathbb{R})$ has a natural right action on the space of imploded framed vectors. \n\nNow fix a covering homomorphism $\\pi : \\mathrm{SU}(2) \\to \\mathrm{SO}(3,\\mathbb{R})$ such that the Cartan subgroup of diagonal matrices $T_{\\mathrm{SU}(2)}$ in $\\mathrm{SU}(2)$\nmaps onto $T_{\\mathrm{SO}(3,\\mathbb{R})}$.\nWe define the space of\nimploded \\emph{spin}-framed vectors, which is topologically\nthe cone $C \\mathrm{SU}(2) \\cong \\mathbb{C}^2$, as the space\n$$\\Big\\{(F,e) \\in \\mathrm{SU}(2) \\times \\mathbb{R}^3 \\mid\n\\text{ $e = t \\pi(F)(\\epsilon_1)$ for some $t \\in \\mathbb{R}_{\\geq 0}$}\\Big\\}$$\nmodulo the equivalence relation\n$(F_1,0) \\sim (F_2,0)$ for all\n$F_1,F_2 \\in \\mathrm{SU}(2)$.\nWe call the equivalence class of $(F,e)$ an ``imploded spin-framed vector''.\nAn \\emph{imploded spin-framed $n$-gon} is an $n$-tuple\n $((F_1,e_1),\\ldots,(F_n,e_n))$ of imploded spin-framed vectors\n such that $e_1 + e_2 + \\cdots + e_n = 0$.\nLet $\\widetilde{P}_n(\\mathrm{SU}(2))$\ndenote the space of imploded spin-framed $n$-gons.\nThere is an\naction of $\\mathrm{SU}(2)$ on $\\widetilde{P}_n(\\mathrm{SU}(2))$ given by\n$$F \\cdot ((F_1,e_1),\\ldots,(F_n,e_n)) =\n((F F_1, \\pi(F)(e_1)),\\ldots,(F F_n, \\pi(F)(e_n))).$$\nWe let $P_n(\\mathrm{SU}(2))$ denote the quotient\nspace. Note that since we may scale the edges of an $n$-gon the\nspace $P_n(\\mathrm{SU}(2))$ is a cone with vertex the zero $n$-gon (all the\nedges are the zero vector in $\\mathbb{R}^3$). Finally, note that \nthere is a natural right action of $T_{\\mathrm{SU}(2)^n}$ on \n$\\widetilde{P}_n(\\mathrm{SU}(2))$, which rotates frames but fixes \nthe vectors:\n$$(t_1,\\ldots,t_n) \\cdot ((F_1,e_1),\\ldots,(F_n,e_n)) = \n((F_1 t_1, e_1),\\ldots,(F_n t_n, e_n)).$$\n\n\\begin{remark}\nWe will see later that $P_n(\\mathrm{SU}(2))$\noccurs naturally in equivariant symplectic\ngeometry, \\cite{GuilleminJeffreySjamaar}: it is the symplectic\nquotient by the left diagonal action of $\\mathrm{SU}(2)$ on the right\nimploded cotangent bundle of $\\mathrm{SU}(2)^n$. The space\n$\\widetilde{P}_n(\\mathrm{SU}(2))$ is the zero level set of the momentum map\nfor the left diagonal action on the right imploded product of\ncotangent bundles. Hence we find that the space $P_n(\\mathrm{SU}(2))$ has a\n(residual) right action of an $n$--torus $T_{\\mathrm{SU}(2)^n}$, the\nmaximal torus in $\\mathrm{SU}(2)^n$ which rotates the imploded spin-frames.\n\\end{remark}\n\nLet $Q_n(\\mathrm{SU}(2))$ be the quotient of the subspace of imploded spin-framed $n$-gons\nof perimeter $1$ by the action of the diagonal embedded circle in\n$T_{\\mathrm{SU}(2)^n}$. In what follows $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ denotes the affine cone over\nthe Grassmannian $\\mathrm{Gr}_2(\\mathbb{C}^n)$ for the Pl\\\"ucker embedding. Thus $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ is the\nsubcone of $\\bigwedge^2(\\mathbb{C}^n)$ consisting of the decomposable\nbivectors (the zero locus of the Pl\\\"ucker equations). The reason\nwhy we consider $Q_n(\\mathrm{SU}(2))$ here is the following theorem (proved in \\S \\ref{HJconstruction})\nwhich gives a polygonal interpretation of $\\mathrm{Gr}_2(\\mathbb{C}^n)$. It is the\nstarting point of our work.\n\n\\begin{theorem}\\label{polygonGrass}\\hfill\n\\begin{enumerate}\n\\item $P_n(\\mathrm{SU}(2))$ and\n$\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ are homeomorphic.\n\\item This homeomorphism induces a homeomorphism between $Q_n(\\mathrm{SU}(2))$ and $\\mathrm{Gr}_2(\\mathbb{C}^n).$\n\\end{enumerate}\n\\end{theorem}\n\n\n\\subsection{Triangulations, trivalent trees, and the construction of Kamiyama-Yoshida}\nLet $P$ denote a fixed convex\nplanar $n$-gon.\nThroughout the paper we will use the\nsymbol $\\mathcal{T}$ to denote either a triangulation of $P$ or its dual\ntrivalent tree. Accordingly we fix a triangulation $\\mathcal{T}$ of $P$.\nThe points of\n$\\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$ are ``imploded spin-framed $n$--gons'' with a\nfixed perimeter in $\\mathbb{R}^3$\nmodulo an equivalence relation called $\\mathcal{T}$-congruence and denoted\n$\\sim_{\\mathcal{T}}$ that depends on the triangulation $\\mathcal{T}$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale = 0.2]{newpoly.eps}\n\\caption{A framed spatial polygon with chosen triangulation.}\\label{fig:TriangulatedPolygon}\n\\end{figure}\n\n\nNow that we have a polygonal interpretation of $\\mathrm{Gr}_2(\\mathbb{C}^n)$ we\nwill impose the equivalence relation of $\\mathcal{T}$-congruence (to be\ndescribed below) on our space of framed polygons and obtain a polygonal\ninterpretation of $\\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$\ncorresponding to the triangulation (tree) $\\mathcal{T}$. We now describe\nthe equivalence relation of $\\mathcal{T}$-congruence.\nHere we will discuss only the case\nof the {\\it the standard triangulation} $\\mathcal{T}_0$, that is the\ntriangulation of $P$ given by drawing the diagonals from the first\nvertex to the remaining nonadjacent vertices. The dual tree to the\nstandard triangulation will be called the caterpillar or fan\n(see Figure \\ref{fig:CaterpillarTree7Leaves}). However we will state \nour theorems in the generality in which they are proved in the paper \nnamely for all trivalent trees $\\mathcal{T}$ with $n$ leaves.\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale = 0.2]{NP3.eps}\n\\caption{The standard triangulation of the model $7$-gon with dual\ncaterpillar tree.}\\label{fig:CaterpillarTree7Leaves}\n\\end{figure}\n\n\n\n\n\n\n\n\n\nThe reader is urged to refer to the pictures below to\nunderstand the following description. The equivalence relation\nfor the standard triangulation (in the case of $n$-gon linkages)\ndescribed below was first introduced in \\cite{KamiyamaYoshida}. We\nhave extended their definition to all triangulations and to\n$n$-gons equipped with imploded spin-frames. Label the diagonals\nof the triangulation counterclockwise by $1$ through $n-3$. For each $S\n\\subset \\{1,2,\\cdots,n-3\\}$ let $\\widetilde{P}_n(\\mathrm{SU}(2))^{[S]}$ denote the\nsubspace of $\\widetilde{P}_n(\\mathrm{SU}(2))$ where the diagonals\ncorresponding to the elements in $S$ are zero and all other diagonals are nonzero.\nLet $\\mathbf{F}=\n((F_1,e_1),(F_2,e_2),\\cdots,(F_n,e_n))$ be a point\nin $\\widetilde{P}_n(\\mathrm{SU}(2))^{[S]}$. Suppose that $|S| = i$.\nSince $i$ diagonals are zero there will be $i+1$ sums of the form\n$e_j + e_{j+1}+ \\cdots + e_{j+k_j}$ that are zero, and the\n$n$-gon underlying $\\mathbf{F}$ will be the wedge of $i+1$ closed\nsubpolygons corresponding to the $i+1$ closed subpolygons in the\ncollapsed reference polygon. Thus we can divide $\\mathbf{F}$ into\n$i+1$ imploded spin-framed closed subpolygons ( in terms of\nformulas we can break up the above $n$-tuple into $i+1$ sub\n$k_j$-tuples of edges for $1 \\leq j \\leq i+1$). We may act on\neach imploded spin-framed subpolygon ($k_j$-tuple) by a copy of\n$\\mathrm{SU}(2)$. We pass to the quotient $\\widetilde{P}_n(\\mathrm{SU}(2))\/\\sim_{\\mathcal{T} _0}$\nby dividing out $\\widetilde{P}_n(\\mathrm{SU}(2))^{[S]}$ by the resulting action\nof $\\mathrm{SU}(2)^{i+1}$. By definition $\\mathcal{T}$-congruence is\nthe equivalence relation induced by the above quotient operation,\nsee Figure \\ref{fig:TreeEquivalence}.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale = 0.2]{newpoly3.eps}\n\\caption{A picture of equivalent polygons in $V_n^\\mathcal{T}$ which\nare not equivalent in $P_n(\\mathrm{SU}(2))$.}\\label{fig:TreeEquivalence}\n\\end{figure}\n\nWe let $V_n^{\\mathcal{T}_0}$ denote the quotient space\n$\\widetilde{P}_n(\\mathrm{SU}(2))\/\\sim_{\\mathcal{T}_0}$. Thus $V_n^{\\mathcal{T}_0}$ is\ndecomposed into the pieces $(V_n^{\\mathcal{T}_0})^{[S]} =\n\\widetilde{P}_n(\\mathrm{SU}(2))^{[S]}\/\\sim_{\\mathcal{T}_0}$. We will call the\nresulting decomposition in the special case of $\\mathcal{T}_0$\nthe Kamiyama-Yoshida decomposition (or\nKY-decomposition). In this paper we use a\nweakened definition of the term {\\it decomposition}; we will refer\nto decompositions of spaces where the pieces are products\nof spaces with isolated singularities.\n\n\n\n\\begin{remark}\nThe equivalence relation of $\\mathcal{T}$-congruence can be defined \nanalogously for\nany triangulation of $P$ (equivalently any trivalent tree with\n$n$-leaves) and induces an equivalence relation on $Q_n(\\mathrm{SU}(2))$\n(and many other spaces associated to spaces of $n$-gons in $\\mathbb{R}^3$,\nfor example, the space of $n$-gons itself or the space of\n$n$-gon linkages). We will use the symbol $\\sim_{\\mathcal{T}}$ to denote\nall such equivalence relations.\n\\end{remark}\n\n\nWe will use $W_n^{\\mathcal{T}}$ to denote the quotient of\n$Q_n(\\mathrm{SU}(2))$ by\nthe equivalence relation $\\sim_{\\mathcal{T}}$. We can now state our\nfirst main result (this is proved in \\S \\ref{homeomorphismVtoP}).\n\n\\begin{theorem} \\label{firstmaintheorem}\\hfill\n\\begin{enumerate}\n\\item The toric fiber of the toric degeneration of\n$\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ corresponding to\nthe trivalent tree $\\mathcal{T}$ is homeomorphic to $V_n^{\\mathcal{T}}$.\n\\item The toric fiber of the toric degeneration of\n$\\mathrm{Gr}_2(\\mathbb{C}^n)$\ncorresponding to the trivalent tree $\\mathcal{T}$ is homeomorphic to\n$W_n^{\\mathcal{T}}$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\nThe quotient map from\n$Q_n(\\mathrm{SU}(2))$ to\n$W_n^{\\mathcal{T}}$ given by passing to $\\mathcal{T}$-congruence classes\nmaps the generic fiber of the toric degeneration onto the special\n(toric) fiber.\n\\end{remark}\n\nWe next describe the space $V_{\\mathbf{r}}^{\\mathcal{T}_0}$ that will be proved\nlater to be homeomorphic to the toric fiber $(M_{\\mathbf{r}})_0^{\\mathcal{T}_0}$.\nAgain we will restrict ourselves to the standard triangulation\nin our description. Starting with the space $\\widetilde{M}_{\\mathbf{r}} =\n\\{(e_1,\\ldots,e_n) \\in (\\mathbb{R}^3)^n \\mid \\sum_i e_i = 0, \\; \\|e_i\\| = r_i\\}$\nof closed $n$-gon linkages with side-lengths $\\mathbf{r}$, we define\n$$V_{\\mathbf{r}}^{\\mathcal{T}_0} = \\widetilde{M}_{\\mathbf{r}}\/\\sim_{\\mathcal{T}_0}.$$\nThis is the construction of \\cite{KamiyamaYoshida} - see below\nfor some pictures.\n\n\n\\begin{theorem} \\label{secondmaintheorem}\nThe toric fiber $(M_{\\mathbf{r}})^{\\mathcal{T}}_0$ of the toric degeneration\nof $M_{\\mathbf{r}}$ corresponding to\nthe trivalent tree $\\mathcal{T}$ is homeomorphic to\n$V_{\\mathbf{r}}^{\\mathcal{T}}$.\n\\end{theorem}\n\n\\begin{remark} The quotient map from $M_{\\mathbf{r}}$ to\n$V_{\\mathbf{r}}^{\\mathcal{T}}$ maps the generic fiber of the toric degeneration onto the special\n(toric) fiber.\n\\end{remark}\n\nThe result in the previous theorem was\nconjectured by Philip Foth and Yi Hu in \\cite{FothHu}.\n\n\\subsection{Bending flows, edge rotations, and the toric structure of $W_{n}^{\\mathcal{T}_0}$}\nThe motivation for the above construction becomes more clear once we introduce the bending flows of\n\\cite{KapovichMillson} and \\cite{Klyachko}. The lengths of the $n-3$ diagonals created are continuous\nfunctions on $M_{\\mathbf{r}}$ and are smooth where they are not zero.\nThey give rise to Hamiltonian flows which were called {\\em bending\nflows} in \\cite{KapovichMillson}. The bending flow associated to a\ngiven diagonal has the following description. The part of the\n$n$-gon to one side of the diagonal does not move, while the other\npart rotates around the diagonal at constant speed. The lengths of\nthe diagonals are action variables which generate the bending\nflows, the conjugate angle variables are the dihedral angles\nbetween the fixed and moving parts. However {\\em the bending flow\nalong the $i$-th diagonal is not defined at those $n$-gons where\nthe $i$-th diagonal is zero}. If the bending flows are everywhere\ndefined (for example if one of the side-lengths is much larger\nthan the rest) then we may apply the theorem of Delzant,\n\\cite{Delzant} to conclude that $M_{\\mathbf{r}}$ is toric. However for\nmany $\\mathbf{r}$ (including the case of regular $n$\n-gons) the bending\nflows are not everywhere defined. The point of\n\\cite{KamiyamaYoshida} was to make the bending flows well-defined\nby dividing out the subspaces of $\\widetilde{M}_{\\mathbf{r}}$ where a\ncollection of $i$ diagonals vanish by $\\mathrm{SO}(3,\\mathbb{R})^{i+1}$. We\nillustrate their construction with two examples.\\vspace {1 mm}\n\n\nFirst, let $\\mathbf{r} = (1,1,1,1,1,1)$ so $M_{\\mathbf{r}}$ is the space of\nregular hexagons with side-lengths all equal to $1$. Let\n$M_{\\mathbf{r}}^{(2)}$\nbe the subspace of $M_{\\mathbf{r}}$\nwhere the middle (second) diagonal vanishes. Thus $M_{\\mathbf{r}}^{(2)}$\nis the space of ``bowties'' (see Figure \\ref{fig:Bowties}) modulo the diagonal action of $\\mathrm{SO}(3,\\mathbb{R})$\non the two equilateral triangles. We can no longer bend on the\nsecond diagonal because we have no axis of rotation. Passing to\n$\\mathcal{T}$-congruence classes\ncollapses the\nspace $M_{\\mathbf{r}}^{(2)}$ to a point by dividing by the action\nof $\\mathrm{SO}(3,\\mathbb{R}) \\times \\mathrm{SO}(3,\\mathbb{R})$. Bending along the second diagonal fixes \nthis point by definition.\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.2]{NP4.eps}\n\\caption{The bowties are all $\\mathcal{T}_0$-congruent\nand so define a single point in $V_\\mathbf{r}^{\\mathcal{T}_0}$.\nHowever in $M_\\mathbf{r}$ the subspace of bowties is homeomorphic to $\\mathrm{SO}(3,\\mathbb{R})$.}\\label{fig:Bowties}\n\\end{figure}\n\n\nFor our second example, we consider the space of regular octagons\nwith all side-lengths equal to $1$ and the subspace $M_{\\mathbf{r}}^{(3)}$ where\nthe middle diagonal vanishes. Thus $M_{\\mathbf{r}}^{(3)}$ is the space of\nwedges of rhombi modulo the diagonal action of $\\mathrm{SO}(3,\\mathbb{R})$ on the two\nrhombi, see Figure \\ref{fig:TwoRhombi}. Passing to $\\mathcal{T}$-congruence classes\namounts to dividing by the action of $\\mathrm{SO}(3,\\mathbb{R}) \\times \\mathrm{SO}(3,\\mathbb{R})$ on the\ntwo rhombi.\n\n\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.2]{NP5.eps}\n\\caption{Here\n$n=8$ and $S= \\{3 \\}$.\nThe middle three components\nof $T^5$ act trivially, and the\nquotient $2$-torus acts by\nbending along the first and fifth diagonals.}\\label{fig:TwoRhombi}\n\\end{figure}\n\n\n\n\n\n\n We obtain the action of the bending flows on $V_n^{\\mathcal{T}_0}$\nas follows. In this case we will be given a lift of the\none-parameter group bending along a diagonal to $\\mathrm{SU}(2)$. The one\nparameter group acts through its quotient in $\\mathrm{SO}(3,\\mathbb{R})$ by bending\nalong the diagonal. If an edge moves under this bending then the\nimploded spin-frame is moved by the one-parameter group in $\\mathrm{SU}(2)$\nin the same way. Hence, one part of the imploded spin-framed\npolygon is fixed and the other moves by a ``rigid motion'' --i.e. all\nthe spin-framed edges of the second part are moved by the same\none-parameter group in $\\mathrm{SU}(2)$. The bendings give rise to an\naction of an $n-3$ torus $T_{bend}$ on $V_n^{\\mathcal{T}}$. There are also ''edge-rotations''\nthat apply a one-parameter group to the imploded spin-frame but do\nnot move the edge. This action on frames is the action of the torus\n$T_{\\mathrm{SU}(2)^n}$ coming from the theory of the imploded cotangent bundle\nof $\\mathrm{SU}(2)^n$. This action is not faithful, the diagonal subtorus acts\ntrivially. The bendings together with the edge rotations give rise to\nan action of a compact $2n-4$ torus $T = T_{bend} \\times T_{\\mathrm{SU}(2)^n}$.\n\n\\subsection{A sketch of the proofs.}\n\nThe main step in proving Theorems \\ref{firstmaintheorem} and \\ref{secondmaintheorem} is to\nproduce a space $P_n^{\\mathcal{T}_0}(\\mathrm{SU}(2))$ that ``interpolates''\nbetween $V_n^{\\mathcal{T}_0}$ and $\\mathrm{Gr}_2(\\mathbb{C}^n)^{\\mathcal{T}_0}_0$.\nOur construction of $P_n^{\\mathcal{T}_0}(\\mathrm{SU}(2))$\nwas motivated by the construction of the toric\ndegeneration of $\\mathrm{SU}(2)$-character varieties of fundamental groups\nof surfaces given by Hurtubise and Jeffrey in\n\\cite{HurtubiseJeffrey}. The connection is that the space\n$M_{\\mathbf{r}}$ can be interpreted as the (relative) character variety\nof the fundamental group of the $n$-punctured two-sphere with\nvalues in the translation subgroup of the Euclidean group\n$\\mathbb{E}_3$ - a small loop around the $i$-th puncture maps to\ntranslations by the $i$-th edge of the polygon (considered as a\nvector in $\\mathbb{R}^3$).\n\nTake the triangulated model (convex planar) $n$-gon $P$ and break\nit apart into $n-2$ triangles $T_1,\\cdots,T_{n-2}$.\nEquivalently we break apart the dual tree $\\mathcal{T}$ into a forest\n$\\mathcal{T}^D$ consisting of $n-2$ tripods.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.2]{newpoly2.eps}\n\\caption{The decomposed polygon and the decomposed dual tree $\\mathcal{T}^D$.}\\label{fig:decomposed_ngon}\n\\end{figure}\n\n\n\nAttach to each of the $3(n-2)$ edges of the $n-2$ triangles\n(or each edge of the forest $\\mathcal{T}^D$) a copy\nof $T^*(\\mathrm{SU}(2))$. Now right-implode each copy of $T^*(\\mathrm{SU}(2))$ so that a\ncopy of $\\mathcal{E} T^\\ast(\\mathrm{SU}(2))) \\cong \\mathbb{C}^2$, the imploded cotangent bundle of\n$\\mathrm{SU}(2)$, is attached to each edge. Since\n$\\mathcal{E} T^\\ast(\\mathrm{SU}(2))$ admits\nan action of the circle (from the right) and $\\mathrm{SU}(2)$ from the left\nthe resulting space admits an action of a torus $\\mathbb{T}$ of\ndimension $3(n-2)$ and a\ncommuting action of $\\mathrm{SU}(2)^{3(n-2)}$. The torus has a product\ndecomposition $\\mathbb{T} = \\mathbb{T}_e \\times\\mathbb{T}_d$\nwhere $\\mathbb {T}_e$ is the product of factors corresponding to\nthe $n$ edges of the polygon $P$ and $\\mathbb{T}_d$ is the\nproduct of factors corresponding to the $n-3$ diagonals of $P$.\nNote that each diagonal of $P$ occurs in two triangles so corresponds to\ntwo edges of $\\mathcal{T}^D$. Hence each diagonal gives\nrise to a two-torus $S^1 \\times S^1$ in $\n\\mathbb{T}$ which we will\nrefer to as the two-torus corresponding to that diagonal. Define a\nsubtorus $\\mathbb{T}_d^-$ of $\\mathbb{T}_d$ of dimension $n-3$ by\ntaking an antidiagonal embedding of $S^1$ in each two-torus\ncorresponding to a diagonal.\n\n\nNow take the symplectic quotient (at level zero) of (the product\nof) the three copies of $\\mathbb{C}^2$ associated to the three sides of\neach triangle by $\\mathrm{SU}(2)$ acting diagonally. For each triangle\n(or each tripod) we\nobtain a resulting copy of $\\bigwedge^2(\\mathbb{C}^3)$. The resulting\nproduct $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ has an induced action of the\ntorus $\\mathbb{T}$. Glue the copies $\\bigwedge^2(\\mathbb{C} ^3)$\nassociated to the triangles together along the edges of the\ntriangles associated to diagonals by taking the symplectic quotient at level zero by the\ntorus $\\mathbb{T}_d^- \\subset \\mathbb{T}_d$ described above. Each of\nthe two previous symplectic quotients has a corresponding GIT quotient.\n Taking both GIT quotients we obtain\na space which is an affine torus quotient of affine space. Hence the combined\nsymplectic quotient is the space underlying the\naffine toric variety\n$$P_n^{\\mathcal{T}_0}(\\mathrm{SU}(2)) =(\\mathrm{\\bigwedge}\n\\space^2(\\mathbb{C}^3))^{n-2}\/\\!\/\n_0 \\mathbb{T}_d^-.$$\nLet $\\mathbf{t}_e(\\lambda)$ be the element in complexified torus $\\underline{\\mathbb{T}} \\cong (\\mathbb{C}^\\ast)^{3n-6}$ of $\\mathbb{T}$ \nsuch that all the edge\ncomponents coincide with $\\lambda \\in \\mathbb{C}^{\\ast}$ and\nall the components corresponding to\ndiagonals are $1$. Then Theorem \\ref{firstmaintheorem} follows by\nputting together items 2 and 4 in the next theorem, which is proved in\n\\S \\ref{homeomorphismVtoP}.\n\n\\begin{theorem} \\label{firstauxiliarytheorem}\n\\hfill\n\\begin{enumerate}\n\\item The toric varieties $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$ \nand $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ are isomorphic as\naffine toric varieties.\n\\item The grading action of $\\lambda \\in\n\\mathbb{C}^*$ on $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$ corresponds to\nthe action of $\\mathbf{t}_e(\\sqrt{\\lambda}^{-1})$ on\n$P_n^{\\mathcal{T} }(\\mathrm{SU}(2))$ (this is well-defined).\nConsequently $ \\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$\nis projectively isomorphic to the quotient of\n$P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ by this $\\mathbb{C}^{\\ast }$ action - we will\ndenote this quotient by $Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\n\\item There is a\nhomeomorphism (that creates imploded spin-frames along the\ndiagonals of the triangulation)\n$$\\Psi_n^{\\mathcal{T}}:V_n^{\\mathcal{T}} \\to P_n^{\\mathcal{T}}(\\mathrm{SU}(2)).$$\n\\item The homeomorphism\n$\\Psi_n^{\\mathcal{T}}$ induces a\nhomeomorphism from $W_n^{\\mathcal{T}}$ to the projective toric variety\n$Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\n\\end{enumerate}\n\\end{theorem}\nThe quotient\n$P_n^{\\mathcal{T}_0}(\\mathrm{SU}(2))=\\bigwedge^2(\\mathbb{C}^3)^{n-2} \/\\!\/ \\mathbb{T}_d^-$\nadmits a residual\naction by the quotient torus $\\mathbb{T}\/ \\mathbb{T}_d^-$. This\nquotient torus contains a factor that can be identified with\n$\\mathbb{T}_e$. The toric fiber $(M_{\\mathbf{r}})_0$ is obtained from\n$P_n^{\\mathcal{T}_0}(\\mathrm{SU}(2))$ by taking the symplectic quotient by\n$\\mathbb{T}_e$ at level $\\mathbf{r}$. Theorem \\ref{secondmaintheorem}\nfollows from the two statements of the following theorem.\n\n\\begin{theorem}\\label{secondauxiliarytheorem}\\hfill\n\\begin{enumerate}\n\\item The toric variety $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))\/\\!\/_{\\mathbf{r}} \\mathbb{T}_e $\nis isomorphic to the toric variety\n$(M_{\\mathbf{r}})_0^{\\mathcal{T}}$.\n\\item For each $\\mathbf{r}$ the homeomorphism $\\Psi_n^{\\mathcal{T}}$ induces a homeomorphism\n$\\Psi_{\\mathbf{r}}:V^{\\mathcal{T}}_{\\mathbf{r}} \\to P_n^{\\mathcal{T}}(\\mathrm{SU}(2))\/\\!\/_{\\mathbf{r}} \\mathbb{T}_e .$\n\\end{enumerate}\\end{theorem}\n\n\nNote that the toric varieties\n$(\\bigwedge^2(\\mathbb{C}^3))^{n-2} \/\\!\/ \\mathbb{T}_d^-$\nresp. $\\bigwedge^2(\\mathbb{C}^3)^{n-2}\n\/\\!\/_{\\mathbf{r},0} (\\mathbb{T}_e \\times\\mathbb{T}_d^-)$ mediate between the\nKamiyama-Yoshida spaces $V_n^{\\mathcal{T}}$ resp. $V_{\n\\mathbf{r} }^{\\mathcal{T}}$ and the toric\nvarieties $\\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$ resp. $(M_{ \\mathbf{r}})_0^{\\mathcal{T}}$.\n\n\\subsection{The Hamiltonian nature of the edge rotations and the bending flows}\nIt remains to place the edge rotations and bending flows in their proper context (in\nterms of symplectic and algebraic geometry). We do this with the following theorems. \nWe note that since the spaces\n$P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ and $P_{\\mathbf{r}}^{\\mathcal{T}}(\\mathrm{SU}(2))$ are quotients of\nthe affine space $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ by tori they inherit symplectic\nstratifications from the orbit type stratification of\n$(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ according to \\cite{SjamaarLerman}. \nWe identify the edge flows and bending flows with the action of\nthe maximal compact subgroups (compact torus) of the complex tori\nthat act holomorphically with open orbits on the toric\nvarieties $\\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$ and\n$(M_{ \\mathbf{r}})_0^{\\mathcal{T}}$. This is accomplished\nby the following theorem, which is proved\nby Theorems \\ref{noncompact} and \\ref{noncompact2}.\n\n\\begin{theorem}\n\\hfill\n\\begin{enumerate}\n\\item The action of the edge rotations corresponds under\n$\\Psi_n^{\\mathcal{T}}$ to the residual action of $\\mathbb{T} \/ \\mathbb{T}_d^-$.\n\\item The action of the bending flows corresponds under\n$\\Psi_n^{\\mathcal{T}}$ (resp. $\\Psi_{\\mathbf{r}}^{\\mathcal{T}}$) to the residual action\nof $\\mathbb{T} \/ \\mathbb{T}_e \\times \\mathbb{T}_d^-$. \n\\end{enumerate}\n\n\\end{theorem}\nWe may now give the edge rotations and the \nbending flows on $V_n^{\\mathcal{T}}$ and $V_{\\mathbf{r}}^{\\mathcal{T}}$ \na natural Hamiltonian interpretation. This is\nproved by the last theorem along with Proposition \\ref{pullback}\nand Theorem \\ref{diagonalpullback}.\n\n\\begin{theorem}\\label{stratification}\n\\hfill\n\\begin{enumerate}\n\\item The edge rotations on $V_n^{\\mathcal{T}}$ are the stratified\nsymplectic Hamiltonian flows (in the sense of \\cite{SjamaarLerman})\nassociated to the lengths of edges in $\\mathcal{T}$-congruence\nclasses of spin-framed polygons. \n\\item The bending flows on $V_n^{\\mathcal{T}}$ (resp. $V_{\\mathbf{r}}^{\\mathcal{T}}$) are the\nstratified Hamiltonian flows associated to the lengths of the diagonals \nof $\\mathcal{T}$-congruence classes of spin-framed polygons (resp. polygonal linkages). \n\\end{enumerate}\n\n\\end{theorem}\n\n\n\n\n\\medskip\n\n{\\bf Acknowledgements.} \nWe thank Bill Goldman, Henry King and Reyer Sjamaar for useful\nconversations. The authors would also like to thank\nAndrew Snowden and Ravi Vakil, a number of ideas from the collaboration\n \\cite{HowardMillsonSnowdenVakil} have reappeared in this paper.\n We thank Philip Foth and Yi Hu for posing the\nproblem solved in this paper. They first observed that the toric\ndegenerations of flag varieties constructed by \\cite{AlexeevBrion}\ncould be descended to the associated weight varieties. We\nespecially thank Philip Foth for pointing out that the\nconstruction of \\cite{AlexeevBrion} gives one toric degeneration\nof $M_{\\mathbf{r}}$ for each triangulation of the model convex\n$n$-gon.\nThis led us to consider triangulations other than the standard\none. We thank Bernd Sturmfels for telling us\nabout \\cite{SpeyerSturmfels} and \\cite{BuczynskaWisniewski} which\nled us to understand from the point of view of tree metrics why\nthere was one toric degeneration for each triangulation.\nWe thank Allen Knutson for pointing out that toric degenerations\nof Grassmannians were first constructed by Sturmfels in\n\\cite{Sturmfels}. Finally we should emphasize that\nthe notion of $\\mathcal{T}$-congruence\nis based on the work of Kamiyama and Yoshida \\cite{KamiyamaYoshida}\nand that the notion of bending flows is based on \\cite{KapovichMillson}\nand \\cite{Klyachko}.\n\n\n\n\n\n\n\\section{The moduli spaces of $n$-gons and $n$-gon linkages in $\\mathbb{R}^3$}\nThroughout this paper the term $n$-gon will mean a closed $n$-gon\nin $\\mathbb{R}^3$ modulo translations. More precisely an $n$-gon $\\mathbf{e}$\nwill be an $n$-tuple $\\mathbf{e} = (e_1,e_2,\\cdots,e_n)$ of vectors\nin $\\mathbb{R}^3$ satisfying the closing condition\n$$e_1 + e_2 + \\cdots + e_n =0.$$\nWe will say the $e_i$ is the $i$-th edge of $\\mathbf{e}$.\nWe will say two $n$-gons $\\mathbf{e}$ and $\\mathbf{e}^{\\prime}$ are congruent\nif there exists a rotation $ g \\in \\mathrm{SO}(3,\\mathbb{R})$ such that\n$$ e_i^{\\prime} = g e_i , 1 \\leq i \\leq n.$$\nWe will let ${\\mathrm{Pol}_ n}$ denote the space of closed $n$-gons in $\\mathbb{R}^3$\nand $\\overline{\\mathrm{Pol}}_ n$ denote the quotient space of $n$-gons modulo congruence.\n\nNow let $\\mathbf{r} = (r_1,r_2,\\cdots,r_n)$ be an $n$-tuple of nonegative\nreal numbers. We will say an $n$-gon $\\mathbf{e}$ is an $n$-gon { \\em linkage}\nwith side-lengths $\\mathbf{r}$ if the $i$-th edge of $\\mathbf{e}$ has length $r_i,\n1 \\leq i \\leq n$. We will say an $n$-gon or $n$-gon linkage is\n{\\it degenerate} if it is contained in a line.\n\n\n\nWe define the configuration space $\\widetilde{M_{\\mathbf{r}}}$ to be\nthe set of $n$--gon linkages with side-lengths $\\mathbf{r}$. We will define the moduli space $M_{\\mathbf{r}}$ of $n$-gon linkages to be the quotient of\nthe configuration space by $\\mathrm{SO}(3,\\mathbb{R})$. The space $M_{\\mathbf{r}}$ is a complex analytic space\n(see \\cite{KapovichMillson} with isolated singularities at the degenerate $n$-gon linkages.\n\nRecall that we have defined a reference convex planar $n$-gon $P$.\nLet $u_i, u_j$ be an ordered pair of nonconsecutive vertices of $P$. For any $n$-gon $\\mathbf{e}\n\\in \\mathbb{R}^3$ we have corresponding vertices $v_i$ and $v_j$ defined up to\nsimultaneous translation. The vector in $\\mathbb{R}^3$ pointing from $v_i$ to $v_j$\nwill be called a {\\em diagonal} of $\\mathbf{e}$. We let $d_{ij}(\\mathbf{e})$\nbe the length of this diagonal.\nIn \\cite{KapovichMillson} and \\cite{Klyachko} the authors described\nthe Hamiltonian flow corresponding to $d_{ij}(\\mathbf{e})$ - see the Introduction.\nIn \\cite{KapovichMillson} these flows were called {\\em bending flows}.\nFurthermore it was proved in \\cite{KapovichMillson} and \\cite{Klyachko} if two such diagonals do not\nintersect then the corresponding bending flows commute.\nSince each triangulation $\\mathcal{T}$ of $P$ contains $n-3 = \\frac{1}{2} \\dim (M_{\\mathbf{r}})$\nnonintersecting diagonals it follows that each one has an integrable\nsystem on $M_{\\mathbf{r}}$ for each such $\\mathcal{T}$. Unfortunately these flows\nare not everywhere defined. The bending flow\ncorresponding to $d_{ij}$ is not well-defined for those $\\mathbf{e}$ where $d_{ij}(\\mathbf{e})$ is zero and the Hamiltonian $d_{ij}$ is not differentiable\nat such $\\mathbf{e}$.\n\n\n\n\n\n\n\n\n\\section{The space of imploded spin-framed Euclidean $n$--gons and the\nGrassmannian of two planes in complex $n$ space} \\label{HJconstruction}\n\nIn this section we will construct the space $P_n(\\mathrm{SU}(2))$ of imploded spin-framed $n$--gons in $\\mathbb{R}^3$ modulo $\\mathrm{SU}(2)$ and prove that this space is\nisomorphic\nto $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ as a symplectic manifold\nand as a complex projective variety. We will first\nconstruct $P_n(\\mathrm{SU}(2))$ using the extension and implosion technique\nof \\cite{HurtubiseJeffrey} without reference to $n$--gons in $\\mathbb{R}^3$,\nthen\nrelate the\nresult to $\\mathrm{Gr}_2(\\mathbb{C}^n)$. Then we will show that a point in $P_n(\\mathrm{SU}(2))$\ncan be interpreted as a Euclidean $n$-gon equipped with an\nimploded spin-frame.\n\n\\subsection{The imploded extended moduli spaces $P_n(G)$ of $n$-gons\nand $n$-gon linkages.}\nIn this subsection we will define the imploded extended moduli\nspace $P_n(G)$ of $n$-gons in $\\mathfrak{g}^*$ for a\ngeneral semisimple Lie group $G$.\nWe have included this subsection to make the connection\nwith \\cite{HurtubiseJeffrey}.\nThroughout we assume that $G$ is semisimple.\\subsubsection{The moduli\nspaces of $n$-gons and $n$-gon linkages}\nTo motivate the definition of the next subsection we briefly recall\ntwo definitions.\\begin{definition}\nAn $n$-gon in $\\mathfrak{g}^*$ is an $n$-tuple of vectors $e_i, 1 \\leq\ni \\leq n, \\in\\mathfrak{g}^*$ satisfying the closing condition$$e_1 +\n\\cdots + e_n = 0.$$ We define the moduli space of $n$-gons to\nbe the set of all $n$-gons modulo the diagonal coadjoint action of\n$G$.\n\\end{definition}\nNow choose $n$ coadjoint orbits $\\mathcal{O}_i, 1 \\leq i \\leq\nn$.\n\n\\begin{definition}\nWe define an $n$-gon {\\em linkage} to be an $n$-gon such that $e_i \\in\n\\mathcal{O}_i, 1 \\leq i \\leq n$. We define the moduli space\nof $n$-gon linkages to be the set of all $n$-gon linkages modulo\nthe diagonal coadjoint action of $G$.\n\\end{definition}\nWe leave the proof of the following lemma to the reader.\n\n\\begin{lemma}\nThe moduli space of $n$-gon linkages is the symplectic quotient$$G \\backslash \\! \\backslash\n(\\prod_{i=1}^n \\mathcal{O}_i).$$\n\\end{lemma}\n\n\n\\begin{remark}\\label{moduliofconnections}\nThe moduli space of $n$-gons is in fact\na moduli space of flat connections modulo gauge\ntransformations (equivalently\na character variety). The moduli space\nof $n$-gon linkages is a moduli space of\nflat connections with the conjugacy classes\nof holonomies fixed in advance (a relative character\nvariety). In this case the\nflat connections are over an $n$-fold punctured $2$-sphere and the\nstructure group is the cotangent bundle $T^*(G) = G \\ltimes \\mathfrak{g}^*$.\nThe holonomy around each puncture is a ``translation'', i.e. a group\nelement of the form $(1,v), v \\in \\mathfrak{g}^*$.\nTo see this connection in more detail the reader is referred to\n\\S 5 of\n\\cite{KapovichMillson}. We will not need this connection in\nwhat follows.\n\\end{remark}\n\n\\subsubsection{The extended moduli spaces of $n$-gons and $n$-gon\nlinkages}\nThe following definition is motivated by the definition of the extended\nmoduli spaces of flat connections of \\cite{Jeffrey}.\n\n\\begin{definition}\nWe define the extended moduli space $M_n(G)$ to be\nthe symplectic quotient of $T^*(G)^n$ by the left diagonal action\nof $G$: $$M_n(G) = G \\backslash \\! \\backslash T^*(G)^n.$$\n\\end{definition}\nThe space $M_n(G)$ has a $G^n$ action coming from right multiplication\non $T^*(G)$.\nWe take each $T^*(G)$ to be identified with $G\\times\\mathfrak{g^*}$ using\nthe\nleft-invariant trivialization.\n\n\\begin{lemma}$M_n(G) = G\\backslash\\{((g_1,\\alpha_1),\n\\ldots,(g_n,\\alpha_n)): \\sum_{i=1}^n Ad_{g_i}(\\alpha_i) = 0\n\\}.$\n\\end{lemma}\n\n\\begin{proof}\nThe momentum mapping associated to the diagonal left action\non $T^*(G)^n$ (identified with $(G \\times \\mathfrak{g}^*)^n$ using the\nleft-invariant trivialization) is\n$$\\mu_L((g_1,\\alpha_1), \\ldots, (g_n,\\alpha_n)) =- \\sum_{i=1}^n\nAd_{g_i}(\\alpha_i).$$\nThe expression on the right above\nclearly corresponds to the 0-momentum level of $\\mu_L$ which\nis $M_n(G)$.\n\\end{proof}\n\n\\begin{remark}\nThe reason for the term {\\em extended} moduli space of $n$-gons is that\nthis space is obtained\nfrom the moduli space of $n$-gons by adding the frames $g_1,g_2,\\cdots,g_n$.\nNote that the moduli space of $n$-gons is embedded in the extended\nmoduli space as the subspace corresponding to $g_1 = g_2 = \\cdots = g_n = e$.\nIf we wish to fix the conjugacy classes of the second components we\nwill call the above the extended moduli space of $n$-gon {\\em\nlinkages}.\n\\end{remark}\n\n\\subsubsection{The imploded extended moduli spaces of $n$-gons and $n$-gon\nlinkages}\nChoose a maximal torus $T_G\\subset G$ and a (closed) Weyl chamber $\\Delta$\ncontained\nin the\nLie algebra $\\mathfrak{t}$ of $T_G$. Note that the action of $G^n$ \non $M_n(G)$ by right multiplication induces an action\nby the torus $T_G^n$ on $M_n(G)$.\nWe now obtain the imploded extended moduli space $P_n(G)$ by {\\it imploding},\nfollowing \\cite{GuilleminJeffreySjamaar}, the extended moduli space $M_n(G)$.\n\n\\begin{definition}\\label{Definitionofimplodedextended moduli space $P_n(G)$} $$P_n(G) = M_n(G)_{impl}.$$\n\\end{definition}\nFollowing \\cite{HurtubiseJeffrey} we will use $\\mathcal{E}T^*(G)$\nto denote the imploded cotangent bundle\n$$\\mathcal{E}T^*(G) = T^*(G)_{impl}.$$\nFor the benefit of the reader we will recall the definition\nof $\\mathcal{E}T^*(G)$. We have\n$$\\mathcal{E}T^*(G) = \\mu_G^{-1}(\\Delta)\/\\sim $$\nHere $\\mu_G$ is the momentum map for the action of $G$\nby right translation and we have identified $\\mathfrak{t}$\n(resp. $\\Delta$) with $\\mathfrak{t}^*$ (resp. a dual Weyl chamber) using the\nKilling form.\nThe equivalence relation $\\sim$ is described as\nfollows.\nLet $F$ be a open face of $\\Delta$ and let $h_F$ be a generic element of $F$.\nLet $G_F$ be the subgroup of $G$ which is the derived subgroup of the\nstabilizer of $h_F$ under the\nadjoint representation. We define $x$ and $y$ in $\\mu_T^{-1}(\\Delta)$\nto be equivalent if $\\mu_T(x)$ and $\\mu(y)$ lie in the same face $F$\nof $\\Delta$ and $x$ and $y$ are in the same orbit under $G_F$. Thus we\ndivide\nout the inverse images of faces by different subgroups of $G$.\nIn what follows we will use the symbol $\\sim$ to denote\nthis equivalence relation (assuming the group $G$, the torus $T$\nand the chamber $\\Delta$ are understood).\n\nWe define the space $E_n(G)$ by\n$$E_n(G) = \\mathcal{E}T^*(G)^n.$$\nNoting that right implosion commutes\nwith left symplectic quotient we have\n$$P_n(G) = G \\backslash E_n(G) \n= G\\backslash \\Big\\{ ([g_1,\\alpha_1], \\ldots, [g_n,\\alpha_n]) \\mid \\alpha_1,\n\\ldots, \\alpha_n \\in \\Delta,\\sum_{i=1}^n Ad_\\mathfrak{g_i}(\\alpha_i) =\n0\\Big\\}.$$\nIn the above $[g_i, \\alpha_i]$ denotes the equivalence class\nin $T^*(G)$ relative to the equivalence relation $\\sim$ above.\nWe will sometimes use $E_n(G)$ to denote the product\n$\\mathcal{E}T^*(G)^n = \\mathcal{E}T^*(G^n)$. In what follows we let\n$\\mathbf{t}(\\lambda)$ denote the element of the complexification maximal torus\n$\\underline{T}_{\\mathrm{SU}(2)^n}$.\n\n\\subsection{The isomorphism of $Q_n(\\mathrm{SU}(2))$ and $\\mathrm{Gr}_2(\\mathbb{C}^n)$}\nIn this subsection we will prove Theorem \\ref{polygonGrass}\nby calculating $P_n(\\mathrm{SU}(2))$, the imploded extended\nmoduli space for the group $\\mathrm{SU}(2)$ and its quotient $Q_n(\\mathrm{SU}(2))$.\nWe will in fact need a slightly more precise version than that\nstated in the Introduction.\n\n\\begin{theorem}\\label{polygonGrasspreciseversion}\n\\hfill\n\\begin{enumerate}\n\\item There exists a homeomorphism\n$\\psi:\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n) \\to P_n(\\mathrm{SU}(2))$.\n\\item The homeomorphism $\\psi$ intertwines the natural action of the maximal torus $T_{\\mathrm{U}(n)}$ of \\ $\\mathrm{U}(n)$ with (the inverse of) that of $T_{\\mathrm{SU}(2)}^n$ acting on imploded frames.\n\\item The homeomorphism $\\psi$ intertwines the grading circle (resp. $\\mathbb{C}^{\\ast}$)\nactions on $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ with the actions of $\\mathbf{t}((\\exp{i\\theta})^{-1\/2})$ (resp. $\\mathbf{t}((\\lambda)^{-1\/2})$)\n\\item The homeomorphism $\\psi$ induces a homeomorphism between $\\mathrm{Gr}_2(\\mathbb{C}^n)$\nand $Q_n(\\mathrm{SU}(2)).$\n\\end{enumerate}\n\n\\end{theorem}\nIn order to prove the theorem we first\nneed to compute $\\mathcal{E}T^*(\\mathrm{SU}(2))$.\n\n\\subsubsection{The imploded cotangent bundle of $\\mathrm{SU}(2)$ }\n\nIn this section we will review the formula\nof \\cite{GuilleminJeffreySjamaar} for the (right) imploded\ncotangent bundle $\\mathcal{E}T^*(\\mathrm{SU}(2))$. Let $T_{\\mathrm{SU}(2)}$ be the maximal\ntorus of $\\mathrm{SU}(2)$ consisting of the diagonal matrices.\nLet $\\mathfrak{t}$ be the Lie algebra of $T_{\\mathrm{SU}(2)}$ and let $\\Delta$ be\nthe\npositive Weyl chamber in $\\mathfrak{t}$ (so $\\Delta$ is a ray in\nthe one-dimensional vector space $\\mathfrak{t}$).\n\n\nWe will take as basis for $\\mathfrak{t}$ the coroot $\\alpha^{\\vee}$\n(multiplied by $i$), that is\n$$\\alpha^{\\vee}= \\begin{pmatrix} i & 0 \\\\ 0 & -i \\\\ \\end{pmatrix}$$\nThen $\\alpha^{\\vee}$ may be\n identified with a\n basis vector over $\\mathbb{Z}$ for the cocharacter lattice $X_*(T_{\\mathrm{SU}(2)})$.\nIt is tangent at the identity to a unique cocharacter.\n Let $\\varpi_1$ be the fundamental weight\nof $\\mathrm{SL}(2,\\mathbb{C})$ thus\n$$ \\varpi_1(\\alpha^{\\vee}) = 1.$$\nThen $\\varpi_1$ may be identifed with a character of $T$ (it\nis the derivative at the identity of a unique character) which is a basis\nfor\nthe character lattice of $T_{\\mathrm{SU}(2)}$.\n\n\n\n\\smallskip\nLet $\\mu_R$ be the momentum map for the action of $\\mathrm{SU}(2)$ on $T^*\\mathrm{SU}(2)$\ninduced by right multiplication. Specializing the\ndefinition of the imploded cotangent bundle to $G = \\mathrm{SU}(2)$ we find that\nthe (right)\nimplosion of $T^*\\mathrm{SU}(2)$ is the set of equivalence\nclasses\n$$\\mathcal{E} T^\\ast (\\mathrm{SU}(2)) = \\mu_R^{-1}(\\Delta)\/\\sim$$\nwhere two points $x,y \\in \\mu_R^{-1}(\\Delta)$ are\nequivalent if\n$$\\mu_R(x) = \\mu_R(y) = 0 \\ \\text{and} \\ x = y \\cdot g \\ \\text{for\nsome}\\ g \\in \\mathrm{SU}(2).$$\nIt follows from the general theory of \\cite{GuilleminJeffreySjamaar}\nthat $\\mathcal{E}T^*(\\mathrm{SU}(2))$ has an induced\nstructure of a stratified symplectic space in the sense of\n\\cite{SjamaarLerman} with induced isometric actions\nof $\\mathrm{SU}(2)$ induced by the left action of $\\mathrm{SU}(2)$ on $T^*\\mathrm{SU}(2)$\nand $T_{\\mathrm{SU}(2)}$ induced by the right action of $T_{\\mathrm{SU}(2)}$ on $T^*(\\mathrm{SU}(2))$.\nHowever the exceptional\nfeature of this special case is that $\\mathcal{E}T^*(\\mathrm{SU}(2))$ is a\n{\\em manifold}. The multiplicative group $\\mathbb{R}_+$ acts on $\\mathcal{E}T^*(\\mathrm{SU}(2))$\nby the formula\n$$\\mu \\cdot [g,\\lambda \\varpi_1] = [g, \\mu \\lambda \\varpi_1]$$\nwhereas the actions of $g_0 \\in \\mathrm{SU}(2)$ and $t \\in T_{\\mathrm{SU}(2)}$ described\nabove are given by\n$$g_0 \\cdot [g, \\lambda \\varpi_1] = [g_0 g, \\lambda\n\\varpi_1] \\ \\text{ and } t \\cdot [g, \\lambda \\varpi_1] = [gt, \\lambda\n\\varpi_1].$$\n\nThe product group $\\mathrm{SU}(2) \\times \\mathbb{R}_+$ fixes the point $[e,0]$\nand acts transitively on the complement of $[e,0]$ in $\\mathcal{E}T^*(\\mathrm{SU}(2))$.\nThe reader will verify, see Example 4.7 of \\cite{GuilleminJeffreySjamaar},\nthat the symplectic form $\\omega$ on $\\mathcal{E}T^*(\\mathrm{SU}(2))$\nis homogeneous of degree one under $\\mathbb{R}_+$.\n\n\nThe following lemma is extracted from Example 4.7\nof \\cite{GuilleminJeffreySjamaar}. Since this lemma is central to what\nfollows\nwe will go into more detail for its proof than is\nin \\cite{GuilleminJeffreySjamaar}.\n\n\\begin{lemma}\\label{implodedcotangentbundle}\nThe map $\\phi: \\mathcal{E} T^\\ast (\\mathrm{SU}(2)) \\to \\mathbb{C}^2$\ngiven by\n$$\\phi([g,\\lambda \\varpi_1]) = \\sqrt{2\\lambda} g\n\\begin{pmatrix}1\\\\0\\end{pmatrix} \\ \\text{and} \\ \\phi([e,0]) =\n\\begin{pmatrix}0\\\\0\\end{pmatrix}$$\ninduces a symplectomorphism of stratified symplectic\nspaces onto $\\mathbb{C}^2$\nwhere $\\mathbb{C}^2$ is given its standard symplectic structure.\nUnder this isomorphism the action of\n$\\mathrm{SU}(2)$ on $\\mathcal{E}T^*(\\mathrm{SU}(2))$\ngoes to the standard action on $\\mathbb{C}^2$ and\nthe action of $T$ on $\\mathcal{E}T^*(\\mathrm{SU}(2))$ goes to the action of $T$\ngiven by $t\\cdot (z,w) = (t^{-1}z, t^{-1}w)$.\n\\end{lemma}\n\n\n\\begin{proof}\nThe inverse to $\\phi$ is given (on nonzero elements of $\\mathbb{C}^2$)\nby $$\\psi((z,w)) = \\Big[g, \\frac{|z|^2 + |w|^2}{2} \\varpi_1\\Big],$$\nwhere $$g= \\frac{1}{\\sqrt{|z|^2 + |w|^2}} \\begin{pmatrix}z &\n-\\overline{w} \\\\w & \\overline{z}\\end{pmatrix}.$$\nIt follows that $\\phi$ is a homeomorphism. It remains to check\nthat it is a symplectomorphism.\nNoting that $\\phi$ is homogeneous under $\\mathbb{R}_+$ of degree one-half\nand the standard symplectic form $\\omega_{\\mathbb{C}^2}$ is homogeneous of\ndegree two it\nfollows (from the transitivity of the action of $\\mathrm{SU}(2) \\times \\mathbb{R}_+$) that\nthe symplectic form $\\phi^{\\ast} \\omega_{\\mathbb{C}^2}$ is a constant\nmultiple of $\\omega$.\nThus is suffices to prove that the two above forms both take value $1$ on\nthe ordered\npair of tangent vectors $(\\partial\/\\partial \\lambda, \\alpha^{\\vee})$. We\nleave\nthis to the reader.\n\\end{proof}\n\n\\begin{remark}\nThe fact that we have inverted the usual action of the circle on the\ncomplex plane will have\nthe effect\nof changing the signs of the torus Hamiltonians that occur in the rest\nof this paper (from {\\it minus} one-half a sum of squares of norms of complex\nnumbers to {\\it plus} one-half of the corresponding sum).\n\\end{remark}\nTaking into account the above remark we obtain a formula for the\nmomentum map for the action of $T_{\\mathrm{SU}(2)}$ on $\\mathcal{E}T^*(\\mathrm{SU}(2))$\nwhich will be\ncritical in what follows.\nNow we have\n\\begin{lemma}\\label{momentummap}\n The momentum map $\\mu_T$ for the action of $T_{\\mathrm{SU}(2)}$ on\n$\\mathcal{E} T^\\ast (\\mathrm{SU}(2))=\\mathbb{C}^2$\n is given by\n $$\\mu_T((z,w)) = (1\/2)(|z|^2 + |w|^2) \\varpi_1.$$\n Consequently, the Hamiltonian $f_{\\alpha^{\\vee}}(z,w)$ for the\nfundamental\n vector field on $\\mathcal{E}T^\\ast (\\mathrm{SU}(2))$ induced by $\\alpha^{\\vee}$\n is given by\n $$f_{\\alpha^{\\vee}}(z,w) = (1\/2)(|z|^2 + |w|^2).$$\n\\end{lemma}\n\n\\subsubsection{The computation of $P_n(\\mathrm{SU}(2))$ and $Q_n(\\mathrm{SU}(2))$}\nWe can now prove Theorem \\ref{polygonGrasspreciseversion}. Recall we have introduced\nthe abbreviation $E_n(\\mathrm{SU}(2))$ for the product $\\mathcal{E} T^\\ast (\\mathrm{SU}(2))^n$.\nWe will use $\\mathbf{E}$ to denote an element of $E_n(\\mathrm{SU}(2))$.\nIt will be convenient to regard $\\mathbf{E}$ as a function from\nthe edges of the reference polygon $P$ into $\\mathcal{E} T^\\ast (\\mathrm{SU}(2))$.\nWe note that there is a map from $E_n(\\mathrm{SU}(2))$ to not necessarily closed\n$n$-gons in $\\mathbb{R}^3$,\nthe map that scales the first element of the frame by $\\lambda$ and forgets\nthe other two elements of the frame this sends\neach equivalence class $[g_i, \\lambda \\varpi_1]$ to $Ad^{\\ast} g_i (\\lambda_i \\varpi_1)$ for $1 \\leq i \\leq n$.\n\n\nBy the previous lemmas we may represent an element of $E_n(\\mathrm{SU}(2))$\nby the $2$ by $n$ matrix\n\\begin{center}$A = \\begin{pmatrix}z_1 & z_2 & \\ldots & z_{n-1} & z_n\n\\\\w_1 & w_2 & \\ldots & w_{n-1} & w_n\\end{pmatrix}$.\\end{center}\nThe left diagonal action of the group $\\mathrm{SU}(2)$ on $E_n(\\mathrm{SU}(2))$\nis then represented\nby the diagonal action of $\\mathrm{SU}(2)$ on the columns of $A$.\nWe let $[A]$ denote the orbit equivalence class of $A$ for this action.\n We let $T_{\\mathrm{U}(n)}$ be the compact $n$ torus\nof diagonal elements in\nin $\\mathrm{U}(n))$. Then $T_{\\mathrm{U}(n)}$ acts by scaling the columns of the matrices $A$.\nThis action coincides with the inverse of the action of the $n$-torus $T_{\\mathrm{SU}(2)^n}$\ncoming from the theory of the imploded cotangent bundle. By this we mean\nthat if an element in $T_{\\mathrm{U}(n)}$ acts by scaling a column by $\\lambda$\nthen the corresponding element acts by scaling the same column by\n$\\lambda^{-1}$.\n\nThe first statement of\nTheorem \\ref{polygonGrasspreciseversion} will be a consequence of the following two lemmas.\nLet $M_{2,n}(\\mathbb{C})$ be the space of $2$ by $n$ complex\nmatrices $A$ as above. In what follows we will let $Z$ denote the first\nrow\nand $W$ the second row of the above matrix $A$. We give $M_{2,n}(\\mathbb{C})$\nthe Hermitian structure given by\n$$( A,B) = Tr (AB^*).$$\nWe will use $\\mu_G$ to denote the momentum map\nfor the action of $\\mathrm{SU}(2)$ on the left of $A$.\n\n\n\\begin{lemma}\\label{levelzero}$$\\mu_G(A) = (1\/2)\\begin{pmatrix}\n(\\|Z\\|^2 - \\|W\\|^2)\/2 & Z \\cdot W \\\\W \\cdot Z & (\\|W\\|^2 -\n\\|Z\\|^2)\/2\\end{pmatrix}$$\nHence $\\mu_G(A) = 0$ $\\iff$ the two rows of $A$ have the same length\nand are orthogonal.\n\\end{lemma}\n\n\\begin{proof}\nFirst note that the momentum map $\\mu$ for the action of $\\mathrm{SU}(2)$\non $\\mathbb{C}^2$ is given by$$\\mu(z,w) = (1\/2)\\begin{pmatrix} (|z|^2 -\n|w|^2)\/2 & z\\overline{w} \\\\w\\overline{z} & (|w|^2 -\n|z|^2)\/2\\end{pmatrix}.$$\nNow add the momentum maps for each of the columns to get the lemma.\n\\end{proof}\nWe find that $$ \\mathrm{SU}(2) \\backslash \\! \\backslash M_{2,n}(\\mathbb{C}) = \\mathrm{SU}(2) \\backslash \\{ (Z,W):\n\\|Z\\|^2 = \\|W\\|^2, Z \\cdot W = 0 \\}.$$\n\n\\begin{lemma}\\label{affinegrass}\n $$\\mathrm{SU}(2) \\backslash \\! \\backslash M_{n,2}(\\mathbb{C}) \\cong \\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n).$$\n\\end{lemma}\n\n\\begin{proof}\nThe map $\\varphi:\\mu^{-1}(0) \\to \\bigwedge^2(\\mathbb{C}^n)$ given by $F(A) = Z\\wedge\nW$ maps onto the decomposable vectors, descends to the quotient\nby $\\mathrm{SU}(2)$, and induces the required isomorphism.\n\\end{proof}\n\nIn what follows it will be important to understand the above result\nin terms of the GIT quotient of $M_{2,n}(\\mathbb{C})$ by $\\mathrm{SL}(2,\\mathbb{C})$\nacting on the\nleft. Since we are taking a quotient of affine space\nby a reductive group the Geometric Invariant Theory\nquotient of $M_{2,n}(\\mathbb{C})$ by $\\mathrm{SL}(2, \\mathbb{C})$\ncoincides with the symplectic quotient by $\\mathrm{SU}(2)$ and consequently is\n$\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$, \\cite{KempfNess}, see also \\cite{Schwarz}, Theorem\n4.2. Note that since $\\mathrm{SU}(2)$ is simple the question of normalizing the\nmomentum map does not arise. It will be\nimportant to\nunderstand this in terms of the\nsubring $\\mathbb{C}[M_{2,n}(\\mathbb{C})]^{\\mathrm{SL}(2, \\mathbb{C})}$ of\ninvariant\npolynomials\non $M_{2,n}(\\mathbb{C})$. In what follows let $Z_{ij} = Z_{ij}(A)$ or $[i,j]$ be the\ndeterminant\nof the $2$ by $2$ submatrix of $A$ given by taking columns $i$ and $j$ of $A$\n( the $ij$-th Pl\\\"ucker coordinate or bracket). The following result on one of\nthe basic results in invariant theory, see \\cite{Dolgachev},Chapter 2.\n\n\\begin{proposition}\\label{PnGGIT}\n$\\mathbb{C}[M_{2,n}(\\mathbb{C})]^{\\mathrm{SL}(2,\n\\mathbb{C})}$ is generated by\nthe $Z_{ij}$ subject to the Pl\\\"ucker relations.\n\\end{proposition}\nBut the above ring is the homogeneous coordinate ring of $\\mathrm{Gr}_2(\\mathbb{C}^n)$.\nThis gives the required invariant-theoretic proof of Theorem \\ref{polygonGrass}.\n\n\\begin{remark}\nIn fact we could do {\\em all} of the previous analysis in terms\nof invariant theory by using Example 6.12 of\n\\cite{GuilleminJeffreySjamaar} to replace the\nimploded cotangent bundle $\\mathcal{E}T^{\\ast}\\mathrm{SU}(2)\\cong \\mathbb{C}^2$ by the\nquotient $\\mathrm{SL}(2,\\mathbb{C})\/\\!\/ N \\cong \\mathbb{C}^2$ where $N$ is the subgroup of $\\mathrm{SL}(2,\\mathbb{C})$ of\nstrictly\nupper-triangular matrices. We leave the details to the reader.\n\\end{remark}\nWe have now proved the first statement in Theorem \\ref{polygonGrasspreciseversion}\nfor both symplectic and GIT quotients. We note the consequence\n(since every element of $\\bigwedge^2(\\mathbb{C}^3)$ is decomposable)\n\n\\begin{equation}\\label{P3}\nP_3(\\mathrm{SU}(2)) \\cong \\mathrm{\\bigwedge} \\space^2(\\mathbb{C}^3).\n\\end{equation}\nIt is clear that $\\psi$ is equivariant as claimed and the second statement\nfollows.\n\nSince the grading action of $\\lambda \\in \\mathbb{C}^{\\ast}$ on \n$\\mathrm{Aff}\\mathrm{Gr}_2(\\mathbb{C}^n)$ is the action that\nscales each Pl\\\"ucker coordinate by $\\lambda$ the third statement is also clear.\nIt remains to prove the fourth statement of Theorem \\ref{polygonGrasspreciseversion}.\nFirst recall the standard $\\mathrm{U}(n)$ invariant positive definite\nHermitian form $( \\ , \\ )$ on $\\bigwedge^2(\\mathbb{C}^n)$ is given on the\nbivector $a \\wedge b$ by\n$$ (a\\wedge b, a \\wedge b ) = det \\begin{pmatrix} (a,a) & (a,b) \\\\\n(b,a) & (b,b)\n\\end{pmatrix}.$$\nWe can obtain $\\mathrm{Gr}_2(\\mathbb{C}^n)$ as a symplectic manifold the cone $\\mathrm{Aff} \\mathrm{Gr}_2(\\mathbb{C}^n)$ of decomposable\nbivectors by taking the subset of decomposable bivectors of norm squared $R$\nthen dividing by the action of the circle by scalar multiplication\n(symplectic quotient by $S^1$ of level $\\frac{R}{2}$). Thus we can find\na manifold diffeomorphic to $\\mathrm{Gr}_2(\\mathbb{C}^n)$ by computing the pull-backs of\nthe function $f(a\\wedge b) = (a \\wedge b, a \\wedge b)$ and scalar multiplication\nof bivectors by the circle under the map $\\varphi$. We represent elements of $P_n(\\mathrm{SU}(2))$\nby left $\\mathrm{SU}(2)$ orbit-equivalence classes $[A]$ of $2$ by $n$ matrices $A$ such that the rows\n$Z$ and $W$ of $A$ are orthogonal and of the same length.\n\n\\begin{lemma}\n$$\\varphi^{\\ast} f (A) = \\Bigg(\\frac{||Z||^2 + ||W||^2}{2}\\Bigg)^2.$$\n\\end{lemma}\n\n\\begin{proof}\nWe have\n$$\\varphi^{\\ast} f (A) = det \\begin{pmatrix} (Z,Z) & (Z,W) \\\\\n(W,Z) & (W,W)\n\\end{pmatrix}.$$\nBut $(Z,Z) = (W,W)$ and $(Z,W) = (W,Z) = 0$.\nHence\n$$\\varphi^{\\ast} f (A) = ||Z||^2 ||W||^2 = \\Bigg(\\frac{||Z||^2 + ||W||^2}{2}\\Bigg)\\Bigg(\\frac{||Z||^2 + ||W||^2}{2}\\Bigg).$$\n\\end{proof}\nNow we ask the reader to look ahead and see that in subsection \\ref{frommatricestopolygons}\nwe define a map $F_n$ from $P_n(\\mathrm{SU}(2))$ to the space of $n$-gons in $\\mathbb{R}^3$.\nFurthermore $F_n([A])$ is the $n$-gon underlying the imploded spin-framed $n$-gon\nrepresented by $[A]$. Accordingly we obtain the following corollary of the\nabove lemma.\n\n\\begin{corollary}\n$\\varphi^{\\ast} f (A)$ is four times the square of the perimeter of the $n$-gon\nin $\\mathbb{R}^3$ underlying the imploded spin-framed $n$-gon represented by $[A]$.\n\\end{corollary}\nNow observe that scaling the bivector $Z \\wedge W$ by $\\exp{i\\theta}$\ncorresponds to scaling the class of the matrix $[A]$ by the square-root of $\\exp{i\\theta}$,\nNote that the choice square-root is irrelevant since\n$$[-A] = [A].$$\nBut scaling the columns of $A$ corresponds to rotating the corresponding imploded\nspin-frames.\nThus fixing the perimeter of the underlying $n$-gon to be one corresponds to\ntaking the symplectic quotient of the cone of decomposable bivectors by $S^1$ at level $2$.\nand we have now completed the proof of the second part of Theorem \\ref{polygonGrass}.\n\n\n\n\n\\subsection{The momentum polytope for the $T_{\\mathrm{U}(n)}=T_{\\mathrm{SU}(2)^n}$ action on $P_n(\\mathrm{SU}(2))$}\nWe have identified the momentum map for the action of $\\mathrm{SU}(2)$\non $M_{2,n}(\\mathbb{C})$. The torus $T_{\\mathrm{U}(n)}$ acts on $M_{2,n}(\\mathbb{C})$ by scaling\nthe\ncolumns.\nWe leave the proof of the following lemma to the reader.\nLet $\\mu_{T_n}$ be the momentum map for the above action of $T_{\\mathrm{U}(n)}$.\n\n\\begin{lemma}$$\\mu_{T_n}(A) = \\Bigg(\\frac{||C_1||^2}{2},\\cdots ,\n\\frac{||C_n||^2}{2}\\Bigg).$$\n\\end{lemma}\nThe action of $T_{\\mathrm{U}(n)}$ descends to the quotient $P_n(\\mathrm{SU}(2))$ with the\nsame momentum map (only now $A$ must satisfy $\\mu_G(A) =0$). Thus we have\na formula for the momentum map for the action of $T_{\\mathrm{U}(n)}$ on the\nGrassmannian\nin terms of special representative matrices $A$ (of $\\mathrm{SU}(2)$-momentum\nlevel zero). It will be important to extend this formula to\nall $A \\in M_{2,n}(\\mathbb{C})$ of rank two. We do this by relating the\nabove formula to\nthe usual formula for momentum map of the action of $T_{\\mathrm{U}(n)}$ on a general\ntwo\nplane in $\\mathbb{C}^n$ represented by a general rank 2 matrix $A \\in M_{2,n}(\\mathbb{C})$.\nThis momentum map\nis described in Proposition 2.1 of \\cite{GGMS}.\nLet $A$ be a $2$ by $n$ matrix of rank two and $[A]$\ndenote the two plane in $\\mathbb{C}^n$ spanned by its columns.\nThen the $i$-th component of the momentum map of $T_{\\mathrm{U}(n)}$ is given\nby$$\\mu_i([A]) = \\frac{1}{\\|A\\|^2}\\sum_{j,j \\neq i} \\|Z_{ij}(A)\n\\|^2.$$\nHere $\\|A\\|^2$ denotes the sum of the squares of the norms of the\nPl\\\"ucker coordinates of $[A]$.\nThe required extension formula is them implied by\n\n\\begin{lemma}\\label{equalityofHamiltonians}\nSuppose that $\\mu_G(A) = 0$. Let $C_i$ be the $i$-th column of $A$.\nThen\n\n$$\\frac{\\|C_i\\|^2 }{2}= \\frac{1}{\\|A\\|^2}\\sum_{j,j\\neq i}\n\\|Z_{ij}(A) \\|^2.$$\n\n\\end{lemma}\n\n\\begin{proof}\nIt suffices to prove the formula in the special case that $i=1$.\nIf the first row of $A$ is zero then both sides of the equation are zero.\nSo assume the first row is not zero. Apply $g$ to $A$ so that the first\nrow of $Ag$ is of the form $(r,0)$ where $r = \\|R_1\\|$. Let $z_i', w_i'$\nbe the $i$-th row of $Ag$. Note\nthat $\\mu_G(Ag)$ is still zero, hence the length of the second column of $Ag$\nis equal to $(1\/2)\\|Ag\\|^2 = (1\/2)\\|A\\|^2$. Also $Z_{ij}(Ag) =\nZ_{ij}(A)$. Now compute\nthe left-hand side of the equation.\nWe have $ Z_{1j}(A) = Z_{1j}(Ag) = r w_j', 2 \\leq j \\leq n$ and hence (noting\nthat $w_1' =0$) we have$$\\sum_{j,j >1} \\|Z_{1j}(A) \\|^2 = r^2\n\\sum_{j=1}^n |w_j'|^2.$$\nNote that the sum on the right-hand side is the length squared of the\nsecond column of $Ag$. The lemma follows.\n\\end{proof}\nLet $D_n \\subset \\Delta^n$ be the cone defined by$$D_n = \\{ \\mathbf{r}\n\\in \\Delta^n : 2 r_i \\leq \\sum_{j=1}^n r_j, 1 \\leq i \\leq n\\}.$$\nThen $D_n$ is the cone on the hypersimplex $\\Delta^2_{n-2}$ of\n\\cite{GGMS} \\S 2.2.\nNote that $D_3$ is the set of nonnegative real numbers satisfying the\nusual\ntriangle inequalities.\nWe can now apply Lemma \\ref{equalityofHamiltonians} and the results of\n\\S 2.2 of \\cite{GGMS} to\ndeduce\n\n\\begin{proposition}\\label{triangleinequalities}$$\\mu_{T_n}(P_n(\\mathrm{SU}(2))\n= D_n.$$\n\\end{proposition}\nWe give another proof of this theorem in terms of the side-lengths\ninequalities\nfor Euclidean $n$-gons in the next subsection.\n\n\n\\subsection{The relation between points of $P_n(\\mathrm{SU}(2))$ and\nframed Euclidean $n$--gons}\\label{frommatricestopolygons}\n\nThe point of this subsection is to show that a point of $P_n(\\mathrm{SU}(2))$\nmay be interpreted as a Euclidean $n$--gon equipped with an imploded\nspin-frame. The critical role in this interpretation is played by\na map $F:\\mathbb{C}^2 \\to \\mathbb{R}^3$. The map $F$ is the Hopf map.\n\n\\subsubsection{The equivariant map $F$ from $\\mathcal{E} T^\\ast (\\mathrm{SU}(2))$ to\n$\\mathbb{R}^3$}\\label{maptoR3}\n\nIn this section we define a map $F$ from\nthe imploded cotangent bundle $\\mathcal{E} T^\\ast (\\mathrm{SU}(2)) \\cong \\mathbb{C}^2$\nto $\\mathbb{R}^3$. This map will give the critical connection between\nthe previous constructions and polygonal linkages in $\\mathbb{R}^3$.\n\nWe first define $\\pi: \\mathcal{E} T^\\ast (\\mathrm{SU}(2)) \\to \\mathfrak{su}^\\ast(2)$\nby$$\\pi([g, \\lambda \\varpi_1]) = Ad^\\ast(g)(\\lambda \\varpi_1),$$\nwhere $\\varpi_1$\ndenotes the fundamental weight of $\\mathrm{SU}(2)$.\nWe leave the following lemma to the reader.\n\n\\begin{lemma}\\label{circlebundle}\n$\\pi$ factors through the action of\n$T_{\\mathrm{SU}(2)}$. The map $\\pi$ restricted\nto\nthe complement of the point $[e,0]$ is a principal $T_{\\mathrm{SU}(2)}$ bundle\n(the Hopf bundle).\n\\end{lemma}\nDefine the normal frame bundle of $\\mathbb{R}^3-{0}$ to be the set\nof triples $\\mathbf{F} = (v,u_1,u_2)$ where $v$ is a nonzero vector in $\\mathbb{R}^3$\nand the pair $u_1,u_2$\nis a properly oriented frame for the normal plane to $v$. The projection\nto $v$ is a principal circle bundle. The normal frame\nbundle is homotopy equivalent to $\\mathrm{SO}(3,\\mathbb{R})$ and admits a unique nontrivial\ntwo-fold cover which again is a principal circle bundle over $\\mathbb{R}^3 -0$\nwhich we call the bundle of normal spin-frames for $\\mathbb{R}^3 -0$.\n\nIn order to prove Proposition \\ref{framedngons} below we will need the\nfollowing lemma. We leave its proof to the reader. In what follows we\nidentify the coadjoint action of $\\mathrm{SU}(2)$ with its action (through its\nquotient $\\mathrm{SO}(3,\\mathbb{R})$) on $\\mathbb{R}^3$.\n\n\\begin{lemma}\\label{framedvectors}\\hfill\n\\begin{enumerate}\\item The map\n$\\pi$ restricted to the subset of $\\mathcal{E}T^\\ast (\\mathrm{SU}(2))$ with\n$\\lambda \\neq 0$ is the spin-frame bundle of $\\mathbb{R}^3 -0$. \\item The\naction of $T_{\\mathrm{SU}(2)}$ on $\\mathcal{E}T^\\ast (\\mathrm{SU}(2))$ preserves the\nset where $\\lambda \\neq 0$ and induces the circle\naction on the normal spin-frames.\n\\end{enumerate}\n\n\\end{lemma}\nWe have that $\\mathcal{E}T^\\ast (\\mathrm{SU}(2))$ is symplectomorphic to $\\mathbb{C}^2$\nand $\\mathfrak{su}^\\ast(2)$ is isomorphic as a Lie algebra to $\\mathbb{R}^3$,\nand so\nthe map $\\pi$ induces a map from $\\mathbb{C}^2$ to $\\mathbb{R}^3$. The induced map\nis the map $F$. We now give details.\n\n\n\nIdentify $\\mathfrak{su}^\\ast(2)$ with the traceless Hermitian matrices\n$\\mathcal{H}_2^0$\nvia the pairing $\\mathcal{H}_2^0 \\times \\mathfrak{su}(2) \\to \\mathbb{R}$ given\nby\n$$\\langle X, Y \\rangle = \\Im (\\mathrm{tr}(XY)).$$\nIn this\nidentification\nwe have\n$$\\varpi_1 = \\begin{pmatrix}\\frac{1}{2} & 0 \\\\0 & -\\frac{1}{2}\n\\\\\\end{pmatrix}.$$\nWe recall there is a Lie algebra isomomorphism $g$ from $\\mathbb{R}^3$ to\nthe traceless skew-Hermitian matrices given by given\nby\n$$g(x_1,x_2,x_3) = \\frac{1}{2}\\begin{pmatrix}ix_1 & ix_2 - x_3\n\\\\ix_2 + x_3 & -ix_1\\end{pmatrix}.$$\nThe isomorphism $f$ dual to $g$ from\nthe traceless Hermitian matrices $\\mathcal{H}_2^0$ to $\\mathbb{R}^3$ is given\nby\n$$ f\\begin{pmatrix}x_1 & x_2 + \\sqrt{-1} x_3 \\\\x_2 - \\sqrt{-1}x_3 &\n-x_1 \\\\\\end{pmatrix} = (x_1,x_2,x_3).$$\nNote that in particular, $f(\\varpi_1) = (1\/2,0,0) $.\n\n\nWe now define the map $F$.\nDefine $h:\\mathbb{C}^2 \\to \\mathcal{H}_2$ as follows. Recall\n$$\\mu_{\\mathrm{U}(2)}(z,w)\n= 1\/2 \\begin{pmatrix}z \\bar{z} & w \\bar{z} \\\\z \\bar{w} & w \\bar{w}\n\\\\\\end{pmatrix}.$$\nThen define $h(z,w) = \\mu_{\\mathrm{U}(2)}(z,w)^0$ where the superscript zero\ndenotes traceless\nprojection. Hence$$ h(z,w) = (1\/4)\\begin{pmatrix}z\\bar{z} - w \\bar{w}\n& 2w\\bar{z} \\\\ 2z \\bar{w} & w\\bar{w} - z \\bar{z} \\\\\\end{pmatrix}.$$\nWe define $F$ by$$F = f \\circ h.$$\n\n\\begin{remark}\nThe map $h$ is the momentum map for the action of $\\mathrm{SU}(2)$ on $\\mathbb{C}^2$\nafter we identify $\\mathcal{H}_2^0$ with the dual of the Lie algebra\n of $\\mathrm{SU}(2)$ using the imaginary part of the trace. Accordingly\n we will use the notations $h$ and $\\mu_{\\mathrm{SU}(2)}$ interchangeably.\n\\end{remark}\nIn the next lemma we note an important equivariance property of $F$.\nLet $\\rho:\\mathrm{SU}(2) \\to \\mathrm{SO}(3,\\mathbb{R})$ be the double cover. We leave its\nproof to the reader.\n\n\\begin{lemma}\\label{equivariance}\nLet $g \\in \\mathrm{SU}(2)$. Then$$F \\circ g = \\rho(g) \\circ F.$$\n\\end{lemma}\nWe next have\n\n\\begin{lemma} We have a commutative diagram\n\\[ \\begin{CD}\\mathcal{E} T^\\ast (\\mathrm{SU}(2)) @>\\pi>> \\mathfrak{su}(2)^{\\ast}\n\\\\@V\\phi VV @VVfV \\\\\\mathbb{C}^2 @>F>> \\mathbb{R}^3\\end{CD}\\]\n\\end{lemma}\n\n\\begin{proof}\nFirst observe that $f\\circ \\pi$ and $F\\circ \\phi$ are homogeneous of\ndegree one with respect to the $\\mathbb{R}_+$ actions on their domain and range\n(note that $\\phi$ is homogeneous of degree $1\/2$ and $F$ is homogeneous\nof degree two). Also both intertwine the $\\mathrm{SU}(2)$ action on $\\mathcal{E}\nT^\\ast (\\mathrm{SU}(2))$\nwith its action through the double cover on $\\mathbb{R}^3$. Since the action\nof $\\mathrm{SU}(2) \\times \\mathbb{R}_+$ on $\\mathcal{E} T^\\ast (\\mathrm{SU}(2))$ is transitive away\nfrom $[e,0]$ it suffices to prove the two above maps coincide at the\npoint $[e,\\varpi_1]$. But it is immediate that both maps take the\nvalue $(1\/2,0,0)$ at this point.\n\\end{proof}\n\n\\begin{remark}\\label{SpinF}\nWe will need the following calculation.\n\n$$F \\circ \\phi([g, \\lambda\\varpi_1]) = \\lambda f( Ad_g^*(\\varpi_1))$$\nIn particular this implies that $\\|F \\circ \\phi([g, \\lambda\\varpi_1])\\| = \\frac{1}{2}\\lambda$.\n\\end{remark}\nThe following lemma is a direct calculation.\n\n\\begin{lemma}\\label{factoroftwo}\nThe formula for $F: \\mathbb{C}^2 \\to \\mathbb{R}^3$ in the usual coordinates is$$F(z,\nw) = \\frac{1}{4}\\Big(z \\bar{z} - w \\bar{w}, \\; 2\\Re(w \\bar{z}), \\;\n2\\Im(w\\bar{z})\\Big).$$\nConsequently, the Euclidean length of the vector $F(z,w) \\in \\mathbb{R}^3$ is\ngiven by$$ \\|F(z,w)\\| = \\frac{1}{4}(|z|^2 + |w|^2) .$$\n\\end{lemma}\n\n\\begin{corollary}\\label{SpinLength}\nNote that the length of $F$ is related to the Hamiltonians\n$f_{\\alpha^{\\vee}}(z,w)$\nfor the infinitesimal action of $\\mathfrak{t}$ by$$||F(z,w)|| = (1\/2)\nf_{\\alpha^{\\vee}}(z,w) .$$\nAlso, by the above remark if $\\phi([g, \\lambda\\varpi_1]) = (z, w)$ then\n$$\\lambda = f_{\\alpha^{\\vee}}(z,w).$$\n\\end{corollary}\nLater we will need the following determination of the fibers of $F$.\n\n\\begin{lemma}\\label{phasefactor}\n$$F(z_1,w_1) = F(z_2,w_2) \\iff z_1 = c\nz_2 \\ \\text{and} \\ w_1 = c w_2 \\ \\text{with} \\ |c|\n=1.$$\n\\end{lemma}\n\n\\begin{proof}\nThe implication $\\Leftarrow$ is immediate. We prove the reverse\nimplication.\nThus we are assuming the equations\n\\begin{align*}\n|z_1|^2 - |w_1|^2 &= |z_2|^2 - |w_2|^2 \\\\\nz_1 \\bar{w_1} &= z_2 \\bar{w_2}\\end{align*}\nSquare each side of the first equation.\nTake four times the norm squared of each side of the second equation and\nadding the\nresulting equation to the new first equation to obtain$$ (|z_1|^2 +\n|w_1|^2)^2 = (|z_2|^2 + |w_2|^2)^2.$$\nHence $|z_1|^2 + |w_1|^2=|z_2|^2 + |w_2|^2$ and consequently$$|z_1|^2\n= |z_2|^2 \\ \\text{and} \\ |w_1|^2 = |w_2|^2.$$\nNow it is an elementary version of the first fundamental theorem\nof invariant theory that if we are given two ordered pairs of vectors\nin the plane so that the lengths of corresponding vectors\nare equal and the symplectic inner products between the two\nvectors in each pair coincide then there is an element in $\\mathrm{SO}(2)$\nthat carries one ordered pair to the other (this is one definition\nof the oriented angle between two vectors).\n\\end{proof}\n\nWe now construct the required map to Euclidean $n$-gons.\nWe define a map $F_n:P_n(\\mathrm{SU}(2)) \\to (\\mathbb{R}^3)^n\/\\mathrm{SO}(3,\\mathbb{R})$ by defining\n$F_n(A)$ to be the orbit of $(F(C_1),\\ldots,F(C_n))$ under the diagonal\naction of $\\mathrm{SO}(3,\\mathbb{R})$. Here $C_i$ is the $i$-th column of $A$.\nLet $\\mathrm{Pol}_ n(\\mathbb{R}^3)$ denote the space of closed $n$-gons in\n$\\mathbb{R}^3$.\n\n\\begin{theorem}\\label{framedngons}\\hfill\n\\begin{enumerate}\n\\item $F_n$ induces a homeomorphism from $P_n(\\mathrm{SU}(2))\/T_{\\mathrm{SU}(2)^n}$ onto\n$\\mathrm{Pol}_ n(\\mathbb{R}^3)\/\\mathrm{SO}(3,\\mathbb{R})$.\n\\item The fiber of $F_n$ over a Euclidean $n$-gon\n is\nnaturally homeomorphic to the set\nof imploded spin framings of the edges of that\n$n$-gon.\n\\item The side-lengths of $F_n(A)$ are related to the norm squared of the columns\nof $A$ by\n$$||e_i(F_n(A))|| = \\frac{||C_i||^2}{4} = \\frac{|z_i|^2 + |w_i|^2}{4}.$$\nHere $e_i(F_n(A)) = F_n(C_i)$ is the $i$-th edge of any $n$-gon in the\ncongruence class represented by $F_n(A)$.\n\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nWe first prove that if $A$ is in the zero level set of $\\mu_G$\nthen $F_n(A)$ is a {\\it closed} $n$-gon.\nRecall $C_i,1 \\leq i \\leq n$ be the $i$-th column of $A$. Then we\nhave$$F_n(A) = (f((C_1C_1^*)^0),\\cdots,f((C_nC_n^*)^0)).$$\nHence the sum $s$ of the edges $F_n(A)$ is given by$$s = f((C_1C_1^* +\n\\cdots + C_nC_n^*)^0).$$\nNow it is a formula in elementary matrix multiplication that we\nhave$$AA^* = C_1C_1^* + \\cdots + C_nC_n^*.$$\nBut by Lemma \\ref{levelzero} we find that $AA^*$ is scalar\nwhence$(AA^*)^0 =0$ and$$s = f((AA^*)^0) = f(0) = 0.$$\nWe next prove that $F_n$ is onto.\nIt is clear that $F_n$ maps $M_{n,2}(\\mathbb{C})$ onto $(\\mathbb{R}^3)^n$ (this may\nbe proved one column at a time). Let $\\mathbf{e}=(e_1,\\cdots,e_n)$\nbe the edges of a closed $n$-gon in $\\mathbb{R}^3$. Choose $A \\in M_{2,n}(\\mathbb{C})$\nsuch that $F_n(A) = \\mathbf{e}$. But by the above$$f((AA^*)^0) = e_1 +\n\\cdots + e_n = 0.$$\nHence $(AA^*)^0 =0$ whence $AA^*$ is scalar and $\\mu_G(A) = 0$.\nWe now prove that $F_n$ is injective.\nSuppose there exists $g \\in \\mathrm{SU}(2)$ such that $F_n(A) = \\rho(g) F_n(A')$.\nThen $F(A) = F(gA')$. Hence by Lemma \\ref{phasefactor} we have $A = g A' t$\nfor some $t \\in T_{\\mathrm{U}(n)}$.\n\nThe second statement follows because the action of $T_{\\mathrm{U}(n)}$ corresponds to\na transitive action on the set\nof imploded spin framings. Indeed using the identification above between\nbetween $\\mathbb{R}^3$ and $\\mathfrak{su}(2)^{\\ast}$ we may replace $F_n$ by the\nmap $\\pi_n :\\mathcal{E}T^\\ast(\\mathrm{SU}(2))^n \\to (\\mathfrak{su}(2)^{\\ast})^n$ given\nby$$\\pi_n([g_1,\\lambda_1 \\varpi_1],\\cdots,[g_n,\\lambda_n \\varpi_i])=\n(Ad^{\\ast}g_1(\\lambda_1 \\varpi_1), \\cdots, Ad^{\\ast}g_n(\\lambda_n\n\\varpi_1)).$$ If no $\\lambda_i$ is zero by Lemma \\ref{framedvectors}\nthe (classes of ) the $n$-tuple $[g_1, \\lambda_1 \\varpi_1],\\cdots,\n[g_n,\\lambda_n \\varpi_1]$ represent the imploded\nspin-frames over the $n$ vectors in the image.\nThen by Lemma \\ref{circlebundle} and Lemma \\ref{framedvectors} two\nsuch $n$--tuples correspond to\nframes over the same image $n$-gon if and only the two $n$--tuples are\nrelated\nby right multiplication by $t_1,\\cdots, t_n$ with $t_i$\nfixing $\\varpi_1$ for all $i$. Hence if no $\\lambda_i$ is zero then\nthe two $n$--tuples are related as above if\nand only if they are in the same $T_{\\mathrm{U}(n)}$ orbit by definition of\nthe $T_{\\mathrm{U}(n)}$ action. Since $T_{\\mathrm{U}(n)}$ acts transitively\non the spin-frames over a given $n$-gon we see that in this case the\nfiber of $\\pi_n$ is the set of spin-frames over the image $n$-gon and the second statement\nis proved. If some subset of the $\\lambda_i$'s is zero then\nwe replace the corresponding $g_i$'s by the identity (this does not\nchange the imploded frame) and procede as above with the remaining\ncomponents.\n\nThe last statement in the theorem is Lemma \\ref{factoroftwo}.\n\n\\end{proof}\n\n\n\n\nLet $\\sigma:\\mathrm{Pol}_ n(\\mathbb{R}^3) \\to \\mathbb{R}_+^n$ be the map that assigns\nto a closed $n$-gon the lengths of its sides. It is standard (see for\nexample the Introduction of \\cite{KapovichMillson}) that\nthe image of $\\sigma$ is the polyhedral cone $D_n$.\nWe can now give another proof of Proposition \\ref{triangleinequalities}\nbased on Euclidean geometry.\nWe restate it for the convenience of the reader.\n\n\\begin{proposition}$$\\mu_{T_n}(P_n(\\mathrm{SU}(2))) = D_n.$$\\end{proposition}\\begin{proof}\nThe proposition follows from the commutative diagram\\[\n\\begin{CD}P_n(\\mathrm{SU}(2)) @>\\mu_{T_n}>> \\mathbb{R}_+^n \\\\@VF_n VV @VVIV\n\\\\ \\mathrm{Pol}_n(\\mathbb{R}^3) @>\\sigma>>\\mathbb{R}_+^n\\end{CD}\\]\\end{proof}\n\n\\subsection{The space $P_n(\\mathrm{SO}(3,\\mathbb{R})$}\n\nWe now compute $P_n(\\mathrm{SO}(3,\\mathbb{R}))$ using the fact that $\\mathrm{SO}(3,\\mathbb{R})$ is covered by $\\mathrm{SU}(2)$.\nLet the semi simple Lie Group $\\bar{G}$ be a quotient of the semi simple Lie\nGroup $G$\nby a finite group $\\Gamma$\n\n$$\\Gamma \\rightarrow G \\xrightarrow{\\pi} \\bar{G}$$\n\nIn Example 4.7 of \\cite{GuilleminJeffreySjamaar}, Guillemin-Jeffrey-Sjamaar, show that\nthis quotient gives a description nomeomorphism $\\phi:\\mathcal{E}(T^*G)\/\\Gamma \\to \\mathcal{E}(T^* \\overline{G}))$. The following lemma is left to\nthe reader to prove.\n\n\\begin{lemma}\\label{quotientimplosion}\nThe homeomorphism $\\phi$ induces a homeomorphism from $\\Gamma \\backslash \\! \\backslash P_n(G)$\nto $P_n(\\overline{G})$.\n\\end{lemma}\n\n\n\\begin{theorem}\\label{compare} The double cover $\\phi:\\mathrm{SU}(2) \\to \\mathrm{SO}(3,\\mathbb{R})$\ninduces a homeomorphism from $H \\backslash \\! \\backslash P_n(\\mathrm{SU}(2))$ to $P_n(\\mathrm{SO}(3,\\mathbb{R}))$.\nHere H is the finite 2-group $Z(\\mathrm{SU}(2))^n$ and $Z(\\mathrm{SU}(2)) \\cong \\mathbb{Z}\/2$\nis the center of $\\mathrm{SU}(2)$.\n\\end{theorem}\n\nWe have the following analogue of Theorem \\ref{framedngons} above.\nWe leave its proof to the reader. We now have orthogonal (imploded) framings\ninstead of imploded spin framings. We note that the map\n$F_n$ above descends to a map $\\overline{F}_n: P_n(\\mathrm{SO}(3,\\mathbb{R})) \\to\n\\mathrm{Pol}_ n(\\mathbb{R}^3)\/\\mathrm{SO}(3,\\mathbb{R})$.\n\n\\begin{theorem}\\label{orthogonalframedngons}\\hfill\n\\begin{enumerate}\n\\item $\\overline{F}_n$ induces a homeomorphism from $P_n(\\mathrm{SO}(3,\\mathbb{R}))\/T_{\\mathrm{SO}(3,\\mathbb{R})^n}$ onto the polygon space\n$\\mathrm{Pol}_ n(\\mathbb{R}^3)\/\\mathrm{SO}(3,\\mathbb{R})$.\n\\item The fiber of $\\overline{F}_n$ over a Euclidean $n$-gon\nof the induced map from $P_n(\\mathrm{SU}(2))$ onto $\\mathrm{Pol}_ n(\\mathbb{R}^3)\/\\mathrm{SO}(3,\\mathbb{R})$ is\nnaturally homeomorphic to the set\nof imploded orthogonal framings of the edges of that\n$n$-gon.\n\\end{enumerate}\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\\section{Toric degenerations associated to trivalent trees}\n\nSuppose one has a planar regular $n$-gon subdivided into\ntriangles. We call this a triangulation of the $n$-gon. The dual\ngraph is a tree with $n$ leaves and $n-2$ internal trivalent nodes\n(see Figure \\ref{fig:triangulated_hexagon}). We will see below\nhow the tree $\\mathcal{T}$ determines a Gr\\\"obner degeneration of the\nGrassmannian $\\mathrm{Gr}_2(\\mathbb{C}^n)$ to a toric variety. These toric\ndegenerations first appeared in \\cite{SpeyerSturmfels}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.2]{NP6.eps}\n\\caption{ A triangulated hexagon with associated dual tree.}\\label{fig:triangulated_hexagon}\n\\end{figure}\n\n\n\\subsection{Toric degenerations of $\\mathrm{Gr}_2(\\mathbb{C}^n)$}\n\nRecall that the standard coordinate ring $R = \\mathbb{C}[\\mathrm{Gr}_2(\\mathbb{C}^n)]$ (for\nthe Pl\\\"ucker embedding) of the Grassmannian is generated by\n$Z_{i,j}$, for $1 \\leq i < j \\leq n$, subject to the\nquadric relations, $Z_{i,j}Z_{k,l} - Z_{i,k}Z_{j,l} + Z_{i,l}Z_{j,k} = 0$ for\n$1 \\leq i < j < k < l \\leq n$. These relations generate the\nPl\\\"ucker ideal $I_{2,n}$. For each pair of indices $i,j$ let\n$w_{i,j}^\\mathcal{T}$ denote the length of the unique path in $\\mathcal{T}$\njoining leaf $i$ to leaf $j$. For example, in Figure\n\\ref{fig:triangulated_hexagon} we have $w_{1,4}^\\mathcal{T} = 5$. To any\nmonomial $m = \\prod_k Z_{i_k, j_k}$ we assign a weight $w^\\mathcal{T}(m) =\n\\sum_k w_{i_k,j_k}^\\mathcal{T}.$ Let $I_{2,n}^\\mathcal{T}$ denote the initial\nideal with respect to the weighting $w^\\mathcal{T}$.\n\nIt is a standard result in the theory of Gr\\\"obner degenerations\nthat one has a flat degeneration of $\\mathbb{C}[\\mathrm{Gr}_2(\\mathbb{C}^n)] =\n\\mathbb{C}[\\{Z_{i,j}\\}_{i < j}] \/ I_{2,n}$ to $\\mathbb{C}[\\{Z_{i,j}\\}_{i 2$. We identify $\\mathrm{Gr}_2(\\mathbb{C}^n)$\nwith the homogeneous space $\\mathrm{SL}(n,\\mathbb{C})\/P$,\nby identifying $gP \\in \\mathrm{SL}(n,\\mathbb{C})\/P$ with\n$g \\cdot (e_1 \\wedge e_2) \\in \\mathrm{Gr}_2(\\mathbb{C}^n)$. Let $\\varpi_2 =\n(1,1,0,\\ldots,0) \\in \\mathbb{Z}^n$ be the second fundamental weight of\n$\\mathrm{SL}(n,C)$. Associated to $\\varpi_2$ is a character\n$\\chi : P \\to \\mathbb{C}^\\ast$ given by $\\chi([a_{ij}]_{1 \\leq i,j \\leq n})\n= \\det\n\\begin{pmatrix}\na_{11} & a_{12} \\\\\na_{21} & a_{22}\n\\end{pmatrix}$.\nFollowing the Borel-Weil construction, we may take the total space\nof $L$ to be the product $\\mathrm{SL}(n,\\mathbb{C}) \\times \\mathbb{C}$ modulo\nthe equivalence relation $(g,z) \\sim (gp, \\chi(p)z)$, for all\n$g \\in \\mathrm{SL}(n,\\mathbb{C})$, $p \\in P$, and $z \\in \\mathbb{C}$. We denote the equivalence\nclass of $(g,z)$ as above by $[g,z]$.\nThe bundle map $\\pi : L \\to \\mathrm{Gr}_2(\\mathbb{C}^n)$ is given by\n$[g,z] \\mapsto gP$. The Plucker coordinate ring of $\\mathrm{Gr}_2(\\mathbb{C}^n)$\nis now $\\bigoplus_{N=0}^\\infty \\Gamma(\\mathrm{Gr}_2(\\mathbb{C}^n),L^{\\otimes N})$, and it is\ngenerated in degree one by the brackets $[i,j]$\nwhich are associated to global sections $s_{ij}$ of $L$, by\n$s_{ij}(gP) := [g, g_{i1}g_{j2} - g_{i2}g_{j1}]$, where\n$g = [g_{kl}]_{1 \\leq k,l \\leq n}$.\n\n\n\nWe suppose that $|\\mathbf{r}| = r_1 + \\cdots + r_n$ is an even integer.\nThe line bundle $L^{\\otimes |\\mathbf{r}|\/2}$ of $L$ may be identified\nwith the product $\\mathrm{SL}(n,\\mathbb{C}) \\times \\mathbb{C}$ modulo the relation\n$(g,z) \\sim (gp, \\chi(p)^{\\otimes |\\mathbf{r}|\/2} z)$.\nWe define an action of $T$ on $L^{\\otimes |\\mathbf{r}|\/2}$ via the character\n$\\chi_\\mathbf{r}$ of $T$, given by\n$\\chi_\\mathbf{r}(t_1,\\ldots,t_n) := \\prod_{i=1}^n t_i^{r_i}$.\nWe define the action of $T$ on $L^{\\otimes |\\mathbf{r}|\/2}$ by\n$$(t_1,\\ldots,t_n) \\cdot [g,z] = [\\mathbf{t} \\cdot g,\n\\chi_\\mathbf{r}(\\mathbf{t})z],$$\nwhere $\\mathbf{t} = (t_1,\\ldots,t_n) \\in T$. We call this the\n$\\mathbf{r}$--linearization of $L^{\\otimes |\\mathbf{r}|\/2}$.\n\nThe space $M_\\mathbf{r}$ is homeomorphic to \n$$\\mathrm{Gr}_2(\\mathbb{C}^n) \/\\!\/_\\mathbf{r} T = \\mathrm{Projm} \\Big(\n\\bigoplus_{N=0}^\\infty \\Gamma(\\mathrm{Gr}_2(\\mathbb{C}^n), L^{\\otimes N |\\mathbf{r}|\/2})^T\n\\Big),$$\nwhere $\\Gamma(\\mathrm{Gr}_2(\\mathbb{C}^n), L^{\\otimes N |\\mathbf{r}|\/2})^T$ indicates\nthe $T$-invariant global sections of $L^{\\otimes N |\\mathbf{r}|\/2}$.\nThe ring \n$$R_\\mathbf{r} = \n\\bigoplus_{N=0}^\\infty \\Gamma(\\mathrm{Gr}_2(\\mathbb{C}^n), L^{\\otimes N |\\mathbf{r}|\/2})^T$$\nis naturally graded by $N$.\nThe invariant sections of $L^{\\otimes N|\\mathbf{r}|\/2}$ are spanned by\nmonomials $m = m_G$ over all graphs $G$ having multi-degree $N\\mathbf{r}$,\ni.e. the valency of vertex $i$ is $r_i$ for each $i$.\nThe degree of $m_G$ is then $N$, if $G$ has multidegree $N \\mathbf{r}$.\n\nRestricting to torus invariants is an exact functor so the flat\ndegenerations of the Grassmannian described above restrict to flat\ndegenerations of $\\mathrm{Gr}_2(\\mathbb{C}^n) \/\\!\/_\\mathbf{r} T$. Furthermore, the\nspecial fiber of this restricted degeneration is toric, since it\nis the $T$-quotient of a toric variety.\nTherefore we obtain a flat degeneration of $\\mathrm{Gr}_2(\\mathbb{C}^n) \/\\!\/_\\mathbf{r} T$ to a\ntoric variety $(\\mathrm{Gr}_2(\\mathbb{C}^n) \/\\!\/_\\mathbf{r} T)^\\mathcal{T}$ for each triangulation $\\mathcal{T}$ of the\nmodel $n$-gon.\nThe associated semigroup\n$\\mathcal{S}^\\mathcal{T}_\\mathbf{r}$ is the set of Kempe graphs having valency a multiple\nof $\\mathbf{r}$; it is a sub-semigroup of $\\mathcal{S}_n^\\mathcal{T}$, however the grading\nof $\\mathcal{S}^\\mathcal{T}_\\mathbf{r}$ is not the same as that of $\\mathcal{S}_n^\\mathcal{T}$. Instead\nthe degree of a Kempe graph $G \\in \\mathcal{S}^\\mathcal{T}_\\mathbf{r}$ of multidegree \n$N \\mathbf{r}$ is $N$, rather than $N |\\mathbf{r}|\/2$ as it would have in $\\mathcal{S}_n^\\mathcal{T}$.\nWe have that\n$$(\\mathrm{Gr}_2(\\mathbb{C}^n) \/\\!\/_\\mathbf{r} T)^\\mathcal{T} = \\mathrm{Projm}(\\mathbb{C}[S^\\mathcal{T}_\\mathbf{r}]),$$\nwith $\\mathcal{S}^\\mathcal{T}_\\mathbf{r}$ graded as mentioned above.\n\n\\begin{definition}\nDefine the graded semigroup $\\mathcal{W}_{\\mathbf{r}}^{\\mathcal{T}}$ to be the graded subsemigroup of\n$\\mathcal{W}^{\\mathcal{T}}_n$ with leaf-edge weights that are integral multiples of $\\mathbf{r}$.\n\\end{definition}\nThe admissible weightings of the tree $\\mathcal{T}$ relating\nto the sub-semigroup $\\mathcal{S}^\\mathcal{T}_\\mathbf{r}$ must satisfy that the weighting of\nthe outer edges $e_1,\\ldots,e_n$ (the edge\n$e_i$ is adjacent to leaf $i$) is some multiple $N$ of $\\mathbf{r}$,\nmeaning that $w(e_i) = Nr_i$ for each $i$,\nand the multiple $N$ is the degree of the weighting.\nThus we have\n\\begin{proposition}\\label{gradedsubsemigroupiso}\nThe isomorphism $\\Omega_n$ of Proposition \\ref{gradedsemigroupiso} induces an isomorphism\n$\\Omega_{\\mathbf{r}}:\\mathcal{S}^{\\mathcal{T}}_{\\mathbf{r}} \\to \\mathcal{W}_{\\mathbf{r}}^{\\mathcal{T}}$.\n\\end{proposition}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{$\\mathcal{T}$-congruence of polygons and polygonal linkages}\\label{KYspace}\n\nIn this section we will define an equivalence relation on polygons and polygonal linkages \nwhich depends on the choice of a trivalent tree $\\mathcal{T}$. First we will collect some results \nabout trivalent trees which will be useful in what follows.\n\n\\subsection{Trivalent trees and their decompositions into forests}\nLet $\\mathcal{T}$ be a trivalent tree with $n$ leaves which we assume is dual\nto a triangulation of $P$. We will\nsay a vertex is {\\it internal} if it is not a leaf. The triangles in the triangulation of $P$ correspond to the internal vertices of $\\mathcal{T}$.\nWe will say an edge is a {\\it leaf edge} or an {\\it outer edge} if it is incident to a leaf. Thus the leaf edges are dual to the edges of $P$.\nAn edge of $\\mathcal{T}$ which does not border a leaf will be called an\n{\\it inner edge}. Thus the inner edges of $\\mathcal{T}$ are dual to the\ndiagonals of the triangulation of $P$.\n\n\\begin{definition}\nWe say two leaves of $\\mathcal{T}$ are a matched pair of leaves if they have a common neighbor.\n\\end{definition}\nThe following technical lemma will be very useful for giving inductive proofs.\n\n\\begin{lemma}\\label{trivalentinduction}\nFor any trivalent tree $\\mathcal{T}$ it is possible\nto find a sequence of subtrees\n\n$$\\mathcal{T}_0 \\subset \\ldots \\subset \\mathcal{T}_{n-3} = \\mathcal{T}$$\nsuch that\n\\begin{itemize}\n\\item The tree $\\mathcal{T}_0$ is a tripod.\n\\item The tree $\\mathcal{T}_i$ can be identified with $\\mathcal{T}_{i-1}$ joined with a tripod along some $e_i$.\n\\item Each internal edge of $\\mathcal{T}$ appears as some $e_i$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathcal{T}$ be a trivalent tree. let $\\mathcal{T'}$ be the (not necessarily trivalent) subtree of $\\mathcal{T}$ with vertex set the internal vertices of $\\mathcal{T}$ and edge set equal to all edges of $\\mathcal{T}$ not connected to leaves. It is easy to see that $\\mathcal{T'}$ is also a tree and therefore has a leaf. Let $n \\in \\mathcal{T}$ be the vertex corresponding to this leaf, then $n$ is trivalent by definition and therefore must be connected to two leaves in $\\mathcal{T}$. This shows that $\\mathcal{T}$ must have a vertex $v$ connected to two leaf edges, $v$ is connected to a third edge, $e$,\nby splitting $\\mathcal{T}$ along $e$ we obtain a tripod and a new trivalent tree. The three items above now follow by induction.\n\\end{proof}\nFor each tree $\\mathcal{T}$ we choose once and for all an ordering on the\ntrivalent vertices of $\\mathcal{T}$ as follows. Pick any matched pair\nof leaves and label their trivalent vertex $\\tau_{n-2}$\nwith $n$ the number of leaves of $\\mathcal{T}$. Then,\nrepeat this proceedure for the tree given by removing $\\tau_{n-2}$\nand its matched pair. Note that this ordering is not unique.\n\n\nWe now discuss the decompositions of $\\mathcal{T}$ obtained\nby splitting certain internal edges.\nWe first describe the decompositions of $P$ which will induce the required\ndecompositions of $\\mathcal{T}$. For a diagonal $d$ in the triangulation $\\mathcal{T}$, \nwe define the operation of splitting $P$ along $d$ by removing a small\ntubular neighborhood of $d$ in $P$. This results in a disjoint union\nof two triangulated polytongs $P' \\cup P'' = P^{\\{d\\}}$. \nWe may generalize this operation by performing the same operation for any subset $S$ \nof the set of diagonals of $P$ to obtain a union of triangulated polygons $P^S$. \nThe dual graph to $P^S$ is a forest $\\mathcal{T}^S$ that may be obtained by removing\na small open interval from each edge corresponding to a diagonal in $S$. \nIf we choose $S$ to be the entire set of diagonals we obtain the decomposition \n$P^D$ of $P$ into triangles and $\\mathcal{T}$ into the forest $\\mathcal{T}^D$ of trinodes.\n\nIt will be important in what follows that we have the quotient map\n$$\\pi^{S}: \\mathcal{T}^S \\to \\mathcal{T}$$\nthat glues together the pairs of vertices that are the boundaries of\nthe open intervals removed from $\\mathcal{T}$.\nWe will say two vertices of $\\mathcal{T}^S$ are equivalent if they have the same\nimage under $\\pi^{S}$ and two edges of $\\mathcal{T}^S$ are equivalent\nif they contain equivalent vertices.\n\nThere are two types of leaf edges of $\\mathcal{T}^S$.\nThe first type correspond to edges of $P$ (leaf edges of $\\mathcal{T}$) and the second correspond to (split)\ndiagonals of $P$. The $n$ edges of $\\mathcal{T}^S$ corresponding to leaf edges of $\\mathcal{T}$ will from now on be referred to as distinguished\nedges, we denote this set $dist(\\mathcal{T}^S)$. For a connected component $C$ of $\\mathcal{T}^S$ we let $dist(C) = dist(\\mathcal{T}^S)\\cap C$.\nNote that these components uniquely partition the set $dist(\\mathcal{T}^S) = \\bigcup_{C_i \\subset \\mathcal{T}^S} dist(C_i)$.\n\n\\subsection{$\\mathcal{T}$-congruence of polygonal linkages}\n\nFix a triangulation of the model n-gon with dual tree $\\mathcal{T}$.\nRecall $\\mathrm{Pol}_ n(\\mathbb{R}^3)$, the space of n-gons in $\\mathbb{R}^3$.\nA polygon $\\mathbf{e} \\in \\mathrm{Pol}_ n(\\mathbb{R}^3)$ comes with a fixed ordering on its edges,\nthese ordered edges are in bijection with the leaves of the tree $\\mathcal{T}$.\nA set of diagonals $S$ corresponds to a set of internal edges of\n$\\mathcal{T}$. We know that such a set defines a\nunique partition of the edges of $\\mathbf{e} $, $E(\\mathbf{e} ) = E_1(\\mathbf{e} ) \\cup \\ldots \\cup E_m(\\mathbf{e} )$\ngiven by grouping together the distinguished edges of $\\mathcal{T}^S$ that lie in a component tree of the forest $\\mathcal{T}^S$.\n\n\\begin{proposition}\nFor $\\mathbf{e} \\in \\mathrm{Pol}_ n(\\mathbb{R}^3)$, if all diagonals in the set $S$ are zero then\nthe edge sets $E_i(\\mathbf{e} )$ define closed polygons.\n\\end{proposition}\n\n\\begin{proof}\nSplit the polygon $P$ along $S$ to obtain a union of polygons.\nChoose a polygon $P_i$ in the union. Then $P_i$ is dual to a component $C_i$ of the forest $\\mathcal{T}^S$.\nThe edges of the polygon $P_i$ are either edges of the\norginal original polygon (so distinguished leaves of the tree $C_i$)\nand hence edges of the Euclidean polygon $\\mathbf{e}$\nor diagonals of $S$ and hence zero diagonals of the Euclidean\npolygon $\\mathbf{e}$. We assume that we have chosen $i$ consistently\nwith the division of the distinguished edges above whence\nthe set of distinguished edges of $C_i$ equals\nthe set $E_i(\\mathbf{e})$. Now since $P_i$ closes up the sum of\nall the vectors in $\\mathbb{R}^3$ associated to the edges of $P_i$\nis zero. But this sum is the sum of the vectors in $\\mathbb{R}^3$ associated\nto the distinguished edges of $C_i$ (that is the elements\nin $E_i(\\mathbf{e})$) and a set of vectors all of which are zero.\n\\end{proof}\nThe following groups will be useful in defining structures on\nspaces of polygons.\n\n\\begin{definition}\nLet $G^{dist(\\mathcal{T}^S)}$ be the group of maps from\nthe set $dist(\\mathcal{T}^S)$ into a group $G$. Define\n$G^{D(S)}$ to be the subgroup of $G^{dist(\\mathcal{T}^S)}$\nof maps which are constant along the distinguished edges\nof each component $C$ of $\\mathcal{T}^S$\n\\end{definition}\nNotice that $G^{D(S)}$ naturally splits as a\nproduct of $G$ over the components of $\\mathcal{T}^S$.\nLet $G^{D(C)}$ be the component corresponding\nto the component $C$. The sets $S$ of zero diagonals define a filtration of the space $\\mathrm{Pol}_ n(\\mathbb{R}^3)$\nwhere the subspace $\\mathrm{Pol}_ n(\\mathbb{R}^3)^S$ in the filtration is defined to be the\nset of all points $\\mathbf{e} $ such that the diagonals in $S$ have zero length.\nThis in turn defines a decomposition of $\\mathrm{Pol}_ n(\\mathbb{R}^3)$ into subspaces.\n$$\\mathrm{Pol}_ n(\\mathbb{R}^3)^{[S]} = \\mathrm{Pol}_ n(\\mathbb{R}^3)^S \\setminus \\cup_{S \\subset J} \\mathrm{Pol}_ n(\\mathbb{R}^3)^J$$\nThe subspace $\\mathrm{Pol}_ n(\\mathbb{R}^3)^{[S]}$ is the collection of all points in $\\mathrm{Pol}_ n(\\mathbb{R}^3)$\nsuch that exactly the diagonals in $S$ have zero length. This also\ninduces a filtration on $M_{\\mathbf{r}}$.\n\n\\begin{definition}\nDefine an action of $\\mathrm{SO}(3,\\mathbb{R})^{D(S)}$ on $\\mathrm{Pol}_ n(\\mathbb{R}^3)^{[S]}$ by\nletting $\\mathrm{SO}(3, \\mathbb{R})^{D(C_i)}$ act diagonally on the edges in $E_i(\\mathbf{e} )$.\nThe equivalence relation given by $\\mathrm{SO}(3,\\mathbb{R})^{D(S)}$ orbit type on $\\mathrm{Pol}_ n(\\mathbb{R}^3)^{[S]}$\nfit together to give an equivalence relation $\\sim_{\\mathcal{T}}$,\nwhich we call $\\mathcal{T}$-congruence, on $\\mathrm{Pol}_ n$.\n\\end{definition}\nWe may describe the above equivalence relation geometrically as follows.\nA polygon $\\mathbf{e} \\in \\mathrm{Pol}_ n(\\mathbb{R}^3)^{[S]}$ is a wedge of $|S| +1$ closed polygons $\\mathbf{e}_i$\nwedged together at certain vertices of the polygon $\\mathbf{e}$. Although each $\\mathbf{e}_i$\nmay contain several wedge points, since the vertices of $\\mathbf{e}$ are ordered there\nwill be a first wedge point $v_i$. Then we apply\na rotation $g_i$ about $v_i$ to each $\\mathbf{e}_i$ for $1 \\leq i \\leq |S|+1$ and identify\npoints in the resulting orbit of $\\mathrm{SO}(3,\\mathbb{R})^{D(S)}$.\n\n$\\mathcal{T}$-congruence for $\\mathrm{Pol}_ n(\\mathbb{R}^3)$ induces an equivalence relation on the space of $n$-gon linkages $\\widetilde{M}_{\\mathbf{r}} \\subset \\mathrm{Pol}_ n(\\mathbb{R}^3)$,\nwhich we also call $\\mathcal{T}$-congruence.\n\n\\begin{definition}\nDefine $$V_{\\mathbf{r}}^{\\mathcal{T}} = \\widetilde{M}_{\\mathbf{r}} \/ \\sim_{\\mathcal{T}}$$\n\\end{definition}\nKamiyama and Yoshida studied the space $V_{\\mathbf{r}}^{\\mathcal{T}}$ for the special case\nwhen $\\mathcal{T}$ was the {\\it fan} tree.\nNote that $V_{\\mathbf{r}}^{\\mathcal{T}}$ inherits a filtration by the subspaces $(V_{\\mathbf{r}}^{\\mathcal{T}})^{S}$.\nLet $(V_{\\mathbf{r}}^{\\mathcal{T}})^{[S]} = (V_{\\mathbf{r}}^{\\mathcal{T}})^{S} \\setminus \\cup_{S \\subset J} (V_{\\mathbf{r}}^{\\mathcal{T}})^J$.\nThe spaces $(V_{\\mathbf{r}}^{\\mathcal{T}})^{[S]}$ define a decomposition of $V_{\\mathbf{r}}^{\\mathcal{T}}$.\nWe can say more about the pieces of this decomposition.\n\n\\begin{theorem}\n$(V_{\\mathbf{r}}^{\\mathcal{T}})^{[S]}$ is canonically homeomorphic to $\\prod_{C_i \\subset \\mathcal{T}^S} M_{\\mathbf{r}_i}^o$\nwhere $\\mathbf{r}_i$ is the subvector of linkage lengths corresponding to the elements in $dist(C_i)$,\nand $M_{\\mathbf{r}_i}^o$ is the dense open subset of $M_{\\mathbf{r}_i}$ corresponding to polygons\nwith no zero diagonals.\n\\end{theorem}\n\n\\begin{proof}\nLet us first describe a map $\\tilde{F}: M_{\\mathbf{r}}^{S} \\to V_{\\mathbf{r}_1} \\times \\ldots \\times V_{\\mathbf{r}_k}$.\nThe diagonals $S$ define a partition of edges sets $E_i(\\mathbf{e})$ for each $\\mathbf{e} \\in \\widetilde{M}_{\\mathbf{r}}^{S}$. By the above proposition, $E_i(\\mathbf{e})$ corresponds to a closed polygon.\nSo we may send a member of the equivalence class of $\\mathbf{e}$ to the product of the equivalence classes\ndefining these closed polygons in $V_{\\mathbf{r}_1} \\times \\ldots \\times V_{\\mathbf{r}_k}$, with the appropriate\n$\\mathbf{r}_i$. This map is certainly onto, and by the definition of $V_{\\mathbf{r}_i}$, it factors through\nthe relation $\\sim{\\mathcal{T}}$. Hence, we get a 1-1 and onto continuous function\n $F: (V_{\\mathbf{r}}^{\\mathcal{T}})^{S} \\to V_{\\mathbf{r}_1} \\times \\ldots \\times V_{\\mathbf{r}_k}$, which is a homeomorphism\nby the compactness of $V_{\\mathbf{r}_1} \\times \\ldots \\times V_{\\mathbf{r}_k}$.\n\n\nWe may restrict this map to the spaces $(V_{\\mathbf{r}}^{\\mathcal{T}})^{[S]}$ which define the induced\ndecomposition of $V_{\\mathbf{r}}^{\\mathcal{T}}$. Recall that these are the polygons with exactly the $S$\ndiagonals zero. We therefore obtain that $(V_{\\mathbf{r}}^{\\mathcal{T}})^{[S]}$ is homeomorphic to $M_{\\mathbf{r}_1}^o \\times \\ldots \\times M_{\\mathbf{r}_k}^o$,\n\\end{proof}\n\n\\begin{remark}\nSince the fibers of $\\pi : M_\\mathbf{r} \\to V^\\mathcal{T}_\\mathbf{r}$ are sometimes\nodd-dimensional, the quotient map $\\pi$ cannot be algebraic\neven when $\\mathbf{r}$ is integral.\n\\end{remark}\n\n\n\n\\subsection{$\\mathcal{T}$-congruence of imploded spin-framed polygons}\n\nIn this section we introduce a generalization of the $\\mathcal{T}$-congruence\nrelation by lifting $\\mathcal{T}$-congruence from $\\mathrm{Pol}_ n(\\mathbb{R}^3)$ to $P_n(\\mathrm{SU}(2))$.\nIn section 3 we constructed a map $F_n: P_n(\\mathrm{SU}(2)) \\rightarrow \\mathrm{Pol}_ n(\\mathbb{R}^3)\/\\mathrm{SO}(3,\\mathbb{R})$. \nPulling back the decomposition of $\\mathrm{Pol}_ n(\\mathbb{R}^3)\/\\mathrm{SO}(3,\\mathbb{R})$ into $\\mathcal{T}$-\ncongruence classes\n by $F_n$ produces a decomposition of $P_n(\\mathrm{SU}(2))$ by the spaces\nwe will denote $P_n(\\mathrm{SU}(2))^{[S]} =F_n^{-1}(\\mathrm{Pol}_ n(\\mathbb{R}^3)^{[S]})$.\n\n\\begin{lemma}\\label{P_nStrat}\nElements of $P_n(\\mathrm{SU}(2))^{[S]}$ are the spin-framed polygons\nin $P_n(\\mathrm{SU}(2))$ such that the following equation holds,\n\n$$\\sum \\lambda_iAd^*_{g_i}(\\varpi_1) = 0$$\nwhere the sum is over all $[g_i, \\lambda_i\\varpi]\\in \\mathbf{E}_j(\\mathbf{e})$, the\nedges in the $j$-th partition defined by $S$, for each $j$.\n\\end{lemma}\n\n\\begin{proof}\nThis follows from Theorem \\ref{framedngons} and Remark \\ref{SpinF}.\n\\end{proof}\nBy the previous lemma we have a decomposition of $\\mathbf{e} \\in P_n(\\mathrm{SU}(2))^{[S]}$ into imploded spin-framed\npolygons\n\n$$\\mathbf{E} = \\mathbf{E}_1(\\mathbf{e} ) \\cup \\ldots \\cup \\mathbf{E}_{|S|+1}(\\mathbf{e} ).$$\nWe define an action of $\\mathrm{SU}(2)^{D(S)}$ on $P_n(\\mathrm{SU}(2))^{[S]}$ by\nletting $\\mathrm{SU}(2)^{D(C)}$ act diagonally on the framed edges associated\nto the component $C$ of $\\mathcal{T}^S$.\n\n\n\\begin{definition}\nThe $\\mathrm{SU}(2)^{D(S)}$-orbits on $P_n(\\mathrm{SU}(2))^{[S]}$\nfit together to give an equivalence relation $\\sim_{\\mathcal{T}}$,\nwhich we again call $\\mathcal{T}$-congruence, on $P_n(\\mathrm{SU}(2))$.\n\\end{definition}\n\n\\begin{definition}\n$$V_n^{\\mathcal{T}} = P_n(\\mathrm{SU}(2))\/ \\sim_{\\mathcal{T}}$$\n\\end{definition}\nNote that this defines a decomposition on $V_n^{\\mathcal{T}}$ into\nsubspaces $(V_n^{\\mathcal{T}})^{[S]} = P_n(\\mathrm{SU}(2))^{[S]}\/\\mathrm{SU}(2)^{D(S)}$.\nWe have seen that $P_n(\\mathrm{SU}(2))$ carries an action of $T_{\\mathrm{SU}(2)^n}$ given\nby the following formula. Let $\\mathbf{t} = (t_1, \\cdots, t_n) \\in T_{\\mathrm{SU}(2)^n}$,\n\n$$ \\mathbf{t} \\circ ([g_1, \\lambda_1\\varpi_1, \\ldots, [g_n, \\lambda_n \\varpi_1) =\n([g_1t_1, \\lambda_1\\varpi_1], \\ldots, [g_nt_n, \\lambda_n\\varpi_1).$$\nIn particular, note that the action of the diagonal $(-1)$ element in $T_{\\mathrm{SU}(2)^n}$ is trivial, because\nthis corresponds to acting on the left by the nontrivial central $\\mathrm{SU}(2)$ element.\nSince $t_i$ fixes $\\varpi_1$ for $1 \\leq i \\leq n$ this action does not change the image of a point under\n$F_n$, hence it fixes each piece of the decomposition. Also this action commutes\nwith the $\\mathrm{SU}(2)^{D(S)}$-action on the piece $P_n(\\mathrm{SU}(2))^{[S]}$, so\nthe action of $T_{\\mathrm{SU}(2)^n}$ must descend to $V_n^{\\mathcal{T}}$.\n\n\n\n\n\n\n\n\\section{The toric varieties $P^{\\mathcal{T}}_{n}(\\mathrm{SU}(2))$ and $Q^{\\mathcal{T}}_{n}(\\mathrm{SU}(2))$}\n\nIn this section we will construct the affine toric variety $P^{\\mathcal{T}}_{n}(\\mathrm{SU}(2))$ of imploded triangulated $\\mathrm{SU}(2)$--framed $n$--gons (without imposing the condition that the side-lengths are $\\mathbf{r}$) and its projective quotient\n$Q^{\\mathcal{T}}_{n}(\\mathrm{SU}(2))$.\n\nWe have tried to follow the notation of \\cite{HurtubiseJeffrey} when possible. In \\cite{HurtubiseJeffrey} the superscript $D$ on $P^D$ refers to a ``pair of pants'' decomposition $D$ of the surface. For us superscript $\\mathcal{T}$ on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ refers to the triangulation of the $n$-gon. The connection is the following. Under the correspondence between moduli spaces of $n$-gons and character varieties briefly explained in Remark \\ref{moduliofconnections}\n(and explained in detail in \\S 5 of \\cite{KapovichMillson}) the standard triangulation corresponds to the following ``pair of pants'' decomposition of the $n$ times punctured two-sphere. Represent the sphere as the complex plane with a point at infinity. Take the punctures to be the points $1,2,\\cdots,n$ on the real line. Draw small circles around the punctures. Now draw n-3 more circles with centers on the $x$-axis, so that the first circle contains the small circles around $1$ and $2$, the next circle contains the circle just drawn and the small circle around 3 and the last circle contains all the previous circles except the small circles around $n-1$ and $n$. The graph dual to the pair of pants decomposition is the tree $\\mathcal{T}$ that is dual to the triangulation.\nFurthermore the decomposed tree $\\mathcal{T}^D$ is dual to the pair-of-pants\ndecomposition of the $n$-punctured sphere obtained by cutting the sphere\napart along the above $2n-3$ circles - we might say that $\\mathcal{T}^D$ is the\npair-of-pants decomposition of $\\mathcal{T}$.\nUsing this correspondence the reader should be able to relate what follows with \\cite{HurtubiseJeffrey} for the case of the $n$-fold punctured sphere.\n\n\n\nIt will be important in what follows that we have earlier defined the quotient map\n$$\\pi^{D}: \\mathcal{T}^D \\to \\mathcal{T}$$\nthat glues together the pairs of vertices that are the boundaries of\nthe open intervals removed from $\\mathcal{T}$.\n\n\n\\subsection{The space $E^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ of imploded framed edges}\n\nWe define $E^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ to be the product of $\\mathcal{E}T^*(\\mathrm{SU}(2))$\nover the edges of $\\mathcal{T}^D$, hence an element $\\mathbf{T} \\in\nE^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ is a map\nfrom the $3(n-2)$ edges of $\\mathcal{T}^D$ into $\\mathcal{E}T^*(\\mathrm{SU}(2))$ or equivalently\nan assignment of an element of $\\mathcal{E}T^*(\\mathrm{SU}(2))$ to each\neach of the $3(n-2)$ edges of the triangles $\\tau_i,1 \\leq n-2$ in the triangulation\nof $P$. It will be important later to note that there is a forgetful\nmap that restricts $\\mathbf{T}$ to the distinguished edges of $\\mathcal{T}^D$ to obtain\nan element $\\mathbf{E}$ of $E_n(\\mathrm{SU}(2))$.\n\n\nIt will be convenient to represent\nthe resulting product $\\mathcal{E}T^*\\mathrm{SU}(2)^{3n-6} \\cong (\\mathbb{C}^2)^{3n-6}$ by a\n$2$ by $3n-6$ matrix. To do this we will linearly order the $3n-6$ edges by\nfirst ordering the $n-2$ triangles (tripods) and then ordering the $3$ edges for\neach triangle (tripod). Now use the lexicographic ordering on pairs $(triangle,edge)$\n($(tripod,edge)$). The point is that in the matrix $A^{\\mathcal{T}}$ below the $(\\mathbb{C}^2)$'s belonging\nto the same triangle (tripod) appear consecutively. Let $(\\tau_i, j)$ denote the\n$j$-th edge of the $i$-th tripod.\n\nWe will use the following notation for the entries in the matrix $A^{\\mathcal{T}}$ to indicate\nthat each successive group of $3$ columns of $A^{\\mathcal{T}}$ belongs to a triangle in the\ntriangulation. \n$$A^{\\mathcal{T}} =\n\\begin{pmatrix}\nz_1(\\tau_1) & z_2(\\tau_1) & z_3(\\tau_1) & z_1(\\tau_2) & z_2(\\tau_2) & z_3(\\tau_2) & \\cdots & z_1(\\tau_{n-2}) & z_2(\\tau_{n-2}) & z_3(\\tau_{n-2}) \\\\\nw_1(\\tau_1) & w_2(\\tau_1) & w_3(\\tau_1) & w_1(\\tau_2) & w_2(\\tau_2) & w_3(\\tau_2) & \\cdots & w_1(\\tau_{n-2}) & w_2(\\tau_{n-2}) & w_3(\\tau_{n-2})\n\\end{pmatrix}$$\nIn what follows we will abbreviate $\\mathcal{E}T^*(\\mathrm{SU}(2))^{3n-6}$\nto $E^{\\mathcal{T}}_n(\\mathrm{SU}(2))$. \n\n\n\n\n\n\n\\subsection{The space $X^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ of imploded framed triangles}\n\nThe action of the group $\\mathrm{SU}(2)^{3n-6}$ on $\\mathcal{E}T^*(\\mathrm{SU}(2))^{3n-6}$ is then\nrepresented by acting on the columns of $A^{\\mathcal{T}}$. We will represent elements $\\mathbf{g}$ of\n$\\mathrm{SU}(2)^{3n-6}$ as $3n-6$-tuples\n\n$$\\mathbf{g} = (f_1(\\tau_1),f_2(\\tau_1),f_3(\\tau_1)|\\cdots|f_1(\\tau_{n-2}),,f_2(\\tau_{n-2}),\nf_3(\\tau_{n-2})).$$\nWe let $\\mathrm{SU}(2)^{n-2}$ denote the ``diagonal'' subgroup of $\\mathrm{SU}(2)^{3n-6}$ defined\nby the condition that for each triangle $T_i$ (tripod $\\tau_i$) we have\n\n$$f_1(\\tau_i) = f_2(\\tau_i) = f_3(\\tau_i).$$\nWe will regard $\\mathrm{SU}(2)^{n-2}$ as the space of mappings $\\mathbf{f}$ from the tripods in $\\mathcal{T}^D$ to $\\mathrm{SU}(2)$.\n\n\n\\begin{definition}\nWe then define the space $X^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ as the symplectic quotient\n$$ X^{\\mathcal{T}}_n(\\mathrm{SU}(2)) = \\mathrm{SU}(2)^{n-2}\\backslash \\! \\backslash (\\mathcal{E}T^*\\mathrm{SU}(2))^{3n-6}.$$\n\\end{definition}\n\nThus $X^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ is obtained by taking the symplectic quotient of each\ntriple of edges belonging to one of the triangles $T_i, 1 \\leq i \\leq n-2$, by\nthe group $\\mathrm{SU}(2)^{n-2}$. The resulting space $X_n(\\mathrm{SU}(2))$ is clearly the product of $n-2$\ncopies of $(P_3(\\mathrm{SU}(2)))^{n-2} \\cong (\\bigwedge^{2}(\\mathbb{C}^3))^{n-2}$.\\\\\n\nWe will often denote an element of $X^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ by $\\mathbf{T}=([T_1],[T_2],\\cdots,[T_{n-2}])$\nwhere $[T_i]$, the $i$-th triangle denotes the equivalence class of the matrix\n\n$A_{\\tau_i} =\n\\begin{pmatrix}\nz_1(\\tau_i) & z_2(\\tau_i) & z_3(\\tau_i)\\\\\nw_1(\\tau_i) & w_2(\\tau_i) & w_3(\\tau_i)\n\\end{pmatrix}$\nsuch that the momentum image of $A_i$ under the momentum map for $\\mathrm{SU}(2)$ is zero\n(equivalently the rows are orthogonal with the same length).\n\n\n\n\\subsection{The affine toric variety $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$}\\label{mastertoricspace}\n\nIn equation (\\ref{P3}) we have seen that\n$$P_3(\\mathrm{SU}(2)) \\cong \\mathrm{\\bigwedge} \\space^2(\\mathbb{C}^3).$$\nSince the space of imploded triangles is the product of\n$n-2$ copies of $P_3(\\mathrm{SU}(2))$ we see that\n$X^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ is the affine space obtained by taking the product of $n-2$ copies\nof $\\bigwedge^2(\\mathbb{C}^3)$. It has an action of a $3n-6$ torus $\\mathbb{T}$ induced\nby the right actions of the torus $\\mathbb{T} = T _{\\mathrm{SU}(2)}^{3n-6}$ on the $3n-6$ copies\nof $\\mathcal{E}T^*(\\mathrm{SU}(2))$. In terms of our matrices $A^{\\mathcal{T}}$ this \namounts to scaling the\ncolumns of $A^{\\mathcal{T}}$ (right multiplication of $A^{\\mathcal{T}}$ by $\\mathbb{T}$).\nHowever note that the entry of an element of $\\mathbb{T}$ corresponding to\nthe edge $e$ of $\\mathcal{T}^D$\nscales the column of $A^{\\mathcal{T}}$ corresponding to $e$ by its {\\it inverse}.\nWe will use the notation $\\underline{\\mathbb{T}}$ to indicate the complexification of \n$\\mathbb{T}$. Similarly for any compact group $G$ that appears in this paper, \nthe notation $\\underline{G}$ indicates the complexification of $G$.\n\nLet $\\mathbb{T}_e$, $\\mathbb{T}_d$, $\\mathbb{T}_d^-$ be as in the introduction.\nNow we glue the diagonals of $P$ back together by taking the\nsymplectic quotient by $\\mathbb{T}_d^-$ at level $0$.\n\n\n\\begin{definition}\n$$P_n^{\\mathcal{T}}(\\mathrm{SU}(2)) = X_n^{\\mathcal{T}}(\\mathrm{SU}(2)\/\/\\mathbb{T}_d^- = (\\mathrm{\\bigwedge} \\space^2(\\mathbb{C}^3))^{n-2}\/\/\\mathbb{T}_d^-.$$\n\\end{definition}\n\n\n\\subsection{The space $P_n^{\\mathcal{T}}(\\mathrm{SO}(3,\\mathbb{R}))$}\\label{orthogonalanalogues}\nThe spaces $E_n^{\\mathcal{T}}(\\mathrm{SU}(2)), X_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ and $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ all have analogues when $\\mathrm{SU}(2))$ is replaced by\n$\\mathrm{SO}(3,\\mathbb{R})$ which are quotients\nby a finite group of the corresponding spaces for $\\mathrm{SU}(2)$. There\nalso analogues of the tori $\\mathbb{T},\\mathbb{T}_e$ and $\\mathbb{T}_d^-$\nfor $\\mathrm{SO}(3,\\mathbb{R})$ which we will denote $\\mathbb{T}(\\mathrm{SO}(3,\\mathbb{R})),\\mathbb{T}_e(\\mathrm{SO}(3,\\mathbb{R}))$ and $\\mathbb{T}_d^-(\\mathrm{SO}(3,\\mathbb{R}))$ respectively which are quotients of the\ncorresponding tori for $\\mathrm{SU}(2)$. We leave the details to the reader.\n\n\\subsection{$P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ as a GIT quotient}\\label{firsttorusquotient}\nSince affine GIT quotients coincide with symplectic quotients, see \\cite{KempfNess} and \\cite{Schwarz},\nTheorem 4.2, we may also obtain $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ as the GIT quotient of $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$\nby the complex torus $\\underline{\\mathbb{T}}^d$. For each triangle $T_k$ (tripod $\\tau_k$) we have a corresponding\n$\\bigwedge^2(\\mathbb{C}^3)$ with Pl\\\"ucker coordinates $Z_{ij}(\\tau_k), 1 \\leq i,j \\leq 3$. The coordinate\nring of the affine variety $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ will be the ring of invariants\n$\\mathbb{C}[(Z_{ij}(\\tau_k))]^{\\underline{\\mathbb{T}}^-_d}$. This ring of invariants will be spanned by the ring of\ninvariant {\\em monomials} which we now determine. There is a technical problem\nhere. We need to know that we have chosen the correctly normalized\nmomentum map for the action $\\mathbb{T}^-_d$. But by Theorem \\ref{twistandshift}\nthe correct normalization is the one that is homogeneous linear in the\nsquares of the norms of the coordinates which is the one we have used here.\n\n\\subsection{The semigroup $\\mathcal{P}^{\\mathcal{T}}_n$}\n\nAs a geometric quotient of affine space by a torus the space\n$P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is an affine toric variety. We now analyze the\naffine coordinate ring of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$. In what follows we label the leaf of the tripod\n$\\tau_i$ incident to the edge $(\\tau_i,k)$ by $k$ - we will do this only when the tripod of which\n$k$ is a vertex is clearly indicated. We leave the proof of the following lemma to the reader.\n\n\n\\begin{lemma}\nThe monomial $f(Z)=\n\\prod_{i=1}^{n-2}Z_{12}(\\tau_i)^{x_{12}(\\tau_i)}Z_{13}(\\tau_i)^{x_{13}(\\tau_i)}\nZ_{23}(\\tau_i)^{x_{23}(\\tau_i)}$ is $\\underline{\\mathbb{T}}_d^-$--invariant if and only if the\nexponents $\\mathbf{x} = (x_{jk}(\\tau_i))$ satisfy the system of equations\n\n$$x_{k,m}(\\tau_i) + x_{k,l}(\\tau_i) = x_{l,k}(\\tau_j) + x_{l,m}(\\tau_j), \\quad \\text{for } \\ (\\tau_i, k) \\ \\text{identified to } (\\tau_j, l) \\ \\text{in $\\mathcal{T}$}.$$\n\\end{lemma}\nBefore stating a corollary of the lemma we need a definition.\n\n\\begin{definition}\nLet $\\mathcal{P}^{\\mathcal{T}}_n$ be the subset of $\\mathbf{x} \\in (\\mathbb{N}^3)^{(n-2)}$ satisfying the equations\nin the above lemma. Clearly $\\mathcal{P}^{\\mathcal{T}}_n$ is a semigroup under addition.\nLet $\\mathbb{C}[\\mathcal{P}_n^{\\mathcal{T}}]$ denote the associated semigroup algebra.\n\\end{definition}\n\n\\begin{corollary}\nThe affine coordinate ring of\n$P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is isomorphic to the semigroup ring $\\mathbb{C}[\\mathcal{P}_n^{\\mathcal{T}}]$.\n\\end{corollary}\n\n\nNow we will relate the semigroup $\\mathcal{P}^{\\mathcal{T}}_n$ and the monomials of the lemma to graphs on vertices of the decomposed tree $\\mathcal{T}^D$. Monomials in the Pl\\\"ucker coordinates $Z_{ij}(\\tau)$ for $\\tau$ fixed correspond to graphs on the vertices $i,j,k$ of the tripod $\\tau$ as follows. We associate to the exponent $x_{ij}(\\tau)$ of $Z_{ij}(\\tau)$ the graph consisting of $x_{ij}(\\tau)$ arcs joining the vertex $i$ of $\\tau$ to the vertex $j$. Each triple $\\{x_{ij}(\\tau)\\}$ for $\\tau$ fixed determines a graph on the leaves of the tripod $\\tau$.\nThus each element $\\mathbf{x} \\in \\mathcal{P}_n^{\\mathcal{T}}$ corresponds to a collection of $n-2$ graphs,\neach on 3 vertices, one for each tripod in $\\mathcal{T}^D$. Also each such element\n$\\mathbf{x}$ corresponds to a product of monomials attached to tripods. This leads to a bijective correspondence\nbetween monomials and graphs.\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.4]{tripod.eps}\n\\caption{A single tripod $\\tau$ with vertex $v$, with $x_{ef}(\\tau) = 3$, $x_{eg}(\\tau)=2$, $x_{fg}(\\tau)=1$. Hence $x_{ef}(\\tau)+x_{eg}(\\tau)=5$, $x_{ef}(\\tau)+x_{fg}(\\tau)=4$, $x_{eg}(\\tau)+x_{fg}(\\tau)=3$. }\n\\label{fig:tripod}\n\\end{figure}\nThe condition\n\n$$x_{km}(\\tau_i) + x_{k\\ell}(\\tau_i) = x_{\\ell,k}(\\tau_j) + x_{\\ell,m}(\\tau_j)$$\nspecifies that the number of arcs associated to the identified edges $(\\tau_i, k)$ of\nthe tripod $\\tau_i$, and $(\\tau_j,\\ell)$ of the tripod $\\tau_j$ must agree. Later, this will allow\nus to glue these ``local'' graphs to obtain a ``global'' graph\non $n$ vertices. In the next section we will associate a weighting of $\\mathcal{T}$ to the $\\{x_{ij}(\\tau_k)\\}$.\n\n\\subsection{The projective toric variety $Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$}\n\nWe define $Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$, a projectivization of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$,\nwhich will be isomorphic to $\\mathrm{Gr}_2(\\mathbb{C}^n)^\\mathcal{T}_0$. The coordinate ring of the affine\ntoric variety $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is the semigroup algebra $\\mathbb{C}[\\mathcal{P}_n^{\\mathcal{T}}]$.\nHence to define the $\\mathbb{C}^*$ action required\nfor projectivization, it suffices to define a grading on $\\mathcal{P}^{\\mathcal{T}}_n$.\nFirst, for $\\mathbf{x} \\in P^{\\mathcal{T}}$ we define the associated weight of the\nedge $(\\tau,i)$ of the tripod $\\tau$ in\n$\\mathcal{T}^D$.\nlet $w_{(\\tau,i)}(\\mathbf{x} ) = x_{ij}(\\tau) + x_{ik}(\\tau)$.\nWe define the degree of an element $\\mathbf{x} \\in \\mathcal{P}_n^{\\mathcal{T}}$ by\n\n$$degree(\\mathbf{x}) = \\frac{1}{2} \\sum_{(\\tau,i) \\in dist(\\mathcal{T}^D)} w_{(\\tau, i)}(\\mathbf{x})$$\nwhere the sum is over all $(\\tau, i)$ which represent leaf edges of $\\mathcal{T}$.\n\n\\begin{remark}\nWe will see below that the under the isomorphism from the semigroup $\\mathcal{P}_n^{\\mathcal{T}}$\nto the semigroups of admissible weightings $\\mathcal{W}_n^{\\mathcal{T}}$ the number $w_{(\\tau,i)}(\\mathbf{x})$ will be the weight assigned to the edge $(\\tau,i)$\nof the tree $\\mathcal{T}$. Thus the degree of $\\mathbf{x}$ is then half the sum of the weights of the leaf edges.\n\\end{remark}\n\n\n\\begin{definition}\nGive $\\mathcal{P}^{\\mathcal{T}}_n$ the grading defined above, then we define\n$$Q_n^{\\mathcal{T}}(\\mathrm{SU}(2)) = \\mathrm{Projm}(\\mathbb{C}[\\mathcal{P}_n^{\\mathcal{T}}])$$\n\\end{definition}\nNote that as defined $Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is a GIT $\\mathbb{C}^*$-quotient of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The toric varieties $Q^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ and $\\mathrm{Gr}_2(\\mathbb{C}^n)^{\\mathcal{T}}_0$ are isomorphic}\n\n\nIn this section we will prove the first statement of Theorem \\ref{firstauxiliarytheorem} namely that the toric variety $Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ constructed in\nthe previous section is the isomorphic to $\\mathrm{Gr}_2(\\mathbb{C}^n)^{\\mathcal{T}}_0$ by proving\nProposition \\label{threeisomorphicsemigroups} below.\n\nRecall that in Section 4 it was shown that the two semigroups $\\mathcal{S}^{\\mathcal{T}}_n$ and $\\mathcal{W}^{\\mathcal{T}}_n$ are isomorphic.\n$\\mathcal{S}^{\\mathcal{T}}_n$ is the semigroup of Kempe Graphs on $n = |edge(\\mathcal{T})|$ vertices, with the product $\\ast_{\\mathcal{T}}$. $\\mathcal{W}^{\\mathcal{T}}_n$ is the semigroup of admissible weightings on $\\mathcal{T}$ under addition. Recall that an admissible weighting is an integer weight satisfying the triangle inequalities about each internal vertex of $\\mathcal{T}$, along with the condition that the sum of the weights about each internal node be even. It was also shown in section 4 that the semigroup algebra of $\\mathcal{S}_n^{\\mathcal{T}} \\cong \\mathcal{W}_n^{\\mathcal{T}}$ is isomorphic to the coordinate ring of $\\mathrm{Gr}_2(\\mathbb{C}^n)_0^{\\mathcal{T}}$.\n\n\\begin{proposition}\\label{twoisomorphicsemigroups}\nThe graded semigroups $\\mathcal{W}^{\\mathcal{T}}_n$ and $\\mathcal{P}^{\\mathcal{T}}_n$\nare isomorphic.\n\\end{proposition}\n\n\\begin{proof}\nIn all that follows we deal with tripods, so we give the unique leaf edge incident to the leaf labelled $i$ the label $i$, and vice-versa.\nWith this in mind, $X_{ij}$ can be thought of as giving the number of arcs in a graph on the\nleaves of a tripod $Y$ between the $i$-th and $j$-th leaves. The number $N_i$ is a natural number\nassigned to the $i$-th edge of a tripod $Y$ obtained by counting the number of arcs which have unique\npath in $Y$ containing the $i$-th edge.\nThe elements of both $\\mathcal{P}^{\\mathcal{T}}_n$ and $\\mathcal{W}^{\\mathcal{T}}_n$ both associate triples of\nintegers to each tripod of ${\\mathcal{T}^D}$ with certain ``gluing conditions''. Let us consider a single tripod $Y \\in \\mathcal{T}^D$. A triple of numbers $N_1, N_2, N_3$ is an admissible weighting of $Y$ if and only if there are integers $X_{ij}$, $i, j \\in \\{1, 2, 3\\}$ such that $X_{ij} + X_{ik} = N_i$. To see this simply note that the equations\n$$X_{ij} = \\frac{N_i + N_j - N_k}{2}$$\nhave natural solutions if and only if $(N_1, N_2, N_3)$ is admissible.\nTherefore we may define a map from $\\mathcal{W}^{\\mathcal{T}}_n$ to $\\mathcal{P}^{\\mathcal{T}}_n$ by solving for $x_{ij}(\\tau)$, with an obvious inverse given by solving for the weighting on the edge $(\\tau, i)$. To see that these maps are well defined, note that the gluing condition\n$$x_{km}(\\tau_i) + x_{k\\ell}(\\tau_i) = x_{\\ell,k}(\\tau_j) + x_{\\ell,m}(\\tau_j), \\quad \\text{for } (\\tau_i, k) \\text{ identified to } (\\tau_j, \\ell) \\text{$\\in \\mathcal{T}$}.$$\nis exactly equivalent to the weights on $(\\tau_i, n)$ and $(\\tau_j, l)$ being equal when these tuples represent the same diagonal in $\\mathcal{T}$. Since both semigroup operations are defined by adding integers, and since both $\\Phi$ and its inverse are linear functions over each trinode, these maps are semigroup isomorphisms.\nFinally, note that the grading on $\\mathcal{P}^{\\mathcal{T}}_n$ was chosen specifically to match the grading on $\\mathcal{S}^{\\mathcal{T}}_n$, we leave\ndirect verification of this to the reader.\n\\end{proof}\n\n\n\n\\subsubsection{An explicit description of the ring isomorphism}\n The semigroup $\\mathcal{S}^{\\mathcal{T}}_n$ is generated as a graded semigroup by the elements corresponding to\ngraphs with exactly one edge. The element corresponding to the graph with one edge, between the $i$ and $j$ vertices corresponds to the Pl\\\"ucker coordinate $Z_{ij}$. Recall that the unique path in the tree $\\mathcal{T}$ joining\nthe leafs $i$ and $j$ has been denoted $\\gamma(ij)$. The path $\\gamma(ij)$\ngives rise to a sequence of edges of the forest $\\mathcal{T}^D$ (where we pass from\none tripod to the next by passing from an edge $(\\tau_i, k)$ to an equivalent\nedge $(\\tau_j,\\ell)$. We let $Z_{\\gamma(ij)}$ be the corresponding\nproduct of Pl\\\"ucker coordinates $Z_{st}(\\tau_k)$. Thus\n$$Z_{\\gamma(ij)} = \\prod Z_{st}(\\tau)$$\nwhere the $(\\tau, s)$ and $(\\tau, t)$ are the edges in the unique path\ndefined by the path $\\gamma(i,j)$ corresponding to $Z_{ij}$ in $\\mathcal{T}$.\nNote that $Z_{\\gamma(ij)}$ is a $\\underline{\\mathbb{T}}_d^-$-invariant monomial\nin the homogeneous coordinate ring of $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ -\nmoreover it is a generator of the ring of invariants.\nThe isomorphism of toric rings $\\Phi$ from the homogeneous\ncoordinate ring of $\\mathrm{Gr}_2(\\mathbb{C}^n)^{\\mathcal{T}}_0$ to $P^{\\mathcal{T}}_n(\\mathrm{SU}(2)$ is then given on generators\nby\n$$\\Phi(Z_{ij}) = Z_{\\gamma(ij)}.$$\n\\begin{remark}\nIt is important to see that the degree assigned to $Z_{\\gamma(ij)} = \\prod Z_{st}(\\tau)$ by the isomorphism $\\Phi$ (namely {\\it one}) is\ndifferent from that given by counting the $Z_{st}(\\tau)$ in the product formula\nfor $Z_{\\gamma(i,j)}$. The latter\ncount is in fact the Speyer-Sturmfels weight $w^{\\mathcal{T}}_{i,j}$ of the Pl\\\"ucker coordinate $Z_{ij}$.\n\\end{remark}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale = 0.3]{invar.eps}\n\\caption{This illustrates $\\Phi(Z_{14}) = Z_{\\gamma(14)} = Z_{13}(\\tau_1)Z_{12}(\\tau_2)Z_{13}(\\tau_4)$ in \nthe case $n=6$ with the symmetric tree.}\\label{fig:tripod2}\n\\end{figure}\n\n\\subsubsection{The grading circle action on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$}\n\nWe verify the second statement of Theorem \\ref{firstauxiliarytheorem}.\nWe have seen that the action of $\\lambda \\in \\mathbb{C}^*$ that gives the grading\nscales each $Z_{\\gamma(i,j)}$ by $\\lambda$.\nClearly this action is induced by the action on the matrix $A^{\\mathcal{T}}$\nthat scales each row corresponding to a leaf edge by $\\sqrt{\\lambda}$,\nin other words by the action of the element $\\mathbf{t}_e((\\sqrt{\\lambda})^{-1})$ as claimed in\nthe second statement of Theorem \\ref{firstauxiliarytheorem}.\nWe conclude by explaining why the actions of $\\mathbf{t}_e(\\sqrt{\\lambda})$\nand $\\mathbf{t}_e(-\\sqrt{\\lambda})$ coincide. It suffices to prove that\n$\\mathbf{t}_e(-1)$ acts trivially. But since the operation\nthat scales all rows of $A^{\\mathcal{T}}$ by $-1$ is induced by\nan element of $\\mathrm{SU}(2)^{n-2}$ it acts trivially on $X_n^{\\mathcal{T}}(\\mathrm{SU}(2))$\nand hence on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$. But also the operation of scaling all\nthe rows of $A^{\\mathcal{T}}$ that belong to nonleaf edges of $\\mathcal{T}^D$\nis induced by an element of $\\underline{\\mathbb{T}}_d^{-}$ hence this\noperation too is trivial on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$. But $\\mathbf{t}_e(-1)$\nis the composition of the two operations just proved to be trivial.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The spaces $W^{\\mathcal{T}}_n$ and $Q_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ are homeomorphic.}\\label{homeomorphismVtoP}\n\nWe now give the proof of Theorem \\ref{firstmaintheorem}.\nWe first note that we have identified an ordered subset of edges of $\\mathcal{T}^D$\nwith the leaf edges of $\\mathcal{T}$ (equivalently the edges of $P$). This\nidentification gives an isomorphism $\\rho:T_{\\mathrm{SU}(2)^n} \\to \\mathbb{T}_e$.\n\n\\subsection{A homeomorphism of affine varieties}\\label{secondtorusquotient}\nWe will prove the following.\n\n\\begin{theorem}\\label{noncompact}\nThe spaces $V^{\\mathcal{T}}_n$ and $P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ are equivariantly homeomorphic with respect to $\\rho$.\n\\end{theorem}\nWe will use the symbol\n$\\Psi^{\\mathcal{T}}_n$ to denote the above equivariant homeomorphism\nand $\\Phi^{\\mathcal{T}}_n$ to denote its inverse.\nBy the results of subsection \\ref{firsttorusquotient} it suffices to\nconstruct a $\\rho$-equivariant homeomorphism (and its inverse) from $V^{\\mathcal{T}}_n$\nto the above symplectic quotient which we will continue to denote\n$P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$. \nWe first construct the $\\rho$-equivariant map \n$\\Phi^{\\mathcal{T}}_n : P_n^{\\mathcal{T}}(\\mathrm{SU}(2)) \\to V^{\\mathcal{T}}_n$.\nThere is a simple idea behind this map. We have an inclusion of\nthe leaf edges of $\\mathcal{T}$ into the edges of $\\mathcal{T}^D$.\nNow an element of $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is a map $\\mathbf{T}$ from the edges\nof $\\mathcal{T}^D$ into $\\mathbb{C}^2$. The map $\\Phi^{\\mathcal{T}}_n$ is induced by the map\n$\\widetilde{\\Phi}^{\\mathcal{T}}_n:E_n^{\\mathcal{T}}(SU(2)) \\to E_n(\\mathrm{SU}(2))$ that\nrestricts $\\mathbf{T}$ to the leaf edges of $\\mathcal{T}$. However we need\nto verify that the image of an element of momentum level zero\nfor $\\mathrm{SU}(2)^{n-2} \\times \\mathbb{T}_d^-$ has $\\mathrm{SU}(2)$--momentum level zero,\nand that the induced map of zero momentum levels descends to the required quotients.\nFor the rest of this discussion, let $\\textbf{T} = (F_1(\\tau_1), \\dots, F_{n-2}(\\tau_{n-2}))$ be an \nelement of $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$. Each $F_i$ is an\nimploded spin framing of the edges of the triangle $\\tau_i$.\nThis means that\n$$F_i = [(g_1(i), \\lambda_1(i)\\varpi_1), (g_2(i), \\lambda_2(i)\\varpi_1), (g_3(i), \\lambda_3(i)\\varpi_1)]$$\nsuch that\n$$\\lambda_1(i)g_1(i)\\varpi_1 + \\lambda_2(i)g_2(i)\\varpi_1 + \\lambda_3(i)g_3(i)\\varpi_1 = 0$$\nWe will henceforth denote $T_i = F_i(\\tau_i)$, thus the symbol\n$T_i$ stands for a triangle together with an imploded spin-frame on its\nedges.\n\n\n\n\\subsubsection{$\\varrho$ flips and normalized framings}\\label{normalizedframings}\n\nLet $\\varrho = \\begin{pmatrix} 0 & 1 \\\\ -1 & 0\\end{pmatrix}$. Note that if\n$t \\in \\mathrm{SU}(2)$ fixes $\\varpi_1$ by conjugation, then $t$ is diagonal and\n$\\varrho t \\varrho^{-1} = t^{-1}$. Furthermore, $\\varrho \\varpi_1 \\varrho^{-1} = - \\varpi_1$.\nLet $[[T_1],[T_2],\\ldots,[T_{n-2}]$ be an element of $P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$.\n\n\\begin{lemma}\\label{rhoflip}\nSuppose that the $k$-th edge of $\\tau_i$ is identified with the $\\ell$-th edge\nof $\\tau_j$ in $\\mathcal{T}$ and that the edge $(\\tau_j, \\ell)$ comes after\nthe edge $(\\tau_i, k)$ in the above ordering of edges of $\\mathcal{T}^D$. Then using the left actions of $\\mathrm{SU}(2)$ we may arrange\nthat\n\n$$g_{(\\tau_j,\\ell)} = g_{(\\tau_i,k)}\\varrho.$$\n\\end{lemma}\n\n\\begin{proof}\nSuppose we have\n\n$$[T_i] = [(a_1, \\ell_1 \\varpi_1),(a_2, \\ell_2\\varpi_1), (h, d \\varpi_1)],$$\nIf the diagonal defined by $(\\tau_i, \\ell)$ and $(\\tau_j, k)$ has length $0$\nthere is nothing to prove since any two frames are equivalent. Suppose then\nthat this diagonal is not $0$. Then by applying $g =h \\varrho (h')^{-1}$ to $T_{j}$,\nwe get\n\n$$g \\cdot T_{j} = [(h \\varrho, d \\varpi_1),(g a_3,\\ell_3 \\varpi_1),(g a_4, \\ell_4 \\varpi_1)],$$\nwhich is equivalent to $T_{j}$. Hence the consecutive diagonal frames (the third\nframe of $T_{i}$ and the first frame of $g \\cdot T_{i+1}$) are now related\nby right multiplication by $\\varrho$.\n\\end{proof}\nWe shall henceforth choose representatives in $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ for\nelements of $P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ so that nonzero frames associated to equivalent edges\nsatisfy this ``$\\varrho$ flip condition''. If the frame associated to\none edge of a pair of equivalent edges is zero then we require that\nthe frame associated to the other edge of the pair is also zero. We say \nsuch elements of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ are {\\it normalized}.\nNote that the definition of normalized depends on the ordering\nof the triangles $T_i$.\n\n\nIt is important to note that if an element $A=([T_1],[T_2],\\cdots,[T_{n-2}])$ is normalized\nthen so is $t\\cdot A$ for any $t \\in \\mathbb{T}_d^-$. This is because $h\\varrho t^{-1} = ht\\varrho$. We leave the details\nto the reader. Thus we may speak of a $\\mathbb{T}_d^-$-equivalence class as being $normalized$.\nWe will define $\\Phi_n^{\\mathcal{T}}$ on such normalized elements.\n\n\n\\begin{lemma}\\label{norm}\nFor any $\\textbf{T} \\in E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ there exist $\\mathbf{f} \\in \\mathrm{SU}(2)^{n-2}$ and a normalized $\\textbf{T}' \\in E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ such that\n$$\\mathbf{f} \\mathbf{T} = \\mathbf{T}'.$$\n\\end{lemma}\n\n\n\\begin{proof}\nLet $\\mathcal{T}' \\subset \\mathcal{T}$ be connected, and let $Y \\subset \\mathcal{T}$\nbe a tripod which shares exactly one edge, $d$ with $\\mathcal{T}'$. Let $(\\tau, i)$\nand $(\\tau', j)$ be the edges of $\\mathcal{T}^D$ which map to $d$.\nSuppose $\\textbf{T} \\in E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is normalized\nat all diagonals corresponding to internal edges of $\\mathcal{T}'$.\nLet $\\mathbf{f} \\in \\mathrm{SU}(2)^{n-2}$ be the element\nwhich is $g_{(\\tau, i)}\\varrho g_{(\\tau', j)}^{-1}$ on\nthe $\\tau'$-th factor, and the identity elsewhere.\nThen $\\textbf{T}' = \\mathbf{f}\\textbf{T}$ is normalized\nat all diagonals corresponding to internal edges of $\\mathcal{T}' \\cup Y$.\nBy lemma \\ref{trivalentinduction} we can always find a sequence\nof connected trees $\\mathcal{T}_i \\subset \\mathcal{T}_{i+1}$\nwith $Y_i = \\mathcal{T}_{i+1}^D \\setminus \\mathcal{T}_i^D$ a tripod\nsuch that each internal edge of $\\mathcal{T}$ appears\nas an edge shared by the images of $Y_i$ and $\\mathcal{T}_i$\nin $\\mathcal{T}$ for some $i$. The lemma now follows by induction.\n\\end{proof}\n\n\n\\subsubsection{Definition of the maps $\\Phi_n^{\\mathcal{T}}$ and $\\Psi_n^{\\mathcal{T}}$}\n\nGiven a normalized element $\\textbf{T} \\in E^{\\mathcal{T}}_n(\\mathrm{SU}(2))$\ndefine $\\widetilde{\\Phi}_n^{\\mathcal{T}}(\\textbf{T})$ to be the element in $E_n(\\mathrm{SU}(2))$ given by projecting on the components\n$(g_{(\\tau_i, j)}, \\lambda_{(\\tau_i, j)}\\varpi_1) \\in \\textbf{T}$ such that $(\\tau_i, j) \\in \\mathcal{T}^D$\nmaps to a leaf edge in $\\mathcal{T}$ under $\\pi_{\\mathcal{T}}$.\n\n\\begin{lemma}\nIf $\\textbf{T} \\in \\mu_{\\mathrm{SU}(2)^{n-2}}^{-1}(0) \\cap \\mu_{\\mathbb{T}_d^-}^{-1}(0)$\nand is normalized then $\\widetilde{\\Phi}_n^{\\mathcal{T}}(\\textbf{T}) \\in\n\\widetilde{P}_n(\\mathrm{SU}(2))$, that is the polygon in $\\mathbb{R}^3$ associated to $\\widetilde{\\Phi}_n^{\\mathcal{T}}(\\textbf{T})$ closes up.\n\\end{lemma}\n\\begin{proof}\nFirst observe that $\\lambda g\\varpi_1 = - \\lambda g\\varrho\\varpi_1$, Now\nbecause each triangle closes up the sum of $g \\lambda \\varpi_1$ over all edges of $\\mathcal{T}^D$ is zero. But by the observation just above the sum over pairs of equivalent edges of $\\mathcal{T}^D$\ncancel. Hence the sum over leaf edges of $\\mathcal{T}$\nis zero as required.\n\\end{proof}\n\n\\begin{lemma}\\label{WDPhi}\n$\\widetilde{\\Phi}_n^{\\mathcal{T}} $ induces a well-defined map on the quotient\n$$\\widehat{\\Phi}_n^{\\mathcal{T}}: \\mathrm{SU}(2)^{n-2} \\backslash \\! \\backslash (\\mu_{\\mathrm{SU}(2)^{n-2}}^{-1}(0) \\cap \\mu_{\\mathbb{T}_d^-}^{-1}(0)) \\to V_n^{\\mathcal{T}}$$\nand a well-defined map\n$$\\Phi_n^{\\mathcal{T}}:P_n^{\\mathcal{T}}(\\mathrm{SU}(2)) \\to V_n^{\\mathcal{T}}.$$\n\\end{lemma}\n\n\\begin{proof}\nTo define $\\widehat{\\Phi}_n^{\\mathcal{T}}$ we need only show that for any pair of normalized $\\textbf{T}$ and $\\textbf{T}'$ in $\\mu_{\\mathrm{SU}(2)^{n-2}}^{-1}(0) \\cap \\mu_{\\mathbb{T}_d^-}^{-1}(0)$ and $\\mathbf{f} = (f_1,\\cdots, f_{n-2}) \\in G$ such that $\\mathbf{f} \\textbf{T}= \\textbf{T}'$\nwe have\n$$\\widetilde{\\Phi}^{\\mathcal{T}}_n(\\textbf{T}) = \\widetilde{\\Phi}_n^{\\mathcal{T}}(\\textbf{T}')$$\nin $V_n^{\\mathcal{T}}$.\n\nThe assumption $\\mathbf{f} \\textbf{T}= \\textbf{T}'$ is equivalent to\n\n$$(f_ig_{(\\tau_i, j)}, \\lambda_{(\\tau_i, j)}\\varpi_1) = (g'_{(\\tau_i, j)}, \\lambda'_{(\\tau_i, j)}\\varpi_1)$$\nfor each $j \\in \\{1, 2, 3\\}$ and $1 \\leq i \\leq n-2$. Hence $\\lambda_{(\\tau_i, j)} = \\lambda'_{(\\tau_i, j)}$, and in particular a (pair of) diagonals of $\\textbf{T}$ vanishes if and only if the same diagonals vanish for $\\textbf{T}'$\nthat is\n\n$$S(\\mathbf{T}) = S(\\mathbf{T}').$$\nHence the forests $\\mathcal{T}^{S(\\mathbf{T})}$ and $\\mathcal{T}^{S(\\mathbf{T}')}$ coincide.\nWe must show that the images of $\\widetilde{\\Phi}_n^{\\mathcal{T}}(\\mathbf{T})$ and $\\widetilde{\\Phi}_n^{\\mathcal{T}}(\\mathbf{T}')$ in $V_n^{\\mathcal{T}}$ coincide, which amounts to proving that the $\\mathbf{f}$ is constant on any connected component in the\nforest $\\mathcal{T}^{S(\\mathbf{T})}$ (recall that we think of $\\mathbf{f}$ as a function from\nthe trivalent vertices of $\\mathcal{T}$ to $\\mathrm{SU}(2)$).\nClearly it suffices to prove that if $(\\tau_i, j)$ and $(\\tau_h, k)$\nare equivalent then\n\n$$f_h = f_i.$$\nSuppose the imploded spin framings on the two edges are\nrespectively\n$(g_{(\\tau_i, j)}, \\lambda_{(\\tau_i, j)}\\varpi_1)$ and\n$(g_{(\\tau_h, k)}, \\lambda_{(\\tau_h, k)}\\varpi_1)$.\nWe have equations\n\n$$f_ig_{(\\tau_i,j)} = g_{(\\tau_i,j)}'$$\n$$f_hg_{(\\tau_h,k)} = g_{(\\tau_h,k)}'$$\nand because we have assumed both $\\textbf{T}$ and $\\textbf{T}'$ are normalized we have\n\n$$g_{(\\tau_h,k)} = g_{(\\tau_i,j)}\\varrho,$$\n$$g_{(\\tau_h,k)}' = g_{(\\tau_i,j)}'\\varrho.$$\nWe can rearrange these equations to get\n\n$$f_ig_{(\\tau_h,k)}\\varrho^{-1} = g_{(\\tau_h,k)}'\\varrho^{-1}.$$\nWe cancel $\\varrho^{-1}$ to obtain $f_ig_{(\\tau_h,k)} = g_{(\\tau_h,k)}'$.\nBut from above $f_hg_{(\\tau_h,k)} = g_{(\\tau_h,k)}'$\nThis proves the first statement of the lemma.\nIt is clear that $\\widehat{\\Phi}_n^{\\mathcal{T}}$ is constant on $\\mathbb{T}_d^-$ orbits, so it descends to give $\\Phi_n^{\\mathcal{T}}$.\n\\end{proof}\nThis shows that $\\Phi_n^{\\mathcal{T}}$ is well-defined and continuous\nbecause of the continuity of $\\widetilde{\\Phi}_n^{\\mathcal{T}}.$ \n\n\\begin{lemma}\n$\\Phi_n^{\\mathcal{T}} $ is proper (hence closed).\n\\end{lemma}\n\n\\begin{proof}\nSuppose $K$ is a compact subset of $P_n(\\mathrm{SU}(2))$. Then the lengths of\nthe columns of any matrix $A$ (so the edge-lengths of the corresponding $n$-gon) representing an element of $K$ are uniformly\nbounded by a constant $C$. We may reinterpret the above uniform bound as a bound\non all edge lengths of all triangles in the given triangulation of $P$ that are\nalso edges of $P$.\nBut we may assume all our matrices $A$ are\nin the zero level sets of the momentum maps for $\\mathrm{SU}(2)^{n-2}$ and\n$\\mathbb{T}_d^-$. It follows by Lemma \\ref{trivalentinduction} and the triangle inequalities that the lengths of all columns of any matrix\n$A^{\\mathcal{T}}$ representing any element in $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ in the inverse image\nof $K$ are bounded by $N(\\mathcal{T})C$ where $N(\\mathcal{T})$ is a positive integer\ndepending only on the tree $\\mathcal{T}$. Then $(\\Phi_n^{\\mathcal{T}})^{-1}(K)$\nis contained in the image of a subset of $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ homeomorphic to the product of $3(n-2)$ copies of the ball\nof radius $N(\\mathcal{T})C$ in $\\mathbb{C}^2$\n\\end{proof}\nWe now construct the map $\\Psi_n^{\\mathcal{T}} : V_n^{\\mathcal{T}} \\to P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ inverse to $\\Phi_n^{\\mathcal{T}}$.\nWe first define $\\widetilde{\\Psi}_n^{\\mathcal{T}}$ on $E_n(\\mathrm{SU}(2))$ then verify that the resulting map descends to $V_n^{\\mathcal{T}}$.\nWe will need to be able to add a diagonal frame on a triangle with two already framed edges.\n\n\n\\begin{lemma}\\label{framing}\nSuppose that two sides of an imploded spin-framed triangle $(g_1, \\lambda_1 \\varpi_1)$, $(g_2, \\lambda_2 \\varpi_1)$ are given. Then we can find $g_3 \\in \\mathrm{SU}(2)$ and $\\lambda_3 \\in \\mathbb{R}$ so that\n$(g_1, \\lambda_1 \\varpi_1)$, $(g_2, \\lambda_2 \\varpi_1)$ $(g_3, \\lambda_3 \\varpi_1)$ is an imploded spin-framed triangle. Precisely we may solve\n\\begin{equation}\\label{closingequation}\n\\lambda_1g_1\\varpi_1 + \\lambda_2g_2\\varpi_1 + \\lambda_3g_3\\varpi_1 = 0.\n\\end{equation}\nMoreover $\\lambda_3$ is uniquely determined by $\\lambda_1$ and $\\lambda_2$ and if\n$\\lambda_3 \\neq 0$ any two choices of $g_3$ for given $g_1$ and $g_2$ are related by right multiplication of $T_{\\mathrm{SU}(2)}$.\n\\end{lemma}\n\n\\begin{proof}\nIf both $\\lambda_1$ and $\\lambda_2$ are zero then we take $\\lambda_3$ to be zero and $g_3 = 1$. Suppose then that exactly one of $\\lambda_1$ and $\\lambda_2$ is nonzero. Without loss of generality, assume $\\lambda_1$ is nonzero. Put $\\lambda_3 = \\lambda_1$ and choose $g_3 = g_1\\varrho$. Hence we have\n$$\\lambda_1g_1\\varpi_1 = -\\lambda_3g_3\\varpi_1.$$\nNote that the map $S^3 \\to S^2$ given by $g \\to g\\varpi_1$ is the Hopf fibration and consequently we may solve the equation $g\\varpi_1 = -g_1\\varpi_1$ locally in a neighborhood of $g_1$, in particular we can solve this equation in a neighborhood of $(g_1, 1)$ of $\\mathrm{SU}(2) \\times \\mathrm{SU}(2)$.\nThe number $\\lambda_3 \\geq 0$ is uniquely determined but $g_3$ is defined only up to right multiplication by an element of $\\mathbb{T}$.\nNow assume that both $\\lambda_1$ and $\\lambda_2$ are nonzero. First assume that the sum $\\lambda_1g_1\\varpi_1 +\\lambda_2g_2\\varpi_1$ is zero, then we are forced to take $\\lambda_3 = 0$ and we may choose any framing data we wish, so we choose $g_3 = 1$. Assume then that $\\lambda_1g_1\\varpi_1 + \\lambda_2g_2\\varpi_1$ is nonzero. We choose $g_3 \\in \\mathrm{SU}(2)$ and $\\lambda_3$ so that $$\\lambda_1g_1\\varpi_1 + \\lambda_2g_2\\varpi_1 = -\\lambda_3g_3\\varpi_1.$$\nAgain, $\\lambda_3 \\neq 0$ is uniquely determined, and $g_3$ is defined up to right multiplication by an element of $\\mathbb{T}$. We define the required framed triangle $T$ by\n$$T = (g_1, \\lambda_1 \\varpi_1), (g_2, \\lambda_2 \\varpi_1), (g_3, \\lambda_3 \\varpi_1)$$\n\\end{proof}\nWe make use of Lemma \\ref{framing} to extend any framing of\nthe edges of $P$ to an element of $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$. The resulting object will be a normalized element\n$\\mathbf{T} \\in \\mu_{\\mathrm{SU}(2)^{n-2}}^{-1}(0) \\cap \\mu_{\\mathbb{T}_d^-}^{-1}(0) \\subset E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\n\n\\begin{lemma}\\label{extend}\nSuppose we are given an imploded spin framing $\\mathbf{E}$ of the edges of the model convex $n$-gon $P$ which is of momentum level zero for the action\nof $\\mathrm{SU}(2)$. Then we may extend the framing by choosing\nimploded spin-frames on the diagonals and an enumeration of the triangles\nof $P$ so that the resulting element $\\mathbf{T} \\in E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$\nis\n\\begin{enumerate}\n\\item normalized\n\\item of momentum level zero for $\\mathrm{SU}(2)^{n-2}$\n\\item of momentum level zero for $\\mathbb{T}_d ^-$.\n\\end{enumerate}\n\nMoreover any two such extensions are equivalent under the action of\nthe torus $\\mathbb{T}_d^-$.\n\n\\end{lemma}\n\n\\begin{proof}\nWe prove the existence of the framing by induction on $n$. We take as base\ncase $n=3$, here there is nothing to prove.\n\nLet $P$ be a model convex $n$-gon with an imploded spin framing on\nits edges. Define $P_{i_1} =P$.\nChoose a triangle $T$ in the triangulation of $P$ which shares two edges with $P$.\nHence two sides of $T$ are framed. Let $e$ be the remaining side (a diagonal\nof $P$). Suppose $e$ has length $\\lambda$. Define $T_{i_1} = T$.\nApply Lemma \\ref{framing} to frame the third side $e$ such that the resulting\nframing is of momentum zero level for $\\mathrm{SU}(2)$. Split $T$ off from $P$\nto obtain an $n-1$--gon $P_{i_2}$. Suppose the framing on $e$\nis $[g, \\lambda \\varpi_1]$. Give the edge $e'$ of $P_{i_2}$ which is not yet\nframed the framing $[g \\varrho, \\lambda \\varpi_1]$. We obtain the existence\npart of the lemma by induction.\n\nNow we prove uniqueness. Suppose that $\\mathbf{T}'$ is another extension of $\\mathbf{E}$, and\nthat $\\mathbf{T}'$ assigns the frame $[h, \\lambda \\varpi_1]$ to $e$ with $h \\neq g$. \nThen again by Lemma \\ref{framing} either $\\lambda = 0$ or there exists an element $t \\in T_{\\mathrm{SU}(2)}$ such that\n$h = gt$. In case $\\lambda \\neq 0$ the frame on $e'$ is necessarily \n$[gt \\varrho, \\lambda \\varpi_1] = [ g \\varrho t^{-1}, \\lambda \\varpi_1]$. Thus the new frames\non the pair of equivalent edges of $\\mathcal{T}^D$ are\n$[gt ,\\lambda \\varpi_1]$ followed by $[g\\varrho t^{-1},\\lambda \\varpi_1]$.\nNote that the framing of $P_{i_2}$ obtained by restriction of $\\mathbf{T}'$ \nsatisfies the three properties in the statement above, and also it is an extension of the framing of its boundary\ninduced by $\\mathbf{T}'$. Hence for the case $\\lambda \\neq 0$\nthe induction step of the uniqueness part of the lemma is completed.\nIn case $\\lambda = 0$ then we may represent both frames by\n$[I ,\\lambda \\varpi_1]$. But since $\\lambda = 0$ we have (by definition of implosion) for any $t \\in\nT_{\\mathrm{SU}(2)}$\n$$[I ,\\lambda \\varpi_1] = [t ,\\lambda \\varpi_1] = [t^{-1} ,\\lambda \\varpi_1].$$\nThis completes the induction step in the uniqueness part of the lemma.\n\\end{proof}\n\n\nBy the above lemma we obtain a well-defined map\n\n$$\\widetilde{\\Psi}_n^{\\mathcal{T}}: E_n(\\mathrm{SU}(2)) \\to (\\mu_{\\mathrm{SU}(2)^{n-2}}^{-1}(0) \\cap \\mu_{\\mathbf{T}_d^-}^{-1}(0))\/\\!\/ \\mathbb{T}_d^-.$$\nWe prove this map descends to $V_n^{\\mathcal{T}}$. We let\n$\\widehat{\\Psi}_n^{\\mathcal{T}}$ denote the map obtained from $\\widetilde{\\Psi}_n^{\\mathcal{T}}$\nby postcomposing it with projection to $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\nLet us note the following obvious refinement of Lemma \\ref{extend}.\nSuppose $P_1$ is a subpolygon of $P$ which meets $P$ along\ndiagonals of length zero in the realization of\n$P$ given by the framing of the edges. Then given any extension of\nthe framing of $P$ to all diagonals of $P_1$ we may find an extension of\nthe framing of $P$ to all diagonals of $P$ agreeing with the given one on $P_1$.\n\n\\begin{lemma}\nThe map $\\widehat{\\Psi}_n^{\\mathcal{T}}$ descends to\n$V_n^{\\mathcal{T}}$.\n\\end{lemma}\n\n\n\\begin{proof}\nWe first verify that $\\widehat{\\Psi}_n^{\\mathcal{T}}$ descends to $P_n(\\mathrm{SU}(2))$.\nIn fact we will show it is equivariant under $\\mathrm{SU}(2)$ where the action\nof $\\mathrm{SU}(2)$ on $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is by the diagonal in $\\mathrm{SU}(2)^{n-2}$.\nLet $f \\in \\mathrm{SU}(2)$. Let $\\mathbf{E}$ be a framing of the\nedges of $P$ and $\\mathbf{T}$ be the extension of Lemma \\ref{extend}.\nApply Lemma \\ref{extend} to extend $f \\mathbf{F}$ to an\nelement $\\mathbf{T}'$ of $E^{\\mathcal{T}}_n(\\mathrm{SU}(2))$\nso that this extension also satisfies the three properties in\nthe statement of Lemma \\ref{extend}. We wish to prove that the resulting element\nof $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is in the\n$\\mathbb{T}_d^-$-orbit of $f\\mathbf{T}$. But since any two extensions satisfying the three properties of Lemma \\ref{extend} are equivalent under $\\mathbb{T}_d^-$\nit suffices to prove that the image of {\\it any one} of the above extensions is \nequal to $f \\mathbf{T}$. But there is an\nextension that is obviously equivalent to $f \\mathbf{T}$ namely\n$f \\mathbf{T}$ itself. \n\nIt remains to check that $\\widehat{\\Psi}_n^{\\mathcal{T}}$ descends to $V_n^{\\mathcal{T}}$.\nLet $\\mathbf{E}$ and $\\mathbf{T}$ be as above. Let $\\mathbf{E}'$ be another framing\nof the edges of $P$ in the same $\\mathcal{T}$-congruence class as\n$\\mathbf{E}$. We have to prove that\n$$\\widehat{\\Psi}_n^{\\mathcal{T}}(\\mathbf{E}) = \\widehat{\\Psi}_n^{\\mathcal{T}}(\\mathbf{E}').$$\nBy transitivity of the $\\mathcal{T}$-congruence relation it suffices to prove this\nfor $\\mathbf{E}'$.\n\nLet $C_1$ be a component of\n$\\mathcal{T}^{S(\\mathbf{T})}$. We may assume that $C_1$ contains leaf edges. \nApply an element $f \\in \\mathrm{SU}(2)$ to the\nframes of the leaf edges of $\\mathcal{T}_1$ to obtain a new framing $\\mathbf{E}'$\nof the edges of $P$. We now prove\n$$\\widehat{\\Psi}_n^{\\mathcal{T}}(\\mathbf{E}) = \\widehat{\\Psi}_n^{\\mathcal{T}}(\\mathbf{E}')$$\nfor this choice of $\\mathbf{E}'$.\nApply Lemma \\ref{extend} to $\\mathbf{E}'$ to obtain an element $\\mathbf{T}'$\nof $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ such that the three properties\nin Lemma \\ref{extend} are satisfied. We must prove that $\\mathbf{T}$\nand $\\mathbf{T}'$ have the same image in $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\nAs before in the paragraph above it suffices to find one extension\n$\\mathbf{T}'$ of $\\mathbf{E}'$ which has the same image as $\\mathbf{T}$.\nWe now construct such an extension $\\mathbf{T}''$\nThe subtree $C_1$ corresponds to a subpolygon $P'$ of $P$ such that\nthe edges of $P'$ that correspond to diagonals of $P$ have zero length. Define\n$\\mathbf{T}''$ by defining the frame on an edge $e$ of $\\mathcal{T}^D$ that\nis in $C_1$ to be $f$ applied to the value of $\\mathbf{T}$ on $e$\nand if $e$ is not in $C_1$ then define the value of $\\mathbf{T}''$\non that edge to be the value of $\\mathbf{T}$ on that edge.\nWe leave to the reader that $\\mathbf{T}''$ satisfies the three properties\nof Lemma \\ref{extend}. Now we construct an element $\\tilde{f} \\in \\mathrm{SU}(2)^{n-2}$\nsuch that\n$$\\mathbf{T}'' = \\tilde{f}\\mathbf{T}.$$\nRecall that an element of $\\mathrm{SU}(2)^{n-2}$ may be regarded as a function\nfrom the trivalent nodes of $\\mathcal{T}$ to $\\mathrm{SU}(2)$.\nWe define $\\tilde{f}$ to be the function that is $I$ if the trivalent node\nis not in $C_1$, and is $f$ if the trinode is in $C_1$. Since an edge $e$ of $\\mathcal{T}$ is in $C_1$\nif and only if the trivalent node incident to $e$ we do indeed have\n$$\\mathbf{T}'' = \\tilde{f}\\mathbf{T}.$$\n\\end{proof}\n\nBecause $\\Psi_n^{\\mathcal{T}}$ is the inverse of a closed map we have\n\n\\begin{lemma}\n$\\Psi_n^{\\mathcal{T}}$ is continuous.\n\\end{lemma}\n\nFinally we have\n\n\\begin{lemma}\n $\\Phi^{\\mathcal{T}}_n$ and $\\Psi^{\\mathcal{T}}_n$ are inverses to one another\n\\end{lemma}\n\\begin{proof}\nIt is obvious that $\\Phi^{\\mathcal{T}}_n \\circ \\Psi^{\\mathcal{T}}_n = I_{P_n(\\mathrm{SU}(2))}$.\nThis is because neither of the two map changes the framing of the edges of $P$.\nNow let $\\mathbf{T}$ be a normalized element of $\\mathrm{SU}(2)^{n-2} \\times \\mathbb{T}_d^-$-momentum level zero in $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$. Let $\\mathbf{F}$ be the restriction\nof $\\mathbf{T}$ to the edges of $P$. Thus $\\mathbf{T}$ is an extension\nof $\\mathbf{F}$ satisfying the three properties of Lemma \\ref{extend}. But\n$\\Psi^{\\mathcal{T}}_n \\circ \\Phi^{\\mathcal{T}}_n(\\mathbf{T})$ also has restriction\nto $P$ given by $\\mathbf{F}$ and also satisfies the three properties of\nLemma \\ref{extend}. Hence $\\mathbf{T}$ and $\\Psi^{\\mathcal{T}}_n \\circ \\Phi^{\\mathcal{T}}_n(\\mathbf{T})$ are in the same $\\mathbb{T}_d^-$-orbit and\nhence their images in $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ coincide.\n\\end{proof}\nIt is clear from the definition of $\\Phi^{\\mathcal{T}}_n$ that this map intertwines the actions of\n$T_{\\mathrm{SU}(2)^n}$ and $\\mathbb{T}_e$.\nThis proves Theorem \\ref{noncompact}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Pulling back Hamiltonian functions from $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ to $V_n^{\\mathcal{T}}$}\n\nIn the previous subsection we constructed a homeomorphism\n$\\Psi_n^{\\mathcal{T}}:V_n^{\\mathcal{T}} \\to P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ with inverse\n$\\Phi_n^{\\mathcal{T}}$. The torus $\\mathbb{T}$ acts on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$\nsuch that the Hamiltonian for the circle factor corresponding to the edge $(\\tau_i,j)$ of $\\mathcal{T}^D$ is given by $f_{\\tau_i,j}(A) = (1\/2)|z_j(\\tau_i)|^2 + |w_j(\\tau_i)|^2$. In spin-framed coordinates this is $f_{\\tau_i,j}(A) = \\lambda_{(\\tau_i, j)}$.\nWe will compute the pullback of these functions to $V_n^{\\mathcal{T}}$ via $\\Psi_n^{\\mathcal{T}}$.\n\n\\begin{definition}\nLet $v_{(\\tau, i)}^{\\mathcal{T}}: E_n^{\\mathcal{T}}(\\mathrm{SU}(2)) \\to \\mathfrak{su}(2)^*$\nby the composition of the projection on the $(\\tau, i)$-th factor\nof $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ with the function $h$ from subsection \\ref{frommatricestopolygons}.\nLet $v_e: E_n(\\mathrm{SU}(2)) \\to \\mathfrak{su}(2)^*$ be the composition of the projection\non the $e$-th factor of $E_n(\\mathrm{SU}(2))$ with the function $h$ from subsection \\ref{frommatricestopolygons}.\n\\end{definition}\n\nFor a distinguished edge $(\\tau, i)$ of $\\mathcal{T}^D$ let $e(\\tau, i)$\nbe associated edge of the model convex planar $n$--gon. Then\nwe have the following identity\n\n$$(\\tilde{\\Phi}_n^{\\mathcal{T}})^*(v_{e(\\tau, i)}) = v_{(\\tau, i)}^{\\mathcal{T}}.$$\n\n\\subsubsection{Pulling back distinguished edge Hamiltonian functions}\n\n\nIn what follows\nwe will need to compute the pull-back of $f_{(\\tau_i,j)}$ to $V_n^{\\mathcal{T}}$\nfor those distinguished edges $(\\tau_i,j)$. Let $f$ be the map from subsection\n\\ref{frommatricestopolygons}, then we have\n\n$$f_{\\tau_i, j}(\\mathbf{E}) = 2 \\| f \\circ v_{(\\tau_i, j)}^{\\mathcal{T}}(\\mathbf{T}) \\|$$\nfor any normalized $\\mathbf{T} \\in \\mu_{\\mathrm{SU}(2)^{n-2} \\times \\mathbb{T}_d^-}^{-1}(0)$ which maps to $\\mathbf{E}$.\n\n\\begin{proposition}\\label{pullback}\nFor $A \\in V_n^{\\mathcal{T}}$ we have\n$$(\\Psi^{\\mathcal{T}}_n)^* f_{\\tau_i,j}(A) = 2 \\|f \\circ v_{e(\\tau_i, j)}(\\tilde{A})\\|.$$\nwhere $\\tilde{A} \\in E_n(\\mathrm{SU}(2))$ maps to $A$.\n\\end{proposition}\n\n\\begin{proof}\nThe proposition follows from\n\n$$(\\Psi^{\\mathcal{T}}_n)^*(f_{\\tau_i, j})(A) = f_{\\tau_i, j}(\\Psi^{\\mathcal{T}}_n(A)) = \n2 \\| f \\circ (\\Psi^{\\mathcal{T}}_n)^*(v_{(\\tau_i, j)}^{\\mathcal{T}})(\\tilde{A}) \\| = \n2 \\| f \\circ v_{e(\\tau_i, j)}(\\tilde{A}) \\|.$$\n\\end{proof}\n\n\\begin{remark}\nNote that $\\| f \\circ v_{e(\\tau_i, j)}(\\tilde{A}) \\|$ is the length\nof the $e(\\tau_i, j)$ edge of $F_n(A)$.\n\\end{remark}\n\n\n\\subsubsection{Pullback of the internal edge Hamiltonian functions}\n\nWe also need to compute the $\\Psi_n^{\\mathcal{T}}$ pullbacks\nof the Hamiltonian functions $f_{(\\tau_j, i)}$ for $(\\tau_j, i)$\nan internal edge of $\\mathcal{T}$, this will be important when we\nwish to indentify the Hamiltonian flows of these pullbacks\nwith the bending flows. Note that $(\\tau_j, i)$ corresponds\nto a diagonal $d(\\tau_j, i)$ in a model $n$-gon $P$.\n\n\\begin{lemma}\nLet $\\mathcal{T}' \\subset \\mathcal{T}$ be a connected subtree\nsuch that every leaf of $\\mathcal{T}'$ except one, say $(\\tau, k)$, is a\nleaf of $\\mathcal{T}$. Then for any element $\\mathbf{E} \\in P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$\nwe can compute\n\n$$f_{\\tau, k}(\\mathbf{E}) = \\| \\sum_{(\\tau_i, j) \\in dist(\\mathcal{T}')} v_{(\\tau_i, j)}^{\\mathcal{T}}(\\mathbf{T}) \\|$$\n\nFor $\\mathbf{T}$ a normalized element of $\\mu_{\\mathrm{SU}(2)^{n-2}\\times\\mathbb{T}_d^-}^{-1}(0)$ which\nmaps to $\\mathbf{E}$.\n\\end{lemma}\n\n\\begin{proof}\nFirst note that we compute $f_{\\tau, k}$ by\ntaking half the length of the $(\\tau, k)$ edge for\nany normalized element $\\mathbf{T} \\in \\mu_{\\mathrm{SU}(2)^{n-2}\\times\\mathbb{T}_d^-}^{-1}(0)$\nwhich maps to $\\mathbf{E}$, this is equal to $\\lambda_{(\\tau, k)}$.\nBy the normalized condition and the closing condition imposed\non $\\mathbf{T}$, the leaf edge of $\\mathcal{T}'$ form a closed polygon,\nwhich implies that $\\lambda_{(\\tau, k)} = \\| \\sum_{(\\tau_i, j) \\in dist(\\mathcal{T}')} \\lambda_{(\\tau_i, j)}(\\mathbf{T}) Ad^*_{g_{(\\tau_i,j)}(\\mathbf{T})}(\\varpi_1) \\|$.\n\\end{proof}\n\nThe previous proposition allows us to conclude the following\ntheorem.\n\n\\begin{theorem}\\label{diagonalpullback}\nLet $d(\\tau_j, i)(A)$ be the associated diagonal in\n$F_n(A)$.\n\n$$(\\Psi_n^{\\mathcal{T}})^*(f_{(\\tau_j, i)})(A) = 2\\| d(\\tau_j, i)(A)\\|$$\n\n\\end{theorem}\n\n\\begin{proof}\nWe have the following equation\n\n$$(\\Psi_n^{\\mathcal{T}})^*(f_{(\\tau_j, i)})(A) = \\| \\sum_{(\\tau_i, j) \\in dist(\\mathcal{T}')} v_{(\\tau_i, j)}^{\\mathcal{T}}(\\tilde{\\Psi}_n^{\\mathcal{T}}(\\tilde{A})) \\|.$$\nThe right hand side of this equation is equal to\n\n$$2\\| \\sum_{(\\tau_i, j) \\in dist(\\mathcal{T}')} f \\circ v_{e(\\tau_i, j)}(\\tilde{A}) \\|.$$\nThis last expression is in turn equal to $2\\|d(\\tau_i, j)(A)\\|$.\n\\end{proof}\n\nThis, along with Proposition \\ref{pullback}, proves Theorem \\ref{stratification}.\n\n\n\n\\subsection{The homeomorphism of projective varieties}\\label{thirdtorusquotient}\nIn this subsection we will descend the homeomorphisms of affine varieties $\\Phi_n^{\\mathcal{T}}$\nand $\\Psi_n^{\\mathcal{T}}$ to their projective quotients completing the proof\nof Theorem \\ref{firstauxiliarytheorem} and hence the proof of\nTheorem \\ref{firstmaintheorem}.\nRecall that we defined elements $\\mathbf{t}(\\lambda) \\in T_{\\mathrm{SU}(2)^n}$\nand $\\mathbf{t}_e(\\lambda) \\in \\mathbb{T}_e$ for $\\lambda \\in S^1$. We note\n$$\\rho(\\mathbf{t}(\\lambda)) = \\mathbf{t}_e(\\lambda).$$\nSince $\\Psi^{\\mathcal{T}}_n$\nis equivariant we have\n\\begin{equation}\\label{degreeequivariance}\n\\Psi^{\\mathcal{T}}_n \\circ \\mathbf{t}(\\lambda) = \\mathbf{t}_e(\\lambda) \\circ\n\\Psi^{\\mathcal{T}}_n.\n\\end{equation}\n\n\\begin{proposition}\nThe map $\\Psi^{\\mathcal{T}}_n$ induces a homeomorphism from $W_n^{\\mathcal{T}}$\nto the symplectic quotient of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ by the grading\ncircle action.\n\\end{proposition}\n\n\\begin{proof}\nIt suffices to prove that the pull-back by $\\Psi_n^{\\mathcal{T}}$ of the grading circle action on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$\nis the grading circle action on $V_n^{\\mathcal{T}}$.\nThis follows\nby replacing $\\lambda$\nby $(\\sqrt{\\lambda})^{-1}$ in equation (\\ref{degreeequivariance}).\n\\end{proof}\n\n\n\\subsection{The symplectic and GIT quotients coincide}\\label{quotientscoincide}\nWe complete the proof of Theorems \\ref{firstauxiliarytheorem}\nand \\ref{firstmaintheorem} by proving\n\n\\begin{proposition}\nThe symplectic quotient of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2)$ by the grading circle\naction coincides with the GIT quotient of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ by the\ngrading $\\mathbb{C}^{\\ast}$-action linearized by acting by the identity on the\nfiber of the trivial complex line bundle over $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$.\n\\end{proposition}\n\nWe are forced to give an indirect argument because $P_n^{\\mathcal{T}}(\\mathrm{SU}(2)$\nis not smooth and there do not seem to be theorems asserting\nthe isomorphism of symplectic and GIT quotients of nonsmooth spaces.\n\nWe first note that the above GIT quotient is canonically isomorphic to the GIT quotient of $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ by the product group\n$\\mathbb{C}^{\\ast} \\times \\underline{\\mathbb{T}}_d^-$ where the first factor acts by\nthe grading action linearized by the identity action on the fiber $\\mathbb{C}$\nof the trivial complex line bundle over $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$.\nThis follows because we can take the ring of invariants for a product\ngroup acting on a ring $R$ by first taking the invariants of one factor and then taking the invariants of the resulting ring by the second factor.\nThe corresponding quotient by stages for symplectic quotients is\nalso true but slightly harder. It is proved in Theorem 4.1 of \\cite{SjamaarLerman}.\n\nThus it remains to prove that the symplectic quotient of $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ by the product of the grading $S^1$-action\nand $\\mathbb{T}_d^-$ is isomorphic to\nthe GIT quotient of $(\\bigwedge^2(\\mathbb{C}^3))^{n-2}$ by the product of\nthe grading $\\mathbb{C}^{\\ast}$ action and the complexified torus \n$\\underline{\\mathbb{T}}_d^-$ where the first factor\nacts by the grading action and the product is linearized by acting by\nthe identity map applied to the projection on the first factor.\nThis follows immediately from Theorem \\ref{twistandshift}\nonce we establish that the momentum map for the product\nof the grading $S^1$-action\nand $\\mathbb{T}_d^-$ is proper.\nWe now prove that the momentum map for the product of the grading circle\naction and $\\mathbb{T}_d^-$ is proper. In fact we prove a different result\nthat turns out to be equivalent to the one we need here because we will need this different result below.\n\n\\begin{proposition}\\label{propernesstheorem}\nThe momentum mapping $\\mu : (\\bigwedge^2 \\mathbb{C}^3)^{n-2} \\to \\mathbb{R}^{2n-3}$\nfor the action of $\\mathbb{T}_e \\times \\mathbb{T}_d^-$ is proper.\n\\end{proposition}\n\n\\begin{proof}\nEarlier we identified $\\bigwedge^2 \\mathbb{C}^3$ with the space of\nframed triangles $P_3(\\mathrm{SU}(2))$. Let\n$(\\ldots, [g_{(\\tau_i, j)}, \\lambda_{(\\tau_i, j)}\\varpi_1], \\ldots)$\nbe an element of $P_3(\\mathrm{SU}(2))^{n-2}$.\nUnder this identification we have that\n$\\mu^{-1}_{\\mathbb{T}_e}(\\ldots, [g_{(\\tau_i, j)}, \\lambda_{(\\tau_i, j)}\\varpi_1], \\ldots)$\nis the vector of elements $\\lambda_{(\\tau_i, j)}$ where $(\\tau_i, j)$\nis a distinguished edge of $\\mathcal{T}^D$. Similarly\n$\\mu_{\\mathbb{T}_d^-}(\\ldots, [g_{(\\tau_i, j)}, \\lambda_{(\\tau_i, j)}\\varpi_1], \\ldots)$\nis the vector of elements $\\lambda_{(\\tau_i, j)} - \\lambda_{(\\tau_k, \\ell)}$\nwhere $(\\tau_i, j)$ and $(\\tau_k, \\ell)$ represent the same internal edge of $\\mathcal{T}$.\nIn order to show that $\\mu_{\\mathbb{T}_e \\times \\mathbb{T}_d^-}$ is a proper map, we must show\nthat if all $\\lambda_{(\\tau_i, j)}$ are bounded for $(\\tau_i, j)$ distinguished,\nand all differences $\\lambda_{(\\tau_i, j)} - \\lambda_{(\\tau_k, \\ell)}$ are bounded\nfor all $(\\tau_i, j)$ and $(\\tau_k, \\ell)$ which represent the same internal edge of $\\mathcal{T}$,\nthen all $\\lambda_{(\\tau_i, j)}$ are bounded.\n\nLet $([g_1, \\lambda_1\\varpi_1], [g_2, \\lambda_2\\varpi_1], [g_3, \\lambda_3\\varpi_1])$\nbe an element of $P_3(\\mathrm{SU}(2))$. Since\n$$\\sum_{i = 1}^3 \\lambda_i Ad_{g_i}^*(\\varpi_1) = 0,$$\nthe $\\lambda_i$ are side-lengths of a closed triangle, hence if two of the three are bounded, so is the third.\nFurthermore, if $(\\tau_i, j)$ and $(\\tau_k, \\ell)$ represent the same internal edge of $\\mathcal{T}$,\nand $\\lambda_{(\\tau_i, j)}$ and the difference $\\lambda_{(\\tau_i, j)} - \\lambda_{(\\tau_k, \\ell)}$ are bounded,\nthen $\\lambda_{(\\tau_k, \\ell)}$ is bounded. The proposition now follows from\ntrivalent induction, see lemma \\ref{trivalentinduction}.\n\\end{proof}\n\n\\begin{corollary}\nThe momentum mapping $\\mu : (\\bigwedge^2 \\mathbb{C}^3)^{n-2} \\to \\mathbb{R}^{2n-3}$\nfor the action of $S^1\\times \\mathbb{T}_d^-$ is proper.\nHere $S^1$ acts by the grading circle action.\n\\end{corollary}\n\\begin{proof}\nThe Hamiltonian for the grading circle action is the half the sum of the\nHamiltonians for the circle $\\mathbb{T}_e$ factors of $\\mathbb{T}_e$.\nBut note that each of the summands is bounded if and only if the sum\nis (since they have the same sign).\n\\end{proof}\n\nThere is one more technical point. We claim that the GIT\nquotient coincides with the symplectic quotient at momentum level\n$(1,0)$. We will prove first that the level for the\n$\\mathbb{T}_d^{-1}$ is zero. But this follows\nbecause the torus $\\mathbb{T}_d^{-1}$ acts linearly on\n$\\bigwedge^{n-2}(\\mathbb{C}^3)$ and trivially on the fiber whereas the grading\ncircle action is twisted by the identity action on the fiber.\n\n\\section{The space $V_{\\mathbf{r}}^{\\mathcal{T}}$ is homeomorphic to the\ntoric variety $(M_{\\mathbf{r}})^{\\mathcal{T}}_0$}\nIn this section we will prove Theorem \\ref{secondmaintheorem}\nby proving Theorem \\ref{secondauxiliarytheorem}.\n\n\\subsection{The toric varieties $(M_{\\mathbf{r}})^{\\mathcal{T}}_0$ and $P^{\\mathcal{T}}_{\\mathbf{r}}(\\mathrm{SU}(2))$ are isomorphic}\nIn this subsection we prove the first statement of Theorem \\ref{secondauxiliarytheorem}.\nBy definition the toric variety $P^{\\mathcal{T}}_{\\mathbf{r}}(\\mathrm{SU}(2))$ is obtained\nas\nthe GIT quotient of $P^{\\mathcal{T}}_{n}(\\mathrm{SU}(2))$ by $\\underline{\\mathbb{T}}_e$\nusing the linearization given by the character\n$$\\chi_{\\mathbf{r}}(\\mathbf{t}(\\mathbf{\\lambda})) = \\lambda_1^{r_1} \\cdots \\lambda_n^{r_n}.$$\nHere $\\mathbf{t}(\\mathbf{\\lambda})$ denotes the element of $\\underline{\\mathbb{T}}_e$\ncorresponding to $\\mathbf{\\lambda} = (\\lambda_1,\\cdots,\\lambda_n) \\in (\\mathbb{C}^{\\ast})^n$.\nWe recall that the graded ring\nof $(M_{\\mathbf{r}})^{\\mathcal{T}}_0$ is the semigroup ring $\\mathbb{C}[\\mathcal{S}^{\\mathcal{T}}_{\\mathbf{r}}]$ where\n$\\mathcal{S}^{\\mathcal{T}}_{\\mathbf{r}}$ is the graded subsemigroup of $\\mathcal{S}^{\\mathcal{T}}_n$ defined by taking\ngraphs with valence $k\\mathbf{r}$ for positive integers $k$.\nDefine the subsemigroup $\\mathcal{P}^{\\mathcal{T}}_{\\mathbf{r}}$ of $\\mathcal{P}^{\\mathcal{T}}_n$\nto be the inverse image of $\\mathcal{W}_{\\mathbf{r}}^{\\mathcal{T}}$ under the isomorphism\nfrom $\\mathcal{P}^{\\mathcal{T}}_n$ to $\\mathcal{W}^{\\mathcal{T}}_n$ under the\nisomorphism of Proposition \\ref{twoisomorphicsemigroups}.\n\n\\begin{lemma}\n\\\nThe $\\underline{\\mathbb{T}}_d^-$-invariant monomial\n$$f(Z)=\n\\prod_{i=1}^{n-2}Z_{12}(\\tau_i)^{x_{12}(\\tau_i)}Z_{13}(\\tau_i)^{x_{13}(\\tau_i)}\nZ_{23}(\\tau_i)^{x_{23}(\\tau_i)}$$\nis $\\underline{\\mathbb{T}}_e$--invariant for the twist\n$\\chi_{\\mathbf{r}}^p$ if and only if the\nexponents $\\{x_{jk}(\\tau_i)\\}$ satisfy the system of equations\n$$x_{k,m}(\\tau) + x_{k,\\ell}(\\tau_i) = p r_{(\\tau_i,k)}, \\quad \\text{for all leaf edges} \\ (\\tau_i, k) \\ \\text{where} \\ k,\\ell,m = 1,2,3.$$\nBy Proposition \\ref{twoisomorphicsemigroups} we may rewrite the\nabove equation as a condition on the leaf weights\n$$w_{(\\tau_i,k)}^{\\mathcal{T}} = p r_{(\\tau_i,k)}, \\ (\\tau_i, k) \\in \\mathcal{L}.$$\n\\end{lemma}\n\n\n\\begin{proof}\nThe lemma follows from the formula for $\\mathbf{t}(\\mathbf{\\lambda})$\nacting on $f$, namely\n$$\\mathbf{t}(\\mathbf{\\lambda}) \\circ f(Z) = \\prod_{(\\tau_i, k) \\in \\mathcal{L}}\n\\lambda_{(\\tau_i,k)}^{[p r_{(\\tau_i,k)} - (x_{k,m}(\\tau_i) + x_{k,\\ell}(\\tau_i))]} f(Z).$$\nHere $\\lambda_{(\\tau_i,k)}$ is the coordinate of $\\underline{\\mathbb{T}}_e$ corresponding to the\nleaf edge $(\\tau_i,k)$ and $k,\\ell,m = 1,2,3$.\n\\end{proof}\n\n\\begin{remark}\nThe set of $\\underline{\\mathbb{T}}_e$-invariant monomials in $Z_{ij}(\\tau)$ {\\it for the twist by $\\chi_{\\mathbf{r}}^p$}\nis the subset of invariants {\\it of degree $p$} (the $p$-th graded piece of the\nassociate semigroup of lattice points).\n\\end{remark}\n\nAs an immediate consequence we have\n\\begin{corollary}\\label{rthsemigroup}\nThe toric variety $P_{\\mathbf{r}}^{\\mathcal{T}}(\\mathrm{SU}(2))$ is the projective toric variety\nassociated to the graded semigroup $\\mathcal{P}^{\\mathcal{T}}_{\\mathbf{r}}$.\n\\end{corollary}\n\n\n\nNow by Proposition \\ref{gradedsubsemigroupiso} we have\nan isomorphism of graded semigroups \n$\\Omega_{\\mathbf{r}}: \\mathcal{S}_{\\mathbf{r}}^{\\mathcal{T}} \\to \\mathcal{W}_{\\mathbf{r}}^{\\mathcal{T}}$. \nSince by definition we have an isomorphism\nof graded semigroups $\\mathcal{P}^{\\mathcal{T}}_{\\mathbf{r}} \\cong \\mathcal{W}_{\\mathbf{r}}^{\\mathcal{T}}$\nwe obtain the required isomorphism of graded semigroups\n$\\mathcal{S}^{\\mathcal{T}}_{\\mathbf{r}} \\cong \\mathcal{P}^{\\mathcal{T}}_{\\mathbf{r}}$\nand we have proved the desired isomorphism of projective toric varieties.\n\n\n\n\\subsection{$V_{\\mathbf{r}}^{\\mathcal{T}}$ is homeomorphic to $P^{\\mathcal{T}}_{\\mathbf{r}}(\\mathrm{SU}(2))$}\nIn this section we will prove the second statement of Theorem \\ref{secondauxiliarytheorem}.\n\n\n\\begin{proposition}\n$V_{\\mathbf{r}}^{\\mathcal{T}}$ is homeomorphic to the symplectic quotient\nof $P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ by $\\mathbb{T}_e$ at level $\\mathbf{r}$.\n\\end{proposition}\n\n\n\n\\begin{proof}\nRecall that in Proposition \\ref{pullback} we proved the following formula.\nLet $\\mathbf{E} \\in V_n^{\\mathcal{T}}$ then\n\n$$(\\Psi^{\\mathcal{T}}_n)^* f_{\\tau_i,j}(A) = 2\\|f \\circ v_{e(\\tau_i, j)}(\\tilde{A})\\| = \\| v_{e(\\tau_i, j)}(\\tilde{A}) \\|.$$\n\nWhere $e(\\tau_i, j)$ is the leaf edge of $\\mathcal{T}$\ncorresponding to the leaf edge $(\\tau_i,j)$ of the decomposed tree $\\mathcal{T}^D$.\nThus $\\Psi_n^{\\mathcal{T}}$ induces a homeomorphism between the $\\mathcal{T}$-congruence classes\nof imploded spin-framed $n$-gons with side-lengths $\\mathbf{r}$ and the $\\mathbf{r}$-th level set for\nthe momentum map of $\\mathbb{T}_e$. But also $\\Psi_n^{\\mathcal{T}}$ is $\\rho$-equivariant,\nhence the above bijection descends to a homeomorphism from the $T_{\\mathrm{SU}(2)^n}$-quotient of the imploded spin-framed $n$-gons\nto the $\\mathbb{T}_e$ symplectic quotient at level $\\mathbf{r}$.\n\\end{proof}\n\n\nIt remains to prove that the symplectic quotient of $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$\nat level $\\mathbf{r}$ coincides with the GIT quotient for the action on the\ntrivial bundle using the twist by $\\chi_{\\mathbf{r}}$.\n\n\\subsubsection{The symplectic quotient coincides with the GIT quotient}\n\\label{fourthtorusquotient}\nLet $L$ be the trivial complex line bundle over $\\bigwedge^{n-2}(\\mathbb{C}^3)$.\nThen $L$ descends to the trivial complex line bundle $\\overline{L}$\nover $P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$. The torus\n$\\underline{\\mathbb{T}}_e$ acts on $\\overline{L}$ by twisting by\nthe character $\\chi_{\\mathbf{r}}$.\nWe need to prove that the GIT quotient of\n$P^{\\mathcal{T}}_n(\\mathrm{SU}(2))$ by $\\underline{\\mathbb{T}}_e$ with linearization the above\naction on $\\overline{L}$ is homeomorphic to the symplectic quotient by the maximal compact subgroup of $\\underline{\\mathbb{T}}_e$ at level $\\mathbf{r}$.\nThe argument is almost the same as that of subsection \\ref{quotientscoincide}.\nIn particular we use reduction in stages to reduce to the\ncorresponding problem for the quotients of $\\bigwedge^{n-2}(\\mathbb{C}^3)$ by\nthe product torus $\\underline{\\mathbb{T}}_e \\times \\underline{\\mathbb{T}}_d^{-}$.\nHere we twist the action by the character that is $\\chi_{\\mathbf{r}}$ on $\\underline{\\mathbb{T}}_e$ and trivial on\n$\\underline{\\mathbb{T}}_d^{-}$.\nSince the momentum map for the action by $\\underline{\\mathbb{T}}_e \\times \\underline{\\mathbb{T}}_d^{-}$ \nis proper by Proposition \\ref{propernesstheorem}\nthe equality of quotients follows by Theorem \\ref{twistandshift}.\nThe momentum level for the product group is then $(\\mathbf{r},0)$.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The residual action of $\\mathbb{T} \/ (\\mathbb{T}_e \\times \\mathbb{T}_d^-)$ and bending flows}\\label{bendingflowsection}\n\nNow we will relate the action of $\\mathbb{T} \/ (\\mathbb{T}_e \\times \\mathbb{T}_d^-)$ on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ and $P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$ to bending flows.\n\n\n\\subsection{A complement to $\\mathbb{T}_d^-$ in $\\mathbb{T}_d$}\nIn this subsection we will define a complement $\\mathbb{T}_d^+$\nto $\\mathbb{T}_d^-$ in $\\mathbb{T}_d$. It will be more convenient\nto work with this complement rather than the quotient\n$\\mathbb{T}_d\/ \\mathbb{T}_d^-$.\n\nWe recall that an element of $\\mathbb{T}$ corresponds to a function\n$f$ from the edges of $\\mathcal{T}^D$ to the circle. The subgroup $\\mathbb{T}_d$\ncorresponds to the functions with value the identity on the distinguished\nedges of $\\mathcal{T}^D$. The nondistinguished edges of $\\mathcal{T}^d$ occur in\nequivalent pairs. The subtorus $\\mathbb{T}_d^-$ of $\\mathbb{T}_d$\nconsists of those $f$ whose values on equivalent pairs of\nnondistinguished edges are inverse to each other.\n\nIn order to construct the complement $\\mathbb{T}_d^+$ we need to choose\nan edge from each distinguished pair. To do this in a systematic way\nwe use the ordering of the trinodes of $\\mathcal{T}^D$ induced by\nthe trivalent induction construction, see Lemma \\ref{trivalentinduction}. Suppose the edges\n$(\\tau, i)$ and $(\\tau', j)$ are equivalent. The two edges occur\nin different trinodes. We relabel the edges by ${\\epsilon}^+$ and\n${\\epsilon}^-$ by labeling the edge that comes in the first trinode\nby (the superscript) plus and the edge that comes in a later trinode\nby minus. Thus every nondistinguished edge is either a plus edge\nor a minus edge. We now define the subtorus $\\mathbb{T}_d^+$ as the subtorus\nof $\\mathbb{T}_d$ consisting of those functions that take value the identity on \nall nondistinguished minus edges. It is clear that $\\mathbb{T}_d^+$ is the required complement.\nIt is also clear that $\\mathbb{T}_d^+$ is a complement to\n$\\mathbb{T}_e \\times \\mathbb{T}_d^-$ in $\\mathbb{T}$. We point out\nhere that the choice of where to place the identity\nin the definition of this complement is irrelevant with respect\nto the Hamiltonian functions - see below. \n\n\n\\subsection{The action of $\\mathbb{T}_d^+$ coincides with bending} \\hfill\n\nWe now prove the following theorem.\n\\begin{theorem}\\label{noncompact2}\n\\hfill\n\\begin{enumerate}\n\\item The homeomorphism $\\Psi^{\\mathcal{T}}_n$ intertwines the bending flows\non $V_n^{\\mathcal{T}}$ with the action of $\\mathbb{T}_d^+$ on $P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$. \\item The homeomorphism $\\Psi^{\\mathcal{T}}_{\\mathbf{r}}$ intertwines the bending flows\non $V_{\\mathbf{r}}^{\\mathcal{T}}(\\mathrm{SU}(2))$ with the action of $\\mathbb{T}_d^+$ on $P_{\\mathbf{r}}^{\\mathcal{T}}(\\mathrm{SU}(2))$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n\nLet $\\mathbf{T} \\in P_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ and $\\mathbf{t} \\in \\mathbb{T}_d^+$.\nLet $d$ be a diagonal of the triangulated $n$-gon $P$ and suppose\nthe triangles $T_i$ and $T_j$ share the diagonal $d$. Suppose\n$d$ divides $P$ into two polygons $P'$ and $P''$ with\n$T \\in P'$ and $T' \\in P''$. Let $\\tau$ and $\\tau'$\nbe the trinodes associated to $T$ and $T'$ respectively,\nand let $[g_{(\\tau, i)}, \\lambda_{(\\tau, i)}\\varpi_1]$ and\n$[g_{(\\tau', j)}, \\lambda_{(\\tau', j)}\\varpi_1]$ be spin-framed\nrepresentatives of the edges of $\\mathbf{T}$ corresponding to\nthe diagonal $d$. Suppose $\\epsilon(d)^- = (\\tau', j)$.\n\nSince the representatives are normalized we have\n$$g_{(\\tau, i)} = g_{(\\tau', j)} \\varrho.$$\nLet $t$ be the $\\epsilon(d)^+$-th component of $\\mathbf{t}$ (the\n$\\epsilon(d)^-$-th component is $1$). Under the action\nof $\\mathbf{t}$ the edge $[g_{(\\tau, i)}, \\lambda_{(\\tau, i)}\\varpi_1]$\nbecomes $[g_{(\\tau, i)}t,\\lambda_{(\\tau, i)}\\varpi_1]$ and all other\ncomponents are unchanged. Thus the resulting\nelement of $E_n^{\\mathcal{T}}(\\mathrm{SU}(2))$ is no longer normalized.\n\nTo normalize this element we multiply all the frames on the edges and diagonals\nof $P''$ by $g_{(\\tau', j)} t g_{(\\tau', j)}^{-1}$. Note that the image of $h=g_{(\\tau', j)} t g_{(\\tau', j)}^{-1} \\in \\mathrm{SU}(2)$.\nis a rotation about the oriented line in $\\mathbb{R}^3$ corresponding to\n$Ad^{\\ast}h(\\varpi_1) \\in \\mathfrak{su}(2)^{\\ast} \\cong \\mathbb{R}^3$ which is the diagonal corresponding to $d$ of\nthe Euclidean $n$-gon $\\mathbf{e}$ underlying the imploded\nframed $n$-gon $\\mathbf{T}$. Thus the Euclidean $n$-gon underlying\nthe imploded framed $n$-gon $\\mathbf{T}$ is bent along the\ndiagonal $d$. Furthermore the frames of all the edges\nand diagonals of $P''$ are transformed by applying the element of the one-parameter group \n$g_{(\\tau', j)} t g_{(\\tau', j)}^{-1}$ in $\\mathrm{SU}(2)$ which covers\nrotation along the diagonal. Applying $\\Phi_n^{\\mathcal{T}}$ we forget the\nframes along the diagonals but the frames on the edges transform\nin the same way as before. This amounts to bending the framed \n$n$-gon $\\Phi_n^{\\mathcal{T}}(\\mathbf{T})$ along the diagonal $d$.\n\nThe first statement implies the second statement because the\nbending flows on $V_{\\mathbf{r}}^{\\mathcal{T}}$ are descended from\nthose on $V_{n}^{\\mathcal{T}}$, and the action of\n$\\mathbb{T}_d^+$ on $P_{\\mathbf{r}}^{\\mathcal{T}}(\\mathrm{SU}(2))$ is descended from the action\non $P_{n}^{\\mathcal{T}}(\\mathrm{SU}(2))$\n\\end{proof}\n\n\\subsection{The Hamiltonians for the residual action}\nIt remains to prove that the action of $\\mathbb{T}_d^+$ is Hamiltonian\nwith the given Hamiltonians.\n\n\nWe claim that $\\mathbb{T}_d^+$ preserves the orbit-type stratification.\nSince $\\mathbb{T} = (\\mathbb{T}_e \\times \\mathbb{T}_d^-) \\times \\mathbb{T}_d^+$ is abelian,\nthe isotropy subgroup $(\\mathbb{T}_e \\times \\mathbb{T}_d^-)_x \\subset \\mathbb{T}_e \\times \\mathbb{T}_d^-$ of\n$x \\in P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$ is equal to the isotropy\ngroup $(\\mathbb{T}_e \\times \\mathbb{T}_d^-)_{t \\cdot x}$ of $t \\cdot x$, for any $t \\in \\mathbb{T}$; in particular this is true for\n$t \\in \\mathbb{T}_d^+$. Although the space $P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$ is possibly\nsingular, we can work in a given symplectic stratum.\nThus it makes sense to say that $\\mathbb{T}_d^+$ acts in a Hamiltonian\nfashion (on each individual stratum), and we may identify\nthe torus action on the whole space\nby the Hamiltonians of the $S^1$ factors.\nThese Hamiltonians are smooth in the sense\nof \\cite{SjamaarLerman}, since these functions are obtained by\nrestricting $\\mathbb{T}_e \\times \\mathbb{T}_d^-$ invariant continuous functions on\n$(\\bigwedge^2 \\mathbb{C}^3)^{n-2}$ to individual strata.\n\nEach $\\mathbb{C}^2 \\subset E_n^\\mathcal{T}(\\mathrm{SU}(2))$ is indexed by\nan edge of $\\mathcal{T}^D$.\nGiven an internal edge $\\epsilon$,\nthe factor $(S^1)_{\\epsilon}$ of $\\mathbb{T}_d^+$\nhas Hamiltonian function\n$\\|(z,w)_{\\epsilon^-}\\|^2\/2 = \\|(z,w)_{\\epsilon^+}\\|^2\/2$\non any given orbit-type stratum.\nThe $(z,w)_{\\epsilon^-}$ corresponds to the diagonal\n$d_{\\epsilon}$ of the associated $n$-gon in\n$\\mathbb{R}^3$.\nBy Lemma \\ref{diagonalpullback} and Lemma \\ref{SpinLength}\nwe have $\\|d_{\\epsilon}\\| = \\frac{1}{4} \\|(z,w)_{\\epsilon^-}\\|^2$,\nand so $\\|d_{\\epsilon}\\|$ is\n\\emph{one-half} that of\nthe Hamiltonian for the $\\epsilon$-th factor $(S^1)_\\epsilon$ of\n$\\mathbb{T}_d^+$.\nIn what follows we will need the following elementary lemma.\n\\begin{lemma}\nSuppose $A: S^1 \\times X \\to X$ is a Hamiltonian action of\nthe circle on a stratified symplectic space $X$. Suppose the action\nis generated by the Hamiltonian potential $f$ and that the element\n$-1 \\in S^1$ acts trivially. Let $\\overline{A}$ be the induced action\nof the quotient circle $\\overline{S^1} = S^1\/\\{\\pm 1\\}$.\nThen the Hamiltonian potential for the $\\overline{A}$ action\nis $\\frac{f}{2}$.\n\\end{lemma}\n\n\\begin{proof}\nWe have\n$$\\overline{A}(\\exp{\\sqrt{-1}t},x) = A(\\exp{\\sqrt{-1}t\/2},x).$$\n\\end{proof}\n\nThe bending flow torus $$T_{bend} = \\prod_{\\text{$d$ diagonal}} (S^1)_d,$$\nacting on $P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$, is such that\nthe $d(\\epsilon)$-th factor has Hamiltonian $\\|d_\\epsilon\\|$.\nTherefore, the $\\mathbb{T}_d^+$ action factors through the action of $T_{bend}$ via\na two to one cover on each component of $\\mathbb{T}_d^+$. Thus we may conclude\nthat $T_{bend}$ and $\\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R}))$ coincide.\n\n\\begin{theorem}\nLet $\\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R})) = \\mathbb{T}_d^+ \/ (\\mathbb{Z}\/2\\mathbb{Z})^{n-3}$ be the image of $\\mathbb{T}_d^+$ in $\\mathrm{SO}(3,\\mathbb{R})^{n-3}$ under the\nsurjection $\\mathrm{SU}(2) \\to \\mathrm{SO}(3,\\mathbb{R})$. The action of $\\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R}))$ on $P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$\ncoincides with the bending flows. Precisely the action is Hamiltonian\nand the circle factor\ncorresponding to $d(\\epsilon)$ has Hamiltonian potential $\\|d_{\\epsilon}\\|$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\pi : \\mathbb{T}_d^+ \\to \\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R}))$ be the quotient map.\nLet $\\epsilon$ be any internal edge of $\\mathcal{T}$. The action of\n$(S^1)_\\epsilon \\subset \\mathbb{T}_d^+$ on $P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$\nfactors through the action of $\\pi((S^1)_\\epsilon) \\in \\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R}))$,\nthrough the map $t \\mapsto t^2$. Consequently by the lemma\nabove the Hamiltonian function\nof $(S^1)_\\epsilon \\subset \\mathbb{T}_d^+$ is twice that of the Hamiltonian\nof $\\pi((S^1)_\\epsilon) \\subset \\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R}))$.\nHence the Hamiltonian for $\\pi((S^1)_\\epsilon) \\subset \\mathbb{T}_d^+(\\mathrm{SO}(3,\\mathbb{R}))$ is\n$\\|d_{\\epsilon}\\|$, as it is for $(S^1)_{d_{\\epsilon}} \\subset T_{bend}$.\nSince the Hamiltonian functions\nof $T_d^+(\\mathrm{SO}(3,\\mathbb{R}))$ and $T_{bend}$ coincide, their actions must coincide on\n$P_\\mathbf{r}^\\mathcal{T}(\\mathrm{SU}(2))$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Appendix : symplectic and GIT quotients of affine space}\\\nThe results in this appendix were obtained with the aid of W. Goldman.\n\n\\subsection{Fiber twists and normalizing the momentum map}\n\\label{fibertwists}\nThe goal of this appendix is to prove Theorem \\ref{twistandshift} below -\nwe match the level for the symplectic quotient with the\ntwist used to define the linearization in forming the GIT quotient.\nIn four places in the paper,\nnamely subsections \\ref{firsttorusquotient},\\ \n\\ref{secondtorusquotient},\\ \n\\ref{thirdtorusquotient} and subsubsection \\ref{fourthtorusquotient} we applied the results of \\cite{Sjamaar} to deduce\nthat for a torus acting on affine space linearized by acting\non the trivial line bundle by a character $\\chi$ the GIT\nquotient is homeomorphic to the symplectic quotient at level\nthe derivative of $\\chi$ at the identity provided the momentum map was proper.\nHowever there is a technical problem \nabout the normalization of the momentum\nmap chosen for the\naction of the torus \n(there is an indeterminancy of an additive\nconstant vector) \nThe correct normalization of\nthe momentum map must depend on the action of the torus on the total space of the line\nbundle. It is given in formula (2.3), page 116 of \\cite{Sjamaar} which\nwe now state for the convenience of the reader. \n\nLet $p:E \\to M$ be a Hermitian line bundle $L$. Let $G$ be a compact group\nacting on $E$ by automorphisms of the Hermitian structure. \nLet $\\xi \\in \\mathfrak{g}, e \\in E$ and $m=p(e)$. In what follows\n$\\xi_E$ denotes the vector field on $E$ induced by $\\xi$,\n$\\xi_M^{hor}$ denotes the horizontal lift of the vector\nfield $\\xi_M$ induced on $M$ and $\\nu_E$ denotes the canonical vertical\nvector field (induced by the $\\mathrm{U}(1)$ action). The connection and curvature\nforms take values in the Lie algebra $\\mathfrak{u}(1) = \\sqrt{-1} \\mathbb{R}$\nof $\\mathrm{U}(1)$. We can now state the formula from \\cite{Sjamaar}:\n\\begin{equation} \\label{normalization}\n\\xi_E(e) = \\xi_M^{hor}(e) + 2\\pi <\\mu(m),\\xi> \\nu_E.\n\\end{equation}\n\n\\begin{definition}\nWe will say a momentum map satisfying equation (\\ref{normalization})\nis {\\em normalized} relative to the linearization (action of $G$\non the total space of the bundle).\n\\end{definition}\n\n\\begin{remark}\nWe can check the conventions involved in equation (\\ref{normalization})\nby applying the connection form $\\theta$ to both sides to obtain\n$$\\theta(\\xi_E(e)) = 2\\pi \\sqrt{-1}<\\mu(m),\\xi>.$$ \nApplying $d$ to each side and Cartan's formula we find that \n$<\\mu(m),\\xi>$ is a Hamiltonian potential for $\\xi_M $ if and only if\nthe symplectic form $\\omega$ and the connection form $\\theta$ are related\nby\n$$\\omega = - \\frac{1}{2\\pi \\sqrt{-1}} d \\theta.$$\n\\end{remark}\n\nSuppose now we twist the action of the torus $G$ on the total\nspace of the line bundle by scaling each fiber by a fixed character\n$\\chi$. This changes the invariant sections and hence changes the\nGIT quotient. Note that the differential of $\\chi$ at the identity\nof $G$ is an element $\\dot{\\chi}$ of $\\mathfrak{g}^*$. Thus we could\nchange the momentum map $\\mu$ by adding $\\dot{\\chi}$ and obtain a\nnew momentum map. The following lemma is an immediate consequence\nof equation (\\ref{normalization}).\n\n\\begin{lemma}\\label{twistshift}\nSuppose we twist the action of $G$ by a character $\\chi$. Then the\nnormalized momentum map for the new action is obtained by adding\n$\\dot{\\chi}$.\n\\end{lemma}\n\nWe now restrict to the case of a torus $T$ acting linearly on a\nsymplectic vector space $V,\\omega$. We assume we have chosen a\n$T$-invariant complex structure $J$ on $V$ so that $\\omega$ is \nof type $(1,1)$ for $J$ (this means $J$ is an isometry of $\\omega$) and \nthe symmetric form $B$ given by $B(v,v) = \\omega(v,Jv)$\nis positive definite.\nWe let $W$ be the subspace of $V \\otimes \\mathbb{C}$ of type $(1,0)$ vectors,\nthat is $W=\\{ v -\\sqrt{-1} Jv: v \\in V\\}$. We define a \npositive-definite Hermitian form $H$ on $W$ by \n$$H(v_1 - \\sqrt{-1}Jv_1, v_2 - \\sqrt{-1}Jv_2)= B(v_1,v_2) - \\sqrt{-1}\n\\omega(v_1,v_2).$$\nWe will abbreviate $\\sqrt{H(v,v)}$ to $\\|v\\|$ in what follows. We define\na symplectic form $A$ on $W$ by $A(w_1,w_2) = - \\Im H(w_1,w_2)$.\nWe note that the map $w \\to \\Re w$ is a symplectomorphism\nfrom $W,A$ to $V,\\omega$.\n\nSuppose that we have chosen an $H$-orthonormal basis for $W$ so we\nhave identified\n$$W \\cong \\mathbb{C}^n.$$\nWe let $T_0 \\cong (S^1)^n$ be the compact torus\nconsisting of the diagonal matrices with unit length elements on\nthe diagonal and $\\underline{T}_0$ be the complexification of $T_0$. We let $\\mathfrak{t}_0^*\n\\cong \\mathbb{R}^n$ be the dual of the Lie algebra of $T_0$. We define\n$\\mu_0:W \\to \\mathfrak{t}_0 ^*$ by\n$$\\mu_0((z_1,\\cdots,z_n)) = (-\\frac{|z_1|^2}{2},\\cdots,-\\frac{|z_n|^2}{2})$$\nThen $\\mu_0$ is a momentum map for the Hamiltonian action of $T_0$\non $W$.\nWe note that all possible momentum maps\nare obtained from $\\mu_0$ by adding a vector $\\mathbf{c} = (c_1,\\cdots,c_n)\n\\in \\mathfrak{t}_0^{\\ast}$. Thus $\\mu_0$ is the unique momentum map\nvanishing at the origin of $W$. \n\nLet $E= W \\times \\mathbb{C}$ be total space of the trivial line bundle $L$ over $W$. We first describe the Hermitian holomorphic structure on $L$.\nWe give $L$ a holomorphic structure by requiring that the nowhere vanishing section $s_0$ of $E$ defined by\n$$s_0(w) = (w,1)$$\nis holomorphic. \nThus if $U$ is an open subset of $W$ and $s$ is a local section over $U$\nthen $s$ is holomorphic if and only if the function $f$ on $U$ defined by\n$$s = f s_0|U$$\nis holomorphic.\n\nWe define a Hermitian structure on $L$ by defining\n$$||s_0||(w) = \\exp{(-\\frac{\\pi}{2}\\|w\\|^2)}.$$ \nHence the section $\\sigma_0$ given by\n$$\\sigma_0(w) = \\exp{(\\frac{\\pi}{2}\\|w\\|^2)} s_0(w)$$\nhas unit length at every point.\nWe note that\n$$ - \\frac{1}{2\\pi \\sqrt{-1}}\\ \\overline{\\partial}\\partial \\ \\log ||s_0||^2 = \\sum _{i=1} ^{n} dx_i \\wedge dy_i$$\nin agreement with \\cite{Sjamaar}, pg. 115.\nLet $E_0$ denote the principal circle bundle of unit length vectors in \n$E$. \n\n\nOur goal is to describe the lift of the action of $T_0$ to $E$\nso that the normalized momentum map corresponding to this lifted\naction is $\\mu_0$. We note there is a distinguished lift given by\n$$t \\circ(w,z) = (tw,z), t \\in T_0, w \\in W, z \\in \\mathbb{C}.$$\nBy definition this lift leaves invariant the holomorphic section $s_0$\nbut it also leaves invariant the unit length section $\\sigma_0$\nbecause the function $\\exp{(-\\frac{\\pi}{2}\\|w\\|^2)}$ is invariant\nunder $\\mathrm{U}(n)$ and hence under $T_0$. Hence $T_0$ leaves fixed the Hermitian structure on $E$\nand $\\underline{T}_0$ leaves the holomorphic section $s_0$ fixed.\nWe will call such lifts {\\it untwisted} and we will say the linearization\nconsisting of the trivial bundle together with the previous action\nis untwisted. We will now prove\n\\begin{proposition}\\label{untwistedstandard}\nThe normalized momentum map of $T_0$ corresponding to the untwisted linearization\nis $\\mu_0$.\n\\end{proposition}\n\nThe proposition will be a consequence of the next lemma and corollary.\n\n\nWe will compute $\\mu_0$ in the trivialization of $E_0$ given by $\\sigma_0$.\nWe note that since $\\sigma_0$ is invariant under $T_0$ the untwisted lift of the compact torus above (relative\nto the trivialization by $s_0$) remains untwisted in the trivialization\nby $\\sigma_0$. We let $\\psi$ denote the coordinate in the fiber circle\nof $E_0$ so $\\nu_{E_0} = \\partial \/ \\partial \\psi$. Hence if $z_i = x_i + \\sqrt{-1}y_i, 1 \\leq i \\leq n$, then\n$x_1,y_1,\\cdots,x_n,y_n, \\psi$ are coordinates in $E_0$.\n\n\n\\begin{lemma}\\label{connectionform}\nThe canonical connection on $E$ is given by\n$$\\nabla \\sigma_0 = - \\pi \\sqrt{-1} (\\sum_{i=1}^n (x_i dy_i - y_i dx_i) \\otimes \\sigma_0.$$\nEquivalently in the above coordinates the connection form $\\theta$ of the canonical connection is given by\n$$\\theta = \\sqrt{-1} d\\psi - \\pi \\sqrt{-1} (\\sum_{i=1}^n (x_i dy_i - y_i dx_i).$$\n\\end{lemma}\n\\begin{proof}\nThe reader will verify that $\\nabla$ satisfies\n$$ \\nabla_{\\partial \/ \\partial \\overline{z}_j} s_0 =0, 1 \\leq j \\leq n, $$\nand has curvature $- 2\\pi \\sqrt{-1}\\omega$. Hence $\\nabla$ is the unique Hermitian connection with\ncurvature $d \\theta = - 2\\pi \\sqrt{-1}\\omega$.\n\\end{proof}\n\nWe then have the following corollary.\n\n\\begin{corollary}\nThe horizontal lift of the vector field $x_i \\partial\/\\partial y_i -\ny_i \\partial\/\\partial x_i$ is $x_i \\partial\/\\partial y_i - y_i\n\\partial\/\\partial x_i + \\pi (x_i^2 + y_i^2)\\partial \/ \\partial \\psi$.\n\\end{corollary}\n\n\\begin{proof}\nBy the formula for $\\theta$ we see that the horizontal lift of\n$\\partial\/\\partial x_i$ is $\\partial\/\\partial x_i - \\pi y_i \\partial \/ \\partial \\psi\n$ and the horizontal lift of $\\partial\/\\partial y_i$ is\n$\\partial\/\\partial y_i + \\pi x_i \\partial \/ \\partial \\psi$. Since the operation\nof taking horizontal lifts is linear over the functions the corollary\nfollows.\n\\end{proof}\n\nProposition \\ref{untwistedstandard} follows from the corollary and equation (\\ref{normalization}).\n\nNow suppose $T$ is a compact torus with complexification $\\underline{T}$ and $T$ acts on $W$ through a\nrepresentation \n$\\rho:T \\to T_0$. Assume further that we linearize the action \nof $\\underline{T}$ on $W$ by the untwisted linearization.\nIt is clear from equation (\\ref{normalization}) that we obtain the {\\it normalized} momentum\nmap $\\mu_T$ for $T$ by restricting the {\\it normalized} momentum map for $T_0$. More precisely\nlet $\\rho^{\\ast} : \\mathfrak{t}_0^{\\ast} \\to \\mathfrak{t}^{\\ast}$\nbe the induced map on dual spaces.\nThen we have\n$$\\mu_T = \\rho^{\\ast} \\circ \\mu_0.$$\nThus the normalized momentum map corresponding to the untwisted linearization\nof the linear action of $T$ is homogeneous linear in the squares of the $|z_i|$'s.\nWe now obtain the desired result in this appendix by\napplying Lemma \\ref{twistshift} and Theorem 2.18 (page 122) of \\cite{Sjamaar}.\n \n\\begin{theorem}\\label{twistandshift}\nLet $\\underline{T}$ be a complex torus with maximal compact subtorus $T$ and let $\\chi$ be a character of $\\mathbb{T}$.\nThen the GIT quotient for a linear action of $\\underline{T}$ on a complex vector space $W$ with linearization given by $W \\times \\mathbb{C}$ and the action \n$$t \\circ (w,z) = (tw, \\chi(t) z)$$\nis homeomorphic to the symplectic quotient by \n$T$ obtained using the momentum\nmap $(\\mu_T)_0 + \\dot{\\chi}$ if the momentum map for the action of $T$\nis proper. Here $(\\mu_T)_0$ is the momentum map that vanishes at the origin\nof $W$.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nThroughout this paper, all spaces are path-connected with homotopy types of $CW$-complexes. \nWe do not distinguish between a map and its homotopy class.\n\nLet $X$ be a connected space, $x_0\\in X$ a base-point and $\\mathbb{S}^1$ the circle. The \\textit{Gottlieb group} $G(X,x_0)$ of $X$ defined in \\cite{gottlieb1} is the subgroup of the fundamental group $\\pi_1(X,x_0)$ consisting of all elements which can be represented by a map $\\alpha \\colon \\mathbb{S}^1\\to X$ such that $\\mathrm{id}_X \\vee \\alpha \\colon X\\vee \\mathbb{S}^1\\to X$ extends (up to homotopy) to a map $F \\colon X\\times \\mathbb{S}^1\\to X$.\nFollowing \\cite{gottlieb1}, we recall that $P(X,x_0)$ is the set of elements of $\\pi_1(X,x_0)$ whose Whitehead products\nwith all elements of all homotopy groups $\\pi_m(X,x_0)$ are zero for $m\\geq 1$. It turns out that $P(X,x_0)$\nforms a subgroup of $\\pi_1(X,x_0)$ called the \\textit{Whitehead center group} and, by \\cite[Theorem~I.4]{gottlieb1}, it holds $G(X,x_0)\\subseteq P(X,x_0)$.\n\nNow, given a map $f \\colon X\\to Y$, in view of \\cite{gottlieb} (see also \\cite{kim}),\nthe \\textit{generalized Gottlieb group} $G^f(Y,f(x_0))$ is defined \nas the subgroup of $\\pi_1(Y,f(x_0))$ consisting of all elements which can be\nrepresented by a map $\\alpha \\colon \\mathbb{S}^1\\to Y$ such that $f\\vee \\alpha \\colon X\\vee \\mathbb{S}^1\\to Y$ extends (up to homotopy) to a map $F \\colon X\\times \\mathbb{S}^1\\to Y$. \n\nThe \\textit{generalized Whitehead center group} $P^f(Y,f(x_0))$ as defined in \\cite{kim} consists of all elements $\\alpha\\in\\pi_1(Y,f(x_0))$ whose Whitehead products $[f\\beta,\\alpha]$ are zero for all $\\beta\\in\\pi_m(X,x_0)$ with $m\\ge 1$. It turns out that $P^f(Y,f(x_0))$ forms a subgroup of $\\pi_1(Y,f(x_0))$ and $G^f(Y,f(x_0))\\subseteq P^f(Y,f(x_0))\\subseteq \\mathcal{Z}_{\\pi_1(Y,f(x_0))}f_*(\\pi_1(X,x_0))$, the centralizer of $f_*(\\pi_1(X,x_0))$ in $\\pi_1(Y,f(x_0))$. \n\nIf $X=Y$ then the group $G^f(Y,f(x_0))$ is considered in \\cite[Chapter~II, 3.5~Definition]{jiang}, denoted by $J(f,x_0)$ and called the \\textit{Jiang subgroup} of the map $f\\colon Y \\to Y$. The role of the $J(f,x_0)$ played in that theory has been intensively studied in the book \\cite{brown} as well. More precisely, it is observed that the group $J(f,x_0)$ acts on the right on the set of all fixed point classes of $f$, and any two equivalent fixed point classes under this action have the same index. Further, Bo-Ju Jiang in \\cite[Chapter~II, 3.1~Definition]{jiang} considered also the group $J(\\tilde{f}_0)$ for a fixed lifting $\\tilde{f}_0$ of $f$ to the universal covering of $Y$ and stressed its importance to the Nielsen--Wecken theory of fixed point classes. \n\nIf $f=\\mathrm{id}_X$ then, by \\cite[Theorem~II.1]{gottlieb1}, the groups $J(f,x_0)$ and $J(\\tilde{f}_0)$ are isomorphic and, according to \\cite[Chapter~II, 3.6~Lemma]{jiang}, the groups $J(f,x_0)$ and $J(\\tilde{f}_0)$ are isomorphic for any self-map $f\\colon X\\to X$ but no proof is given.\n\n\nThe aim of this paper is to follow the proof of \\cite[Theorem~II.1]{gottlieb1} and give not only a proof of \\cite[Chapter~II, 3.6~Lemma]{jiang} but also present a proof of its generalized version for any map $f\\colon X \\to Y$.\n\nThe paper is divided into two sections. Section~\\ref{sec.1} follows some results from \\cite{gottlieb1} and deals with some properties of fibre-preserving maps and deck transformations used in the sequel. In particular, we show the functoriality of the fundamental group via deck transformations. Section~\\ref{sec.2} takes up the systematic study of the group $G^f(Y,f(x_0))$. If $X=Y$, $f=\\mathrm{id}_X$ and $x_0\\in X$ is a base-point then the group $G^f(X,x_0)=G(X,x_0)$ has been described in \\cite[Theorem~II.1]{gottlieb1} via the deck transformation group of $X$ and by \\cite[Chapter~II, 3.6~Lemma]{jiang} the groups $G^f(X,x_0)=J(f,x_0)$ and $J(\\tilde{f}_0)$ are isomorphic for a self-map $f\\colon X\\to X$.\n\nDenote by $\\mathcal{D}(Y)$ the group of all deck transformations of a space $Y$. Given a map $f\\colon X \\to Y$, write $\\mathcal{L}^f(Y)$ for the set of all liftings $\\tilde{f}\\colon \\tilde{X}\\to \\tilde{Y}$ of $f$ with respect to universal coverings of $X$ and $Y$, respectively. Now, for a fixed $\\tilde{f}_0\\in \\mathcal{L}^f(Y)$, we denote by $\\mathcal{D}^{\\tilde{f}_0}(Y)$ the set (being a group) of all elements $h\\in \\mathcal{D}(Y)$ such that $\\tilde{f}_0\\simeq_{\\tilde{H}} h\\tilde{f}_0$, where $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ is a fibre-preserving homotopy with respect to the universal covering maps $p\\colon \\tilde{X}\\to X$ and $q\\colon \\tilde{Y}\\to Y$. Then, the main result, Theorem~\\ref{qq} generalizes \\cite[Theorem~II.1]{gottlieb1} and \\cite[Chapter~II, 3.6~Lemma]{jiang} as follows:\n\n\\textit{Given $f\\colon X \\to Y$, the canonical isomorphism $\\pi_1(Y,f(x_0))\\xrightarrow{\\approx}\\mathcal{D}(Y)$ restricts to an isomorphism $G^f(Y,f(x_0))\\xrightarrow{\\approx} \\mathcal{D}^{\\tilde{f}_0}(Y)$.}\n\n\\section{Preliminaries}\\label{sec.1}\n\nLet $p\\colon X\\to A$ and $q\\colon Y\\to B$ be maps. We say that $f\\colon X\\to Y$ is a \\textit{fibre-preserving map} with respect to $p,q$ provided $p(x)=p(x')$ implies $qf(x)=qf(x')$ for any $x,x'\\in X$. \n\nWe say that $H\\colon X\\times I \\to Y$ is a \\textit{fibre-preserving homotopy} with respect to $p,q$ if $H$ is a fibre-preserving map with respect to $p\\times \\mathrm{id}_I\\colon X\\times I \\to A\\times I$ and $q\\colon Y \\to B$.\n\nIt is clear that the commutativity of a diagram \\[ \\xymatrix{ X\\ar[r]^{f} \\ar[d]_p & Y\\ar[d]^q \\\\ A \\ar[r]^g& B }\\] guarantee that $f$ is a fibre-preserving map. \n\n\n\\begin{remark}\n(1) Let $p,q,f$ be maps as above. If $p\\colon X \\to A$ is surjective then there exists a map $g\\colon A\\to B$ such that $qf=g p$. In addition, if $p$ is a quotient map, then $g$ is continuous.\n\n(2) Given discrete groups $H$ and $K$, consider actions $H\\times X \\to X$ and $K\\times Y \\to Y$ and write $p\\colon X \\to X\/H$ and $q\\colon Y \\to Y\/K$ for the quotient maps. If $f\\colon X \\to Y$ is a $\\varphi$-equivariant map for a homomorphism $\\varphi\\colon H \\to K$ then $f$ is a fibre-preserving map with respect to $p$ and $q$.\n\\end{remark}\n\nIf $f\\colon X \\to Y$ is a fibre-preserving map and $g=\\mathrm{id}_A$ then the map $f$ is a fibrewise map in the sense of \\cite[Chapter~1]{james}. But, the reciprocal of that does not hold, as it is shown below:\n\n\n\\begin{example}\nLet $p=q\\colon \\mathbb{S}^1\\times I \\to \\mathbb{S}^1$ be the projection. Fix $1\\neq \\lambda\\in \\mathbb{S}^1$ and define $f\\colon \\mathbb{S}^1\\times I\\to \\mathbb{S}^1\\times I$ by $f(z,t)= (\\lambda z,t)$ for $(z,t)\\in \\mathbb{S}^1\\times I$. Then, $f$ is a fibre-preserving map but clearly $qf\\neq p$.\n\\end{example}\n\nWrite $\\mathcal{D}(X)$ for the group of all deck transformations of $X$ and recall that there is an isomorphism $\\mathcal{D}(X)\\approx \\pi_1(X,x_0)$. Next, given a map $f\\colon X\\to Y$, consider the set $\\mathcal{L}^f(Y)$ of all maps $\\tilde{f}\\colon \\tilde{X}\\to \\tilde{Y}$ such that the diagram \\[ \\xymatrix{ \\tilde{X}\\ar[r]^{\\tilde{f}} \\ar[d]_p & \\tilde{Y}\\ar[d]^q \\\\ X \\ar[r]^f & Y }\\] is commutative, where $p,q$ are universal covering maps. \n\nFixing $\\tilde{f}_0\\in \\mathcal{L}^f(Y)$, we follow \\cite[Chapter~I, 1.2~Proposition]{jiang} to show:\n\n\\begin{proposition}\\label{star}If $f\\colon X\\to Y$ then for any lifting $\\tilde{f}\\colon \\tilde{X}\\to \\tilde{Y}$ of $f$ there is a unique $h\\in \\mathcal{D}(Y)$ such that $\\tilde{f}=h\\tilde{f}_0$. \n\\end{proposition}\n\\begin{proof} First, fix $x_0\\in X$ and $\\tilde{x}_0\\in p^{-1}(x_0)$, and\nwrite $y_0=f(x_0)$, $\\tilde{y}_0=\\tilde{f}_0(\\tilde{x}_0)$, and\n$\\tilde{y}=\\tilde{f}(\\tilde{x}_0)$. Obviously, $\\tilde{y}_0,\\tilde{y}\\in\nq^{-1}(y_0)$. Then, there exists a unique $h\\in \\mathcal{D}(Y)$ with\n$h(\\tilde{y}_0)=\\tilde{y}$, that is, $h\\tilde{f}_0(\\tilde{x}_0)=\n\\tilde{f}(\\tilde{x}_0)$. Since both $\\tilde{f}$ and $h\\tilde{f}_0$ are\nlifts of $fp\\colon \\tilde{X}\\to Y$, the unique lifting property\nguarantees that $\\tilde{f}=h\\tilde{f}_0$. \n\\par Now, suppose that $\\tilde{f}=h\\tilde{f}_0=h'\\tilde{f}_0$ for some $h,h'\\in \\mathcal{D}(Y)$. Then, $h\\tilde{f}_0(\\tilde{x}_0)=h'\\tilde{f}_0(\\tilde{x}_0)$ implies $h(\\tilde{y}_0)=h'(\\tilde{y}_0)$. Consequently, $h=h'$ and the proof is complete.\n\\end{proof}\n\nFor a deck transformation $l\\in \\mathcal{D}(X)$, we notice that $\\tilde{f}_0l$ is also a lifting of $f$. By Proposition~\\ref{star}, there exists a unique $h_l\\in \\mathcal{D}(Y)$ such that $\\tilde{f}_0l=h_l\\tilde{f}_0$. \nThen, we define \\[f_*\\colon \\mathcal{D}(X)\\to \\mathcal{D}(Y)\\] by $f_*(l)=h_l$ for any $l\\in \\mathcal{D}(X)$. Obviously, the map $f_*$ is a homomorphism. Notice that the map $f_*$ has been already defined in \\cite[Chapter~II, 1.1~Definition]{jiang} for any self-map $f\\colon X \\to X$.\n\n\nGiven $\\tilde{f}_1,\\tilde{f}_2\\in \\mathcal{L}^f(Y)$, we define $\\tilde{f}_1\\ast\\tilde{f}_2=h_1h_2\\tilde{f}_0$, where $\\tilde{f}_1=h_1\\tilde{f}_0$ and $\\tilde{f}_2=h_2\\tilde{f}_0$ for $h_1,h_2\\in \\mathcal{D}(Y)$ as in Proposition~\\ref{star}. This leads to a group structure on $\\mathcal{L}^f(Y)$ with $\\tilde{f}_0$ as the identity element. Notice that the groups $\\mathcal{L}^f(Y)$ and $\\mathcal{D}(Y)$ are isomorphic. In the sequel we identify those two groups, if necessary.\n\nFor a homotopy $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$, we write $\\tilde{H}_t=\\tilde{H}(-,t)$ with $t\\in I$.\n\n\\begin{lemma}\\label{Ht}Let $f\\colon X \\to Y$. A homotopy $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ with $\\tilde{H}_0=\\tilde{f}_0$ is a fibre-preserving homotopy if and only if for any $l\\in \\mathcal{D}(X)$ and $t\\in I$ the following diagram \n\\[ \\xymatrix@C=1.5cm{ \\tilde{X}\\ar[r]^{\\tilde{H}_t} \\ar[d]_-{l} & \\tilde{Y}\\ar[d]^-{f_*(l)} \\\\ \\tilde{X} \\ar[r]^{\\tilde{H}_t} & \\tilde{Y} } \\] commutes.\n\\end{lemma}\n\n\\begin{proof}Let $(\\tilde{x},t),(\\tilde{x}',t')\\in \\tilde{X}\\times I$ with $(p\\times \\mathrm{id}_I)(\\tilde{x},t)=(p\\times \\mathrm{id}_I)(\\tilde{x}',t')$. Then, $p(\\tilde{x})=p(\\tilde{x}')$ and $t=t'$. Next, consider $l\\in \\mathcal{D}(X)$ such that $l(\\tilde{x})=\\tilde{x}'$. Because $\\tilde{H}_t=f_*(l)\\tilde{H}_tl^{-1}$, we conclude that $q\\tilde{H}_t(\\tilde{x})=qf_*(l)\\tilde{H}_tl^{-1}(\\tilde{x})=qf_*(l)\\tilde{H}_t(\\tilde{x}')=q\\tilde{H}_t(\\tilde{x}')$. Hence $\\tilde{H}$ is fibre-preserving.\n\nSuppose $\\tilde{H}$ is fibre-preserving, $l\\in \\mathcal{D}(X)$ and take $\\tilde{x}\\in \\tilde{X}$, $t\\in I$. Then, $\\tilde{x}$ and $ l(\\tilde{x})$ are in the same fibre of $p$. Since $\\tilde{H}_t(\\tilde{x})$ and $\\tilde{H}_t(l(\\tilde{x}))$ are in the same fibre of $q$, there exists a unique $h\\in \\mathcal{D}(Y)$ such that $h\\tilde{H}_t(\\tilde{x})=\\tilde{H}_tl(\\tilde{x})$. If $\\varepsilon>0$ is sufficiently small, $h\\tilde{H}_{t-\\varepsilon}(\\tilde{x})=\\tilde{H}_{t-\\varepsilon}l(\\tilde{x})$. Thus, the greatest lower bound of the set of $t$'s such that $h\\tilde{H}_t(\\tilde{x})=\\tilde{H}_tl(\\tilde{x})$ must occur when $t=0$. Therefore, by continuity, $h\\tilde{H}_0(\\tilde{x})=\\tilde{H}_0l(\\tilde{x})$. But $\\tilde{H}_0=\\tilde{f}_0$, so we get $h\\tilde{f}_0(\\tilde{x})=\\tilde{f}_0l(\\tilde{x})=f_*(l)\\tilde{f}_0(\\tilde{x})$. This can occur only when $h=f_*(l)$. Consequently, $\\tilde{H}_tl=f_*(l)H_t$ and the proof is complete.\n\\end{proof}\n\nNow, fix $\\tilde{f}_0\\in \\mathcal{L}^{f}(Y)$ and consider the subset $\\mathcal{D}^{\\tilde{f}_0}(Y)$ of elements $h\\in \\mathcal{D}(Y)$ such that $\\tilde{f}_0\\simeq_{\\tilde{H}} h\\tilde{f}_0$, where $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ is a fibre-preserving homotopy with respect to the universal covering maps $p,q$. \nEquivalently, in view of Lemma~\\ref{Ht}, the set $\\mathcal{D}^{\\tilde{f}_0}(Y)$ coincides with the set of all elements $h\\in \\mathcal{D}(Y)$ for which there is a homotopy $f\\simeq_H f$ which lifts to a homotopy $\\tilde{H}$ with $\\tilde{f}_0\\simeq_{\\tilde{H}}h\\tilde{f}_0$.\n\nNext, write $\\mathcal{Z}_{\\mathcal{D}(Y)} f_*(\\mathcal{D}(X))$ for the centralizer of $ f_*(\\mathcal{D}(X))$ in $\\mathcal{D}(Y)$. Then, the result below generalizes \\cite[Chapter~II, 3.2~Proposition, 3.3~Lemma]{jiang} as follows:\n\n\\begin{proposition}\\label{prop.center}The subset $\\mathcal{D}^{\\tilde{f}_0}(Y)$ is contained in $\\mathcal{Z}_{\\mathcal{D}(Y)} f_*(\\mathcal{D}(X))$ and is a subgroup of $\\mathcal{D}(Y)$.\n\\end{proposition}\n\n\\begin{proof}\nLet $h\\in \\mathcal{D}^{\\tilde{f}_0}(Y)$. Then, there is a fibre-preserving homotopy $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ with $\\tilde{H}_0=\\tilde{f}_0$ and $\\tilde{H}_1=h\\tilde{f}_0$. But, by Lemma~\\ref{Ht}, it holds $f_*(l)\\tilde{H}_t=\\tilde{H}_tl$ for any $l\\in \\mathcal{D}(X)$ and $t\\in I$. Hence, for $t=0,1$ we get $f_*(l)\\tilde{f}_0=\\tilde{f}_0l$ and $f_*(l)h\\tilde{f}_0=h\\tilde{f}_0l=hf_*(l)\\tilde{f}_0$. Consequently, $f_*(l)h=hf_*(l)$ and we get that $h\\in \\mathcal{Z}_{\\mathcal{D}(Y)} f_*(\\mathcal{D}(X))$.\n\n\nTo show the second part, take $h,h'\\in \\mathcal{D}^{\\tilde{f}_0}(Y)$. Then, there are fibre-preserving homotopies $\\tilde{H},\\tilde{H}'\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ with $\\tilde{H}_0=\\tilde{H}_0'=\\tilde{f}_0$, $\\tilde{H}_1=h\\tilde{f}_0$ and $\\tilde{H}'_1=h'\\tilde{f}_0$. Next, consider the map $\\tilde{H}''\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ given by $\\tilde{H}''(\\tilde{x},t)=hh'^{-1}\\tilde{H}'(\\tilde{x},1-t)$ for $(\\tilde{x},t)\\in \\tilde{X}\\times I$ and notice that $\\tilde{H}''$ is a fibre-preserving homotopy with $\\tilde{H}''_0=h\\tilde{f}_0$ and $\\tilde{H}''_1=hh'^{-1}\\tilde{f}_0$. Finally, the concatenation $\\tilde{H}\\bullet \\tilde{H}'' \\colon \\tilde{X}\\times I \\to \\tilde{Y}$ is a fibre-preserving homotopy with $(\\tilde{H}\\bullet \\tilde{H}'')_0=\\tilde{f}_0$ and $(\\tilde{H}\\bullet \\tilde{H}'')_1=hh'^{-1}\\tilde{f}_0$. Consequently, $hh'^{-1}\\in \\mathcal{D}^{\\tilde{f}_0}(Y)$ and the proof is complete.\n\\end{proof}\n\nNotice that if $f\\colon X \\to X$ is a self-map then the group $\\mathcal{D}^{\\tilde{f}_0}(X)$ coincides with the group $J(\\tilde{f}_0)$ defined in \\cite[Chapter~II, 3.1~Definition]{jiang}. \n\n\n\\section{Main result}\\label{sec.2}\n\nGiven spaces $X$ and $Y$, write $Y^X$ for the space of continuous maps from $X$ into $Y$ with the compact-open topology. Next, consider the evaluation map $\\mathrm{ev} \\colon Y^X\\to Y$, i.e., $\\mathrm{ev}(f)=f(x_0)$ for $f\\in Y^X$ and the base-point $x_0\\in X$. \nThen, it holds \\[G^f(Y,f(x_0))=\\operatorname{Im}\\bigl(\\mathrm{ev}_\\ast \\colon \\pi_1(Y^X,f)\\to \\pi_1(Y,f(x_0))\\bigr).\\] \nCertainly, $G^f(X,f(x_0))$ coincides with the group $J(f,x_0)$ defined in \\cite[Chapter~II, 3.5~Definition]{jiang} for a self-map $f\\colon X\\to X$.\n\\par Now, we follow \\textit{mutatis mutandis} the result \\cite[Theorem~II.1]{gottlieb1} to generalize \\cite[Chapter~II, 3.6~Lemma]{jiang} as follows: \n\n\\begin{theorem}\\label{qq}Given $f\\colon X \\to Y$, the canonical isomorphism $\\pi_1(Y,f(x_0))\\xrightarrow{\\approx} \\mathcal{D}(Y)$ restricts to an isomorphism $G^f(Y,f(x_0))\\xrightarrow{\\approx} \\mathcal{D}^{\\tilde{f}_0}(Y)$.\n\\end{theorem}\n\n\\begin{proof}Let $\\alpha\\in G^f(Y,f(x_0))$ and $h\\in \\mathcal{D}(Y)$ be the corresponding deck transformation. Then, there is a homotopy $H\\colon X\\times I \\to Y$ such that $H_0=H_1=f$ and $H(x_0,-)=\\alpha$, where $x_0\\in X$ is a base-point. Next, consider the commutative diagram\n\\[ \\xymatrix@C=1.5cm{\n\\tilde{X} \\ar[d]_{i_0}\\ar[rr]^{\\tilde{f}_0} && \\tilde{Y} \\ar[d]^q \\\\\n\\tilde{X}\\times I \\ar[r]^{p\\times \\mathrm{id}_I} & X\\times I \\ar[r]^H & Y\\rlap{.}\n} \\] Then, by the lifting homotopy property there is a map $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ such that $\\tilde{H}i_0=\\tilde{H}_0=\\tilde{f}_0$ and $q\\tilde{H}=H(p\\times \\mathrm{id}_I)$. This implies that $\\tilde{H}$ is a fibre-preserving homotopy. Further, because $H_0=H_1=f$, we also derive that $\\tilde{H}_0,\\tilde{H}_1\\in \\mathcal{L}^f(Y)$.\n\nNow, since the path $\\tilde{\\tau}\\colon I\\to \\tilde{Y}$ defined by $\\tilde{\\tau}=\\tilde{H}(\\tilde{x}_0,-)$ runs from $\\tilde{f}_0(\\tilde{x}_0)$ to $\\tilde{H}_1(\\tilde{x}_0)$, we derive that $\\alpha=q\\tilde{\\tau}$. Consequently, by means of Proposition~\\ref{star} we get $\\tilde{H}_1=h\\tilde{f}_0$ and so $\\tilde{H}$ is the required fibre-preserving homotopy with $\\tilde{f}_0\\simeq_{\\tilde{H}} h\\tilde{f}_0$. \n\nConversely, given $h\\in \\mathcal{D}^{\\tilde{f}_0}(Y)$, there is a fibre-preserving homotopy $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ with $\\tilde{f}_0\\simeq_{\\tilde{H}} h\\tilde{f}_0$. This implies a homotopy $H\\colon X\\times I \\to Y$ such that $H_0=H_1=f$ and $q\\tilde{H}=H(p\\times \\mathrm{id}_I)$. Then, the path $\\tau\\colon I\\to Y$ given by $\\tau=H(x_0,-)$ leads to the required loop in $G^f(Y,f(x_0))$.\n\\end{proof}\nNotice that by Theorem~\\ref{qq} the group $\\mathcal{D}^{\\tilde{f}_0}(Y)$ is independent of the lifting $\\tilde{f}_0\\in\\mathcal{L}^f(Y)$. Further, the advantage of $G^f(Y,f(x_0))$ over $\\mathcal{D}^{\\tilde{f}_0}(Y)$ is that it does not involve the covering spaces $\\tilde{X}$ and $\\tilde{Y}$ explicitly, hence it is easier to handle.\n\nNext, Lemma~\\ref{Ht} and Theorem~\\ref{qq} yield:\n\\begin{corollary}\\label{cor2}If $f\\colon X \\to Y$ then $G^f(Y,f(x_0))$ is isomorphic to the subgroup of $\\mathcal{D}(Y)$ given by those deck transformations $h$ for which there are homotopies $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ such that $\\tilde{f}_0\\simeq_{\\tilde{H}} h\\tilde{f}_0$ and the diagrams \n\\[ \\xymatrix@C=1.5cm{ \\tilde{X}\\ar[r]^{\\tilde{H}_t} \\ar[d]_-{l} & \\tilde{Y}\\ar[d]^-{f_*(l)} \\\\ \\tilde{X} \\ar[r]^{\\tilde{H}_t} & \\tilde{Y} } \\] commute for any $l\\in \\mathcal{D}(X)$ and $t\\in I$. Equivalently, the homotopies $\\tilde{H}\\colon \\tilde{X}\\times I \\to \\tilde{Y}$ are $f_*$-equivariant.\n\\end{corollary}\n\nLet $\\mathcal{H}^{\\tilde{f}_0}(Y)$ be the subset of all $h\\in\\mathcal{Z}_{\\mathcal{D}(Y)}f_*(\\mathcal{D}(X))$ such that $\\tilde{f}_0\\simeq h\\tilde{f}_0$. By similar arguments as in the proof of Proposition~\\ref{prop.center}, it is easy to verify that $\\mathcal{H}^{\\tilde{f}_0}(Y)$ is a subgroup of $\\mathcal{Z}_{\\mathcal{D}(Y)}f_*(\\mathcal{D}(X))$. \n\nNow, we process as in the proof of \\cite[Theorem~II.6]{gottlieb1} to show:\n\n\\begin{proposition}\\label{pp}Given $f\\colon X \\to Y$, there are inclusions \\[ G^f(Y,f(x_0))\\subseteq \\mathcal{H}^{\\tilde{f}_0}(Y)\\subseteq P^f(Y,f(x_0)). \\]\n\\end{proposition}\n\n\\begin{proof} Certainly, the inclusion $G^f(Y,f(x_0))\\subseteq \\mathcal{H}^{\\tilde{f}_0}(Y)$ is a direct consequence of Proposition~\\ref{prop.center} and Theorem~\\ref{qq}.\n\nNow, let $h\\in \\mathcal{H}^{\\tilde{f}_0}(Y)$ and $\\tilde{H} \\colon \\tilde{X}\\times I\\to \\tilde{Y}$ be a homotopy with $\\tilde{f}_0\\simeq_{\\tilde{H}} h\\tilde{f}_0$. Next, consider the path $\\tilde{\\phi} \\colon I\\to \\tilde{Y}$ defined by $\\tilde{\\phi}=\\tilde{H}(\\tilde{x}_0,-)$, where $p(\\tilde{x}_0)=x_0$. Then, the loop $\\phi=q\\tilde{\\phi}$ corresponds to $h$.\n\nNotice that $\\phi$ acts trivially on $f_\\ast(\\pi_m(X,x_0))$ for $m>1$ if and only if there is a map $F \\colon \\mathbb{S}^m\\times \\mathbb{S}^1\\to Y$ such that the diagram\n\\[ \\xymatrix@C=2cm{\\mathbb{S}^m\\vee \\mathbb{S}^1\\ar@{^{(}->}[d] \\ar[r]^-{f\\alpha\\vee \\phi} & Y \\\\ \n\\mathbb{S}^m\\times \\mathbb{S}^1 \\ar@{-->}[ru]_F &} \\]\ncommutes (up to homotopy) for any $\\alpha\\in \\pi_m(X,x_0)$.\\\n\nGiven $\\alpha\\in \\pi_m(X,x_0)$ with $m>1$, there exists $\\tilde{\\alpha}\\in\\pi_m(\\tilde{X},\\tilde{x}_0)$ such that $p\\tilde{\\alpha}=\\alpha$. Thus, we define a map \\[F' \\colon \\mathbb{S}^m\\times I\\xrightarrow{\\tilde{\\alpha}\\times \\mathrm{id}_I}\\tilde{X}\\times I\\xrightarrow{\\tilde{H}}\\tilde{Y}\\xrightarrow{q}Y.\\]\nBecause $F'(s,0)=q\\tilde{H}(\\tilde{\\alpha}(s),0)=q\\tilde{f}_0(\\tilde{\\alpha}(s))=fp\\tilde{\\alpha}(s)$ and\n$F'(s,1)=q\\tilde{H}(\\tilde{\\alpha}(s),1)=qh\\tilde{f}_0(\\tilde{\\alpha}(s))=q\\tilde{f}_0(\\tilde{\\alpha}(s))=fp\\tilde{\\alpha}(s)$ for $s\\in \\mathbb{S}^m$, the map $F'$\nimplies the required map $F\\colon \\mathbb{S}^m\\times \\mathbb{S}^1\\to Y$. \n\nSince $\\mathcal{H}^{\\tilde{f}_0}(Y)\\subseteq \\mathcal{Z}_{\\mathcal{D}(Y)}f_*(\\mathcal{D}(X))$, we derive that $\\phi$ acts trivially also on $f_\\ast(\\mathcal{D}(X))$. This gives the inclusion $\\mathcal{H}^{\\tilde{f}_0}(Y)\\subseteq P^f(Y,f(x_0))$ and the proof is complete.\n\\end{proof}\n\nLet $H$ be a finite group acting freely on a $(2n+1)$-homotopy sphere $\\Sigma(2n+1)$.\nIf $\\Sigma(2n+1)\/H$ is the corresponding space form then, following \\cite[Chapter~VII, Proposition~10.2]{brown}, the action of $H=\\mathcal{D}(\\Sigma(2n+1)\/H)$ on $\\pi_m(\\Sigma(2n+1)\/H,y_0)$ is trivial for $m>1$. In particular, $H$ acts trivially on $\\pi_{2n+1}(\\Sigma(2n+1)\/H,y_0)\\approx \\pi_{2n+1}(\\Sigma(2n+1),\\tilde{y}_0)$. This implies that for any $h\\in H$, the induced homeomorphism $h_*\\colon \\Sigma(2n+1)\\to \\Sigma(2n+1)$ is homotopic to $\\mathrm{id}_{\\Sigma(2n+1)}$. Consequently, if $f\\colon X\\to \\Sigma(2n+1)\/H$ is a map then $\\mathcal{H}^{\\tilde{f}_0}(\\Sigma(2n+1)\/H)=\\mathcal{Z}_{H}f_*(\\mathcal{D}(X))$. Because $P^f(\\Sigma(2n+1)\/H,f(x_0))\\subseteq \\mathcal{Z}_{H}f_*(\\mathcal{D}(X))$, Proposition~\\ref{pp} yields \\[\\mathcal{H}^{\\tilde{f}_0}(\\Sigma(2n+1)\/H)=P^f(\\Sigma(2n+1)\/H,f(x_0))=\\mathcal{Z}_{H}f_*(\\mathcal{D}(X)). \\]\n\nFurther, the result \\cite[Theorem~1.17]{gol}, Theorem~\\ref{qq} and Proposition~\\ref{pp} lead to:\n\n\n\\begin{corollary}\nIf $f\\colon X\\to\\Sigma(2n+1)\/H $ is a map as in \\cite[Theorem~1.14]{gol} then $\\mathcal{D}^{\\tilde{f}_0}(\\Sigma(2n+1)\/H)=\\mathcal{H}^{\\tilde{f}_0}(\\Sigma(2n+1)\/H)=\\mathcal{Z}_Hf_*(\\mathcal{D}(X))$. In particular, $J(f,x_0)=\\mathcal{Z}_Hf_*(H)$ for any self-map $f\\colon \\Sigma(2n+1)\/H\\to \\Sigma(2n+1)\/H$.\n\\end{corollary}\n\nGiven a free action of a finite group $H$ on $\\mathbb{S}^{2n+1}$, Oprea \\cite[\\textsc{Theorem~A}]{oprea} has shown that $G(\\mathbb{S}^{2n+1}\/H,y_0)=\\mathcal{Z}H$, the center of $H$.\nIn the special case of a free linear action of $H$ on $\\mathbb{S}^{2n+1}$, the description of $G(\\mathbb{S}^{2n+1}\/H,y_0)$ via deck transformations presented in \\cite[Theorem~II.1]{gottlieb1} has been applied in \\cite{bro} to get a very nice representation-theoretic proof of \\cite[\\textsc{Theorem~A}]{oprea}. The result stated in Theorem~\\ref{qq} might be applied to extend the methods from \\cite{bro} to simplify the proof of \\cite[Theorem~1.17]{gol} on the case of a free linear action of $H$ on $\\mathbb{S}^{2n+1}$ as well.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe solar wind is a plasma flow, electrically neutral, which emanates from the basis of the solar corona, permeating and shaping the whole heliosphere. During solar activity minima, when the meridional branches of the polar coronal holes (CH hereafter) reach the equatorial regions of the sun, an observer located in the ecliptic plane would record a repeated occurrence of fast (700-800 km\/s) and slow (300-400 km\/s) wind samples. The balance between the two classes of wind would change during the evolution of the 11-year solar cycle depending on changes in the topology of the heliomagnetic equator and its inclination on the ecliptic plane. The fast wind originates from unipolar open field line regions, typical of the coronal holes \\citep{Schatten69, Hassler99}. The slow wind origin is much more uncertain, although we know this class of wind is generated within regions mainly characterized by closed field lines configuration that likely inhibit the escape of the wind \\citep{Wang90, Antonucci2005, Bavassano97, Wu2000, Hick99}. Indeed, the different composition and mass flux of the slow wind and the different degree of elemental fractionation with respect to the corresponding photospheric regions, strongly suggest that such slow plasma flow could be initially magnetically trapped and then released \\citep{Geiss95a, Geiss95b}. Interchange reconnection process plays a fundamental role in opening up part of the closed field lines linked to the convective cells at photospheric level \\citep{Fisk99, Fisk2001, Schwadron2002, Schwadron2005, Wu2000, Fisk2020}. This process should preferentially develop close to the border of the CH, influencing as well the neighbour regions, i.e. the coronal hole boundary layer (hereafter CHBL), and the closed loop corona. It has been empirically proven that the expansion factor of the magnetic field line configuration is anti-correlated to the wind speed \\citep{Wang90}. A smaller expansion factor, typical of the central region of the coronal hole, would therefore produce a faster wind while larger expansion factor, typical of the border of the coronal hole, would produce instead a slower wind \\citep{Wang90, Wang2006, D'Amicis2015}. Several other contributions to the slow wind come from more specific processes like, e.g. the release of plasmoids by helmet streamers or the continuous leakage of plasma from the highest regions of the large coronal loops where magnetic pressure is no longer largely dominating the thermal pressure \\citep[among others]{Bavassano97, Hick99, Antonucci2005}. \nDuring the wind expansion, magnetic field lines from coronal holes of opposite polarities are stretched into the heliosphere by fast wind streams being separated by the heliospheric current sheet, an ideal plane, magnetically neutral, permeated by slow wind plasma. During the expansion into the interplanetary space, because of solar rotation, the high speed plasma would overtake and impact on any slower plasma ahead of it creating a compression region \\citep{Hundhausen1972}.\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=7cm]{Fig1_Stream_sketc.png}\n\t\\caption{Sketch of a stream structure in the ecliptic plane, adapted from Hundhausen 1972. The spiral structure is a consequence of solar rotation. The spiral inclination changes as the solar wind velocity changes. When the high-speed stream (in blue) compresses the slower ambient solar wind (in black), a compression region downstream (in red) and a rarefaction region upstream (in pink) are formed.}\n\t\\label{fig01}\n\\end{figure}\nAs a consequence of this interaction, a so called co-rotating interaction region (CIR), characterized by a rather rapid increase in the wind speed, from values typical of the slow wind (300-400 km\/s) to values typical of fast wind (700-800 km\/s), forms at the stream-stream interface. The CIR is characterized by strong compressive phenomena (red zone in Figure \\ref{fig01}) affecting both magnetic field and plasma \\citep{Richardson2018}. The CIR is followed by a region where the wind speed persist at its highest values (blue zone in Figure \\ref{fig01}), resembling a sort of plateau, while the other plasma and magnetic field parameters rapidly decrease to remain rather stable across its extension. Beyond this region, highlighted at times by a a sharp knee in the radial velocity, the wind speed decreases monotonically to reach values typical of the following slow wind (black zone in Figure \\ref{fig01}), which permeates the interplanetary current sheet. This decreasing wind speed region is slightly more rarefied than the high speed region which immediately follows the CIR and is commonly called rarefaction region (pink zone in Figure \\ref{fig01}). The three regions described above can be considered the imprint in the interplanetary medium of the coronal structure from which the wind originated \\citep{Hundhausen1972}. The fast wind plateau corresponds to the core of the CH while CIR and rarefaction region correspond to the same CHBL encircling the CH, though within the CIR the CHBL is compressed by the dynamical interaction described earlier between fast and slow wind while, within the rarefaction region the CHBL is stretched by the wind expansion into the interplanetary medium.\nBeyond a few solar radii, the solar wind (both fast and slow) becomes supersonic and super-Alfv\\'enic. Fluctuations of interplanetary magnetic field and plasma parameters start to develop their turbulent character since the first tens of solar radii \\citep{Kasper2019} and, in particular, turbulence features within fast and slow wind differ dramatically \\citep[and references therein]{BrunoCarbone2013}. Fast wind is rather uncompressive and mostly characterized by strong Alfv\\'enic correlations between velocity and magnetic field fluctuations, though carrying also compressive fast and slow magnetosonic modes \\citep{Marsch90,Marsch93,Klein2012, Howes2012, Verscharen2019}. The origin of Alfv\\'enic and compressive fluctuations is rather different. The former appear to be generated at large scales through the shuffling of the magnetic field lines foot-points (by photospheric motion \\citep{Hollweg2006}) and at small scales by magnetic reconnection processes together with fast\/slow magnetosonic modes \\citep{Parker57, Lazarian2015, Kigure2010, Cranmer2018}. On the other hand, following \\cite{Marsch93} and \\cite{Tu94} compressive modes may also originate during the wind expansion from a superposition of pressure balanced structures of solar origin and magnetoacoustic fluctuations generated by the dynamical interaction of contiguous flux-tubes. Uncompressive and compressive fluctuations eventually experience a non-linear cascade to smaller and smaller scales.\nAlfv\\'{e}nicity in the slow wind is in general much lower with compressive effects affecting more the plasma dynamics and reflecting the complex magnetic and plasma structure of its source regions, close to the heliomagnetic equator\n\nThis paper deals with a peculiar phenomenon that we observed in the Alfv\\'{e}nic turbulence when passing from the fast wind plateau to the rarefaction region. The border between these two regions is often marked by a rather sharp knee in the velocity profile. As detailed in the following, We noticed that, while downstream the knee, fluctuations have large amplitude and are highly Alfv\\'enic, in the upstream region, fluctuations dramatically reduce their amplitude and the Alfv\u00e9nic correlation weakens, beginning to fluctuate between positive and negative values. Interesting enough, the region where the remarkable depletion of these fluctuations takes place can be very short in time, of the order of tens of minutes, and is located around the velocity knee.\nTo our knowledge, no specific analyses, before the present work, have ever been devoted to understand the reason of such a sudden change of the Alfv\\'enic turbulence across the velocity knee. \\par \nAs we will explain in the following section, after analyzing the turbulent character of the stream, we first looked for the presence of a tangential discontinuity between the fast wind region and the rarefaction region. Then, we made a composition analysis of the minor ions of the solar wind. Finally, we suggested a mechanism that we believe may explain the rapid depletion of fluctuations and Alfv\u00e9nicity observed at the beginning of the rarefaction region.\n\\section{Data analysis}\n\\label{sec:data}\n\nWe analyzed 3 second averages of magnetic field and plasma observed at 1AU from the WIND spacecraft and 2 hour averages minor ions parameters from the ACE spacecraft during the last minimum of solar activity. The data refers to the Heliocentric Earth Ecliptic (HEE) system, which has an X axis pointing along the Sun-Earth line, and a Z axis pointing along the ecliptic north pole \\citep{Thompson2006}. In particular, after a visual inspection of several co-rotating high speed streams towards the end of solar cycle 24, we focused on a series of wind streams in which the phenomenon described above appears, though for sake of brevity here we report results related to the high speed stream observed on 3-9 August 2017, representative of those streams showing a similar fluctuations reduction phenomenon. We also show, for comparison, a case in which the reduction in fluctuations and Alfv\u00e9nicity occurs more slowly, without being able to identify a clear velocity knee in this case.\nSince the magnetic field and plasma data collected by WIND are not provided with synchronized time stamps, we re-sampled the time series with a 6 second cadence in order to build up a single merged data-set with a uniform time base to be used to characterize the statistics of the turbulent fluctuations of the solar wind parameters under study. \nFor the minor ions analysis we were forced to employ the 2 hour cadence being the observations with the highest time resolution available during the selected streams.\n\nTo characterize the turbulence in the solar wind we built the Els\\\"asser variables $ \\bf{z^\\pm} = \\bf{v} \\pm \\bf{b} $ \\citep{Elsasser1950}, where $\\bf{v}$ is the velocity vector and $\\bf{b}$ is the magnetic field vector expressed in Alfv\\'en units \\citep{Tu95, BrunoCarbone2013}. \n\nThe second order moments linked to Els\\\"asser variables are \\citep{BrunoCarbone2013} :\n\\begin{subequations}\n\\begin{align}\n e^{\\pm} &=\\frac{1}{2} \\langle \\left(\\bf{z^{\\pm}} \\right)^2 \\rangle & \\: (energies\\; related\\; to\\; \\mathbf{z^+} \\;and\\; \\mathbf{z^-})\\\\\n e_{v}&=\\frac{1}{2} \\langle v^2 \\rangle & (kinetic\\; energy)\\\\\n e_{b}&=\\frac{1}{2} \\langle b^2 \\rangle & (magnetic\\; energy)\\\\\n e_c &= \\frac{1}{2} \\langle \\bf{v} \\cdot \\bf{b} \\rangle & (cross-helicity)\n\\end{align}\n\\end{subequations}\nwhere angular brackets indicate averaging process over the established time range. In our case $e^{\\pm}$, $e_{v}$, $e_{b}$ and $e_{c}$ represent the variances calculated on hourly moving window for the whole stream.\nIn order to describe the degree of correlation between v and b, it is convenient to use normalized quantities:\\\\\n\\begin{subequations}\n\\begin{align}\n\\sigma_c &= \\frac{e^+ - e^-}{e^+ + e^-} & \\;\\; (normalized \\;cross-helicity) \n\\label{def_sigma_c} \\\\\n\\sigma_r &= \\frac{e_v - e_b}{e_v + e_b} & \\;\\; (normalized \\;residual \\; energy) \\label{def_sigma_r}\n\\end{align}\n\\end{subequations}\nWhere $-1 \\leq \\sigma_c \\leq 1 $ and $-1 \\leq \\sigma_r \\leq 1 $.\n\n\\vspace{7mm}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura2_A17_Stream.png}\n\t\\caption{Temporal profiles of solar wind parameters from $3^{th}$ to $9^{th}$ August 2017, at 6 seconds time resolution. Panels show from top to bottom: proton velocity components in HEE (Heliocentric Earth Ecliptic) reference system, respectively $v_x$ (panel a), $v_y$ (panel b), $v_z$ (panel c); proton number density (d); proton temperature (e); magnetic field magnitude (f); magnetic field components in HEE, respectively $B_x$ (panel g), $B_y$ (panel h), $B_z$ (panel i); angle between the average magnetic field direction and the radial direction (j). In the latest panel the black dashed curve represent the local Parker spiral direction, referred to the high-speed region and rarefaction region. The red dashed line in each panel denotes the velocity knee between fast wind region and rarefaction region. The green dashed line in each panel denotes the location of the Tangential Discontinuity found at almost the end of the rarefaction region (see section \\ref{sec:data}). On the higher part of the figure (outside the panels) are indicated the different solar wind region, vertically interpreted, with color-correspondence to sketch of Figure \\ref{fig01}.}\n\t\\label{fig02}\n\\end{figure}\nThe multiple panels of Figure \\ref{fig02} show some of the plasma and magnetic field features characterizing the particular stream we chose observed by WIND on August 2017. From top to bottom we show the time profiles of the three velocity components $v_x$, $v_y$ and $v_z$ (panels a to c), proton number density (panel d), proton temperature (panel e), magnetic field intensity (panel f), magnetic field components $B_x$, $B_y$ and $B_z$ (panels g to i) and, at the bottom (panel j), the angle $ \\theta = cos^{-1} \\left(|B_{x}| \/ |B| \\right) $ (limited between $0$ and $90^{\\circ}$) between the local field and the radial direction, respectively. This is a canonical corotating high-speed stream in which it can be easily recognized an interaction region, roughly occurring around noon of August the $4^{th}$ and characterized by strong magnetic field intensity and plasma number density enhancements followed by a clear proton temperature increase, as expected for a compressive region, because of the dynamical interaction of this stream with the slow wind ahead of it \\citep{BrunoCarbone2013}. This interaction region, within which the wind speed rapidly increases from about 400 km\/s to about 700 km\/s, is followed by a speed plateau about two days long. The latter is characterized by fluctuations in velocity and magnetic field components of remarkable amplitude and rather constant values of number density \\begin{math} { n \\sim 3 \\; cm^{-3} } \\end{math} and field intensity \\begin{math}{|B| \\sim 5 \\; nT }\\end{math}. In addition, the temperature profile (panel e), although lower than that within the preceding interaction region (CIR), is much higher than in the surrounding low speed regions. The high speed plateau is then followed by a slowly decreasing wind speed profile identifying the rarefaction region of this corotating stream. These features are typical for most of the high speed corotating streams we examined. What is less common is the abrupt depletion of large amplitude fluctuations detected in the high speed plateau, across the velocity knee (defined later, when describing Figure \\ref{figura5}) indicated in Figure \\ref{fig02} by the vertical dashed red line around day 218 (6 August) at noon. This phenomenon is highlighted by the profile of the curve in the bottom panel j related to the angle $\\theta$ between the magnetic field vector and the radial direction. The difference in the behavior of this parameter before and after the velocity knee is dramatic and, as shown later on, this transition is rather fast, of the order of minutes.\nLarge angular fluctuations between the magnetic field vector and the radial direction are due to the presence of large amplitude, uncompressive Alfv\\'{e}nic fluctuations populating the high speed plateau that force the tip of the magnetic field vector to randomly move on the surface of a hemisphere centered around the background mean field direction \\citep[e.g. see figure 4 in ][]{Bruno2001}.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura3_M17.png}\n\t\\caption{Temporal profiles of solar wind parameters from $28^{th}$ February to $15^{th}$ March 2017, at 6 seconds time resolution. Panels show from top to bottom: proton velocity components in HEE (Heliocentric Earth Ecliptic) reference system, respectively $v_x$ (panel a), $v_y$ (panel b), $v_z$ (panel c); proton number density (d); proton temperature (e); magnetic field magnitude (f); magnetic field components in HEE, respectively $B_x$ (panel g), $B_y$ (panel h), $B_z$ (panel i); angle between the average magnetic field direction and the radial direction (j). In the latest panel the black dashed curve represent the local Parker spiral direction referred to the high-speed region and rarefaction region.}\n\t\\label{fig03}\n\\end{figure}\n\nOn the other hand, not all streams have the same abrupt depletion of fluctuations after the fast wind plateau; an example is the stream of March 2017 shown in Figure \\ref{fig03}. In the multiple panels there are: velocity components $v_x$, $v_y$ and $v_z$ (panels a to c), proton density (panel d), proton temperature (panel e), magnetic field magnitude (panel f), magnetic field components $B_x$, $B_y$ and $B_z$ (panels g to i) and the angle $\\theta$ (panel j) between the local field and the radial direction. We can clearly identify the interaction region characterized by a compression (enhancement of proton density in panel d), followed by an increase in magnetic field, proton temperature and velocity of the solar wind, typical of high speed corotating streams.\nImmediately after, there are wide fluctuations in velocity and magnetic field components in correspondence of the fast wind plateau of this stream, that has a longer duration with respect to the previous stream. In this case, unlike the previous one, even though there is a decrease in velocity and magnetic fluctuations, it occurs gradually over time (days). This gradual decreasing trend is also present in the temperature profile (panel e). Moreover, the angular fluctuations between the local field and the radial direction (panel j) change rapidly over time, without having a clear trend in the rarefaction region, unlike the previous stream. Hence the magnetic field vector fluctuates continuously around the radial direction for almost the entire duration of this second stream.\n\nComparing the time profiles of the angle between the average direction of the magnetic field with respect to the radial direction (panels h of Figures \\ref{fig02} and \\ref{fig03}), we note that in correspondence of the fast wind region of both August (Figure \\ref{fig02} h) and March (Figure \\ref{fig03} h) streams, most of the values are above the black solid curve, which indicates the predicted Parker spiral angle based on the wind speed, computed as $\\Psi = \\arctan(\\Omega R \/ |v|)$ where R is the distance from the Sun to the Earth and $\\Omega$ is the sidereal angular velocity of the Sun at the equator, i.e. $2.97 \\cdot 10^{-6} rad\/s$.\nAs a consequence, before the velocity knee, the background magnetic field is over-wound \\citep{BrunoBavassano1997}, most likely as a consequence of the dynamical interaction of the fast stream with the slow wind ahead \\citep{Schwenn90}. Within the rarefaction region, the time profile of this angle for the August stream shows a smaller variability with an average direction closer to the radial one and smaller than the local Parker spiral direction, as expected for a rarefaction region \\citep{SchwadronMcComas2005}. This region maps in the interplanetary space the coronal hole boundary layer (CHBL) encircling the fast wind source region at the sun \\citep{McComas2003}. The solar wind observed in space and emanating from this source region, because of the solar rotation, is characterized by compressive phenomena when detected ahead of the high velocity stream and rarefaction phenomena when it is observed following the high speed plateau. The existence of this sub-Parker spiral orientation of the background magnetic field depends on the rate of the motion of open magnetic field foot points across the coronal hole boundary at photospheric level \\citep{SchwadronMcComas2005}. Such motion is responsible for connecting and stretching magnetic field lines from fast and slow wind sources across the CHBL, as explained in the model by \\citep{SchwadronMcComas2005}.\n\nIt is interesting to remark how different are the profiles of $\\theta$ for these two streams within the rarefaction regions. For the stream of August, the $B_x$ component is generally positive and the profile of $\\theta$ is largely confined to values below $45 ^{\\circ} $ with sporadic jumps to larger values. For the March stream, the $B_x$ profile is much more structured and $\\theta$ continuously fluctuates between $0^{\\circ}$ and $90^{\\circ}$. Moreover, for this last stream, $B_x$ continuously jumps from positive to negative values suggesting that we must be close to the heliomagnetic equator.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9.2cm]{Fig4_Backprojection_A17_M17.png}\n\t\\caption{Source Surface Synoptic Charts at $3.25 R_s$ related to July-August 2017 on the left side and February-March 2017 on the right side (Source: Wilcox Solar Observatory).The level curves indicate values of constant magnetic field (at $3.25 R_s$), the black line represents the heliomagnetic equator (current sheet), the two different shades of gray indicate the magnetic field polarity: dark gray and red lines correspond to negative polarity while light gray and blue lines correspond to positive polarity. The orange dashed line represent the Earth trajectory projected to the Solar surface at $3.25 R_s$ . The different purple lines correspond to the periods of the studied streams. In the bottom two panels are shown the velocity profiles of August 2017 stream (on the left) and March 2017 stream (on the right).}\n\t\\label{figura4}\n\\end{figure}\nAs a matter of fact, in our analysis we have observed many cases like the one (August 2017) described, on the other hand we also observed several other cases where the disappearance of the Alfv\\'enic fluctuations is not characterized by a sudden event like the one observed here on August the $6^{th}$. Individual streams can be different from each other and one cannot simply infer their turbulence properties without proper analysis.\nAs observed by \\cite{Shi2021}, structures such as heliospheric current sheets can play an important role in modifying the properties of turbulence in the solar wind, also supported by 2D MHD simulations of Alfv\u00e9nic waves on a current sheet \\citep{Malara1996}, which showed that initially high values of $\\sigma_c$ are quickly destroyed near a current sheet.\nWe observed that the sudden depletion of the Alfv\\'{e}nic fluctuations beyond the velocity knee might be related to the pitch angle with which an observer, in the ecliptic plane, crosses the heliomagnetic equator. \nIn the upper panel of the left-hand-side of Figure \\ref{figura4} we show the Source Surface Synoptic Chart relative to July-August 2017 taken from Wilcox Solar Observatory. The level curves indicate values of constant magnetic field (at 3:25Rs), the black line represents the heliomagnetic equator, which runs in the middle of the heliospheric current sheet. The two different shades of gray indicate the magnetic field polarity: dark gray and red lines correspond to negative polarity while light gray and blue lines correspond to positive polarity. The orange dashed line represent the Earth trajectory projected to the Solar surface at 3:25Rs. The purple lines indicate the beginning and the end of the high speed stream studied in this work, whose speed profile is shown in the lower panel. In this case (4-8 August 2017), the heliospheric current sheet is highly inclined with respect to the trajectory of the observer, i.e. WIND s\/c. In the right-hand-side of this Figure, we show, in the same format as of the left-hand-side, one of those cases for which we do not see a clear and abrupt depletion of the fluctuations. This is a high speed stream observed on March 2017. In this case, the speed profile shows a less clear velocity knee and a slow and progressive disappearance of the fluctuations without any abrupt event like the one observed in the previous stream. The main difference, in this case, is that the heliospheric current sheet is very flat and the observer experiences a sort of surfing along this structure instead of a clear and rapid crossing. \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura5_completa_A17.png}\n\t\\caption{Fast Wind region and rarefaction region of the August 2017 stream. From top to bottom: solar wind speed at 6 seconds time resolution (a), values, computed on 1 hour moving window, of kinetic energy in semi-logarithmic scale (b), magnetic energy in semi-logarithmic scale (c), density compressibility in semi-logarithmic scale (d), compressibility of the magnetic field magnitude in semi-logarithmic scale (e), normalized cross-helicity (f) and normalized residual-energy (g). The bottom panel (h) shows an enlargement of the total pressure (yellow), magnetic pressure(orange) and thermal pressure (blue) trend in a small region of few hours close to the velocity knee, the latter represented by the red dashed line in each panel. The green dashed line, in panels from (a) to (g), denotes the location of the tangential discontinuity (TD) founded at almost the end of the rarefaction region (see section \\ref{sec:data}).}\n\t\\label{figura5}\n\\end{figure}\n\nAs anticipated, the high speed plateau of the stream under study is characterized by large amplitude Alfv\\'{e}nic fluctuations as shown in Figure \\ref{figura5}. The top panel (a) shows part of the speed profile of our corotating high speed stream. The location of the velocity knee, beyond which large amplitude fluctuations seems to turn off, is indicated by the vertical dashed red line and determined as follows. Panels b and c show hourly values of kinetic and magnetic energy, respectively, on a semi-logarithmic vertical scale.\nThe sharp variation in these two parameters, at the end of the speed plateau, identifies what we have defined as velocity knee. They highlight remarkably the different physical situation before and after the velocity knee and the abrupt change across it. We notice that\nkinetic energy is the parameter that experiences the largest decrease, of a factor $\\sim 20$. As a consequence, fluctuations within this region become strongly magnetically dominated. In addition, this decrease happens in correspondence of a strong temperature reduction as shown in the previous Figure \\ref{fig02} by the thermal energy parameter. This strong depletion of proton temperature reflects the thermal pressure decrease (Figure \\ref{figura5} h) that starts at the velocity knee together with a corresponding increase of the magnetic pressure such to maintain the wind in a pressure balanced status, achieved at the edge of the Alfv\\'{e}nic surface during the wind initial expansion \\citep{SchwadronMcComas2005}. Figure \\ref{figura5} (d) shows the density compressibility over time in a semi-logarithmic vertical scale, computed as ${\\sigma_{n}}^2 \/ n^{2}$; its value remains quite low (on average $3 \\cdot 10^{-3}$) both during the fast wind region and the rarefaction region, indicative of a fairly incompressible wind, given that the density remains approximately constant in both regions.\nOn the other hand, the compressibility of magnetic field magnitude, computed as ${\\sigma_{B}}^2 \/ |B|^{2}$ and shown in panel \\ref{figura5} (e) on a semi-logarithmic vertical scale, denotes that magnetic fluctuations, in modulus, are more compressive in correspondence of the fast wind plateau rather than in the rarefaction region; there is a sudden reduction at the velocity knee of about an order of magnitude. The two other panels, f and g, highlight the Alfv\\'{e}nic nature of these fluctuations within the high speed plateau. Panel f shows high values of the normalized cross helicity $\\sigma_C$ while panel g shows that the normalized residual energy, although dominated by magnetic energy, is not far from the equipartition expected for Alfv\\'enic fluctuations. Beyond the velocity knee, these two parameters experience a considerable change towards a much lower Alfv\\'enicity although the transition is not as abrupt as the one relative to the amplitude of these fluctuations, as discussed above. \\\\\nThe behaviour of $\\sigma_C$ is similar to that recently observed in the SW from Parker Solar Probe near perihelions of the orbits $(R \\le 0.3 - 0.4 \\;AU) $, as described by \\cite{Shi2021}, where $\\sigma_c$ is usually close to $1$, implying a status of dominating outward-propagating Alfv\u00e9n waves. There are also periods when $\\sigma_C$ oscillates and becomes negative; these periods correspond to Parker Solar Probe observing heliospheric large-scale inhomogeneous structures, such as the heliospheric current sheet.\\\\\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9.4cm]{Figura6_M17.png}\n\t\\caption{Fast Wind region and rarefaction region of the March 2017 stream. From top to bottom: solar wind speed at 6 seconds time resolution (a), values computed on 1 hour moving window, of kinetic energy in semi-logarithmic scale (b), magnetic energy in semi-logarithmic scale (c), density compressibility in semi-logarithmic scale (d), compressibility of the magnetic field magnitude in semi-logarithmic scale (e), normalized cross-helicity (f) and normalized residual-energy (g).}\n\t\\label{figura6}\n\\end{figure}\nThe same analysis of Figure \\ref{figura5} for the August 2017 stream was also carried out for the March 2017 stream and shown in Figure \\ref{figura6}. In this case the decrease in fluctuations and Alfv\u00e9nicity occurs very gradually over time (few days). The velocity profile (panel a) of the March 2017 stream does not allow us to define a velocity knee in this case. Furthermore, the magnetic energy (panel b) and kinetic energy (panel c) decrease more slowly in the rarefaction region of the stream, with respect to the previous one. The Alfv\u00e9nic character of the fast wind remains evident, as shown by the profile of $\\sigma_c$ in panel e and $\\sigma_r$ in panel f. Even in this case the density compressibility (panel d) oscillates around very low values, of the order of $10^{-3}$, while in this stream the compressibility of magnetic field magnitude (panel e) does not show an effective decrease after the velocity plateau, unlike the previous case (August 2017 stream).\\\\\nThe MHD simulations of an ensemble of Alfv\u00e9nic fluctuations propagating in an expanding solar wind including the presence of fast and slow solar wind streams, made by \\cite{Shi2020}, showed that the decrease in $\\sigma_c$ is more significant in the compression and rarefaction regions of the stream (as we observed in Figure \\ref{figura5}) than within the fast and slow streams. In particular, in the rarefaction region of such simulations $\\sigma_c$ rapidly decreases from 1 to 0.6 at $ R \\approx 80 R_s $ and remains around this value until the end of the simulation. While leaving open the hypothesis that the phenomenon under study may develop during the radial expansion of the solar wind, the very rapid (few minutes) reduction of the fluctuations and Alfv\u00e9nicity between the fast wind plateau and the rarefaction region makes us lean towards the hypothesis that this phenomenon should originate at the source regions of the solar wind. \\\\\nThe sudden changes in some of the parameters described above for the August stream, between the end of the fast wind plateau and the beginning of the rarefaction region, suggested to look for magnetic structures like tangential discontinuities (TD, hereafter), which might represent a barrier for the propagation of the Alfv\\'enic fluctuations. The presence of a TD, where by definition there is no magnetic component normal to the discontinuity surface, could explain the depletion of Alfv\u00e9n waves, that propagate along the magnetic field direction \\citep{Hundhausen1972}. In fact, these fluctuations would not propagate across this kind of discontinuity since the magnetic component perpendicular to the discontinuity plane vanishes. This would prevent their propagation into the CHBL or rarefaction region and would produce the observed sudden depletion of the Alfv\\'enic fluctuations. \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura7.png}\n\t\\caption{Top three panels: time profile of velocity components (blue lines) and magnetic field components in Alfv\u00e9n units (black lines) at 6 seconds (panels a, b, c), in about an 8 hr time interval. Bottom panel (d): 5 minute values of magnetic $e_b$ (black line) and kinetic $e_v$ (blue line) energies within a narrower time interval of almost 5 hours around the estimated velocity knee (red dashed line).}\n\t\\label{figura7}\n\\end{figure}\n\nTo better determine the starting time of the depletion of magnetic and kinetic fluctuations, we show in Figure \\ref{figura7} the time profile of 6 sec averages of magnetic field components, expressed in Alfv\\'en units, and velocity components within a short time interval, slightly longer than 8 hrs and roughly centered around the previously estimated time location of the velocity knee. There is a clear change in the amplitude of the fluctuations around the location of the dashed line. This phenomenon is highlighted by the time profile of magnetic and kinetic energies, $e_b$ and $e_v$, respectively, computed within 5 min running window, shown in the bottom panel (panel d). The sharp profile of these curves allows to better estimate the time when fluctuations are abrupt reduced, that is on day $218^d13^h24^m$ (i.e. 6 August).\nHowever, we did not find any relevant TD around the estimated time, which could be associated with the observed phenomenon. Thus TD can not be a valid explanation of the rapid decrease of the Alfv\u00e9nic fluctuations at the velocity knee, leaving open the possibility that this phenomenon could be due to a different evolution of the solar wind during its expansion or to a possible reconnection mechanism at the base of the Solar Corona, as we will explain in details in the following..\nOn the other hand we point out the presence of a TD two days later the velocity knee, on August the $8^{th}$ at $14^h34^m$, easily recognizable in Figure \\ref{fig02} by looking at the switchback in the $B_x$ component and highlighted in Figure \\ref{figura8}. \n\n \\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura8_minvar.png}\n\t\\caption{Left-hand-side: panels from top to bottom show the three components of magnetic field (panels (a), (b), (c))and velocity (panels (d), (e), (f)), rotated in the magnetic field minimum variance reference system, and the total pressure (panel g). The 3D figures in the right-hand-side panels show magnetic field (panel h) and velocity components (panel i) rotated in the field minimum variance reference system. All these quantities are plotted with data at 6 sec time resolution. The green dashed line, in panels from (a) to (g), denotes the location of the tangential discontinuity (TD) founded at almost the end of the rarefaction region (see section \\ref{sec:data}).}\n\t\\label{figura8}\n\\end{figure}\nIn the latter, panels on the left hand side show (from top to bottom): magnetic field (panels a , b and c) and velocity components (panels d, e and f), and total pressure (panel g). Magnetic field and velocity components are rotated in the magnetic field minimum variance reference system, as better shown in the 3D plots in the right hand side panels. A TD discontinuity is an MHD structure for which the total pressure is conserved, as shown in Figure \\ref{figura8} panel (g), the component of the magnetic field normal to the discontinuity plane is identically zero on both sides (Figure \\ref{figura8} panel c) and number density, thermal pressure and tangential component of the magnetic field vector can be discontinuous across the layer. In this particular TD, the magnetic field rotates in less than 1 minute in the minimum variance plane shown on the right-hand-side panel and characterized by robust eigenvalues ratios: $\\Lambda_1 \/ \\Lambda_3 = 0.015 $ and $\\Lambda_2 \/ \\Lambda_3 = 0.10 $.\n\n\n\nTotal pressure remains roughly constant across the discontinuity and there is no relevant mass flow across the latter, because the velocity component normal to the discontinuity plane is quite constant throughout the whole time interval of Figure \\ref{figura8}, indicating that it is at rest in the plasma reference frame. Looking back at Figure \\ref{fig02}, magnetic field and plasma parameters rapidly become more variable upstream the TD, approaching the slow wind. This TD seems to mark the physical border between the rarefaction region and the surrounding slow wind. In other words, it could be the coronal discontinuity (CD) observed by \\cite{McComas98} in Ulysses data during March 1995, when Ulysses was rapidly crossing the low-latitudinal region of the solar wind towards the fast polar wind flow and modeled by \\cite{SchwadronMcComas2005}. The relevant difference would be that this is the first time this particular structure is recognized between the rarefaction region of an high speed stream and the following slow solar wind.\n\nMinor ions components of the solar wind carry relevant information about its source region, i.e. the CHBL. This is the source of a sort of transition region between the fast coronal wind and the slow wind. As a matter of fact, minor ions parameters like freezing-in temperature and relevance of low-FIP elements \\citep{Geiss95a, Geiss95b, McComas2003} monotonically increase from fast to slow wind.\nAt this point we searched for possible signatures in plasma composition and charge state assuming the rapid depletion of fluctuations and Alfv\u00e9nicity might have been due to differences in the source regions (at the basis of the solar Corona) of the stream high speed plateau and the subsequent rarefaction region. As a matter of fact, fast and slow wind largely differ for ion composition, charge state and freezing-in temperature (\\cite{Bochsler IoniMinori}) and the region in between, i.e. the rarefaction region, might be a sort of wind largely mixed with the slower wind, and this would be highlighted by the composition analysis. \\cite{Geiss95a} and \\cite{Phillips95}, based on Ulysses observations, found several dramatic differences in ions composition between fast and slow wind. One of the most relevant is the strong bias of the slow wind in favor of low FIP (first ionization potential) elements with respect to the fast wind. \\cite{Geiss95b}, using charge state ratios for C6+\/C5+ and O7+\/O6+ estimated the corresponding freezing-in temperatures for these elements and found that slow wind sources at the sun are quite hotter than fast wind sources. Finally, fast and slow wind greatly differ also for the higher abundance of He++ within fast and hot wind with respect to slow and cold wind \\citep{Kasper2012}. \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura9_ioni_A17.png}\n\t\\caption{Temporal profiles of ions compositions during the August 2017 stream. In the top panel(a): Alpha particles over protons ratio $ {He^{++}}\/{p+}$ at 6 sec time resolution (in blue) and its hourly moving average (in red). Second panel b: charge state ratio, at 2 hours time resolution, of ${C^{+6}}\/{C^{+5}}$ in blue, ${O^{+7}}\/{O^{+6}}$ in red and ${O^{+8}}\/{O^{+6}}$ in yellow. Third panel c: iron average charge state, at 2 hours time resolution. Bottom panel d: iron over Oxygen abundance ratio, at 2 hours time resolution. The red dashed line denotes the velocity knee between fast wind region and rarefaction region.The green dashed line denotes the location of the tangential discontinuity (TD) founded at almost the end of the rarefaction region (see section \\ref{sec:data}). }\n\t\\label{figura9}\n\\end{figure}\nThe four panels of Figure \\ref{figura9} show, from top to bottom, the relative abundance of He++\/p+ (panel a), some charge state ratios for carbon and oxygen (panel b), charge state for iron (panel c) and relative abundance of iron over oxygen (panel d) respectively. This figure (\\ref{figura9}) refers to the entire stream, i.e. the compressive region, followed by the high-speed plateau, followed by the rarefaction region and finally by the slow wind (for comparison keep the Figure \\ref{fig02} in mind). As expected, the last three panels of Figure \\ref{figura9} show a gradual increase from the fast wind plateau to the rarefaction region, but without a sharp jump across the velocity knee. \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9cm]{Figura10_ioni_M17.png}\n\t\\caption{Temporal profiles of ions compositions during the March 2017 stream. In the top panel(a): Alpha particles over protons ratio $ {He^{++}}\/{p+}$ at 64 sec time resolution (in blue) and its hourly moving average (in red). Second panel(b): charge state ratio, at 2 hours time resolution, of ${C^{+6}}\/{C^{+5}}$ in blue, ${O^{+7}}\/{O^{+6}}$ in red and ${O^{+8}}\/{O^{+6}}$ in yellow. Third panel(c): iron average charge state, at 2 hours time resolution. Bottom panel(d): iron over Oxygen abundance ratio, at 2 hours time resolution.}\n\t\\label{figura10}\n\\end{figure}\nA very similar result can be seen in Figure \\ref{figura10} that is the equivalent result of minor ions analysis of Figure \\ref{figura9}, but referred to entire March 2017 stream. Looking at the minor ions analysis (both Figure \\ref{figura9} and \\ref{figura10}) we can clearly see that there are no jumps in the quantities in correspondence of the end of the fast wind plateau, nor are there large differences between the two streams of August and March.\nAs a matter of fact, although the time resolution of minor ions parameters provided by ACE related to the selected time period is not very high to allow to estimate small scale structures, but it's enough to show that there are not strong discontinuities in ions compositions at the location of the velocity knee such to justify, in terms of differences in the source region, abrupt changes in the fluctuations. The top panel shows the He++\/p+ ratio and, although higher values for this parameter are found within the high speed plateau, the transition across the velocity knee is quite smooth. Thus, the results from this composition analysis point towards a smooth transition from fast to slow wind and did not unravel dramatic changes to be associated with the observed fluctuations abrupt depletion. Therefore, these similarities strongly suggest that the reason for the sudden depletion of turbulence observed within the stream of August 2017 and not observed within the stream of March 2017 can not be hidden in differences in the source region at photospheric level.\nMoreover, the charge state ratio profile of $C_{+6}\/C_{+5}$ (see the blue curve on panel b of both Figures \\ref{figura9} and \\ref{figura10}) is in agreement with the results shown by \\cite{Schwadron2005}. They modeled a solar minimum configuration that gives rise to a co-rotating region considering the differential motion of magnetic field foot points at the Sun across the CHBL. In their model, they started the simulation at $30 R_s$ ($ \\sim 0.14$ AU ) with the presence of a discontinuity at the stream interface (that comes before the fast wind plateau) and another discontinuity at the CHBL. They showed that the discontinuity at the stream interface remains whereas the discontinuity at the CHBL, in the rarefaction region, is eroded as the distance from the Sun increases (they have extended the simulation up to 5 AU). This result is in agreement with what we observe in the carbon charge ratio (panel b of Figure \\ref{figura9} and Figure \\ref{figura10}): we can notice a rapid decrease at the stream interface, corresponding to CH discontinuity before the fast wind plateau and, more important, we also observe a gradual increase of $C_{+6}\/C_{+5}$ in the rarefaction region, corresponding to the CHBL described in model of \\cite{Schwadron2005}.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=9.2cm]{Figura11_Riconnessione_AB.png}\n\t\\caption{Proposed mechanism of interchange reconnection invoked to explain the abrupt depletion of Alfv\u00e9nic fluctuations often observed in co-rotating high velocity streams. More details in the text.}\n\t\\label{figura11}\n\\end{figure}\n\nAt this point we invoke a simple mechanism, as shown in Figure \\ref{figura11}, which might solve this puzzle, taking into account that there is no TD there, nor is there an abrupt change in the composition analysis of the SW. The left-hand panel (Figure \\ref{figura11} A) sketches open magnetic field lines rooted inside a coronal hole, indicated by CH, and coronal loops rooted inside a coronal hole boundary layer, indicated by CHBL, encircling the CH. Field lines inside the CH are populated by Alfv\\'enic fluctuations while coronal loops are not. \nAlthough it is expected that Alfv\u00e9nic fluctuations largely populate the photospheric magnetic carpet \\citep{Title1998} transverse Alfv\u00e9nic fluctuations would experience a different fate in open or closed magnetic structures since in the latter ones the presence of counter-propagating Alfv\u00e9nic fluctuations would ignite turbulence process with consequent transfer of energy to smaller and smaller scales to eventually dissipate and heat the plasma. Thus, we would expect smaller amplitude and weaker Alfv\u00e9nic character in those fluctuations coming from originally closed field regions and larger amplitude and stronger Alfv\u00e9nic character for those coming from open-field-line regions.\nInterchange reconnection between open and closed field lines plays a major role in the regions bordering the coronal hole \\citep{Fisk99, Fisk2001, Fisk2005, Fisk2020,Schwadron2005}. This mechanism governs the diffusion of open magnetic flux outside the CH. The motion of foot-points of magnetic field lines due to photospheric dynamics is at the basis of interchange reconnection, which tends to mix open field line regions with nearby closed loop regions. Within this framework, an open field line region, populated by propagating Alfv\\'enic fluctuations, might reconnect with a closed loop region and cause a sudden disappearance of these fluctuations as shown in the right-hand panel (Figure \\ref{figura11} B). This phenomenon might happen at the base of the corona during the initial phase of the wind expansion \\citep{Fisk2005}. \n\n\n\\section{Summary}\n\nCo-rotating high speed streams are generally characterized by a high speed plateau followed by a slow velocity decrease, commonly identified as the stream rarefaction region. These two portions of the stream, in some cases, are separated by a clear velocity knee. In these cases, the knee marks also the point beyond which Alfv\\'enic fluctuations, which generally populate the high speed plateau, are dramatically and rapidly depleted. Interesting enough, this thin border region is not characterized by any relevant TD that could be responsible for such an abrupt depletion of the fluctuations. On the other hand, we could not find either clue in the composition analysis which might have indicated a clear different origin in the source region between the wind observed in the high speed plateau and that observed in the rarefaction region. Thus, the origin of the observed rapid depletion of the fluctuations does not seem to be related neither to differences at the source region nor to a TD separating the high speed plateau from the following rarefaction region. \nNot excluding the hypothesis that this reduction may occur during the expansion of the solar wind, the rapid time scales over which this abrupt decrease in fluctuations (few minutes) and Alfv\u00e9nicity (less than one hour) is observed make us lean towards the hypothesis that a magnetic reconnection event during the initial phase of the wind expansion, within the lowest layers of the solar atmosphere, might be the mechanism responsible for what we observed and described in this paper. Phenomena of interchange magnetic reconnection favored by the motion of the foot-points of magnetic field lines due to photospheric dynamics would mix together field lines originating from open field line regions, robustly populated by large amplitude Alfv\\'enic fluctuations, with magnetic loops, characteristic of surrounding closed field line regions, populated by smaller amplitude and less Alfv\\'enic fluctuations. \nThis suggested mechanism, which we cannot prove but propose as an hypothesis at the present, will be tested in the near future as soon as the nominal phase of the Solar Orbiter mission will start and it will become possible to directly link in-situ and remote observations. This sort of behavior could be related to the topology of the heliospheric current sheet at the moment of the crossing by the observer. Highly inclined current sheet crossings seem to result in an abrupt depletion of the fluctuations whereas during flat current sheet crossings we observe neither a clear velocity knee nor an abrupt reduction of the fluctuations.\n\n\\begin{acknowledgements}\nThis work was partially supported by the Italian Space Agency (ASI) under contract ACCORDO ATTUATIVO n. 2018-30-HH.O. Results from the data analysis presented in this paper are directly available from the authors. WIND data have been accessed through the NASA SPDF-Coordinated Data Analysis Web. G.C. acknowledges support from the Laboratoire de M\u00e9canique des Fluides et d'Acoustique at the \u00c9cole centrale de Lyon, France. G.C. acknowledges SWICo's recognition through the award \"Premio Mariani\" 2021.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nModification and external control of the electronic structure of two-dimensional materials via the proximity effect is of great experimental and technological interest for designing systems with novel magnetic and spin properties~\\cite{Han2014,Sierra2021,RevModPhys.92.021003}. Of particular importance is the introduction of \nspin interactions in graphene, whose Dirac electrons exhibit only weak spin-orbit coupling. Placing graphene in proximity to transition metal dichalcogenides (TMDCs) is a viable route, resulting in spin-orbit couplings on the meV scale~\\cite{Gmitra2015,Wang2015, Gmitra2016, Gani2020},\nalso tunable by twisting \\cite{Li2019,David2019,naimer2021twistangle,pezo2021manipulation}. While the proximity spin-orbit coupling is in general sublattice dependent \\cite{Frank2016, Kochan2017}, \nTMDC substrates induce valley-Zeeman and Rashba couplings, providing uniform pseudospin spin-orbit fields, opposite at $K$ and $K'$~\\cite{Gmitra2015, Wang2015,PhysRevB.94.241106}. \nSimilar proximity spin-orbit physics and the appearance of the valley-Zeeman coupling has been predicted for graphene on Bi$_2$Se$_3$-family of topological insulators \\cite{Song2018,PhysRevB.100.165141}, but also for \nbilayer Jacutingaite\\cite{Rademaker2021}. There already exists a significant body of experimental evidence demonstrating the presence of valley-Zeeman coupling in proximitized graphene~\\cite{Garcia2018,Island2019,Karpiak2019,Tang2020,Hoque2021,Ghiasi2017,Benitez2018,Ghiasi2019,Safeer2019,Herling2020,Bentez2020,Avsar2014,Banszerus2020,Khokhriakov2020, Wakamura2019, Wakamura2020,amann2021gatetunable,2021boosting,InglaAyns2021}.\n\nIn proximitized graphene with valley-Zeeman coupling, two groups of edge states---pseudohelical (intervelley) and intravalley states---form within the spin-orbit gap. At each edge there are those two pairs of states, conforming to the trivial topology of the system \\cite{Frank2018}. However, for ribbons on the nanoscale, with the widths less than a micron, the intravalley states are gapped out by the confinement-induced hybridization and only the lone pseudohelical pair remains at each edge. This pair is fully protected against backscattering by time-reversal defects, similarly to helical states of the spin quantum Hall effect \\cite{Kane2005}. But, unlike helical states, the pseudohelical states in a flake change the helicity from one sigzag edge to the other, flipping the spin along the armchair edge at which perfect tunneling of the states occurs. In large flakes, where also intravalley states propagate, the backscattering protection by time-reversal symmetry is lifted, as the pseudohelical states reflect at armchair edges to intravalley states at the same edge. \n\nIs there a way to reinstate the lone pseudopotential pair also in larger ribbons and flakes, where there are nominally intravelley states as well. We show in this paper that placing proximitized graphene in a perpendicular magnetic field achieves exactly that. Magnetic effects are typicall twofold: Zeeman-like, providing spin imbalance, and orbital effects, leading to Landau quantization. Zeeman effects in proximitized graphene were investigated earlier, demonstrating the appearance of the quantum anomalous Hall effect \\cite{Hogl2020} and chiral Majorana modes \\cite{Hogl2020a}. Magnetic orbital response of helical edge states to magnetic fields was studied in previous works \\cite{Luo2020,Frank2020,Yang2016,Li2018,DeMartino2011,Lado2015,Delplace2010,Gani2020}. Rather surprisingly, the quantum spin Hall edge states, which are generated by uniform intrinsic (Kane-Mele) spin-orbit coupling, are not necessarily destroyed by the cyclotron effect~\\cite{Shevtsov2012, Goldman2012}, which can theoretically be used to switch between the quantum spin Hall and quantum Hall regimes by gating. However, there can be a crossover between topological and trivial regimes in the presence of a perpendicular magnetic field ~\\cite{Scharf2015,Bottcher2019}. \n\nIn this paper we study theoretically the response of the pseudohelical and intravalley edge states in proximitized graphene (using realistic paramaters for a graphene\/WSe2 heterostucture) to an external perpendicular magnetic field, as depicted in Fig. 1.(a). We employ a tight-binding model supplemented with Peierl's substitution to study the electronic structure of graphene zigzag nanoribbons and finite flakes. The Landau levels calculated by the tight-binding approach are in excellent agreement with the bulk Landau level predictions \\cite{Frank2020}. The pairs of pseudohelical edge states, see Fig.\\ref{fig:schema}(b), are preserved even if time-reversal symmetry is broken by the magnetic field. \nThis is \nsimilar to what happens in the quantum spin Hall effect, where helical edge states are preserved in an applied field~\\cite{Shevtsov2012}.\n\nHowever, the intravalley states (originating from Rashba SOC~\\cite{Frank2018}) disappear once the magnetic field increases beyond some critical value $B_c$. At $B > B_c$, intravalley edge states merge with the conduction and valence bands, opening an intravalley gap. \nInside the gap one finds a lone pair of pseudohelical states, at each zigzag edge. Effectively, the magnetic field gaps out the weakly localized intravalley states, mimicking finite-size confinement. \\cite{Frank2018}. \n\nThe paper is organized as follows. In Sec. II. we introduce a tight-binding model Hamiltonian for graphene with proximity induced spin-orbit coupling, and present a scaling method to obtain the energy spectrum for microscopic structures. In Sec. III. we analyze the electronic structures of zigzag ribbons and finite flakes states in the presence of an external perpendicular magnetic field. We also discuss the crossover at which the intravalley edges states are gapped, referring to bulk Landau level results. Finally, in Sec. IV., we briefly summarize the main results.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{1.pdf}\n\\end{center}\n\t\\caption{(a) Sketch of a graphene ribbon or flake on a substrate such as a TMDC which yields valley-Zeeman spin-orbit and Rashba couplings. A perpendicular magnetic field is applied to affect the orbital states of the Dirac electrons. (b) Two types of edge states near the Fermi level in graphene with valley-Zeeman spin-orbit interaction: (1-4) spin-polarized pseudohelical (intervalley) states, and (5) intravalley states. (c) Lone pairs of pseudohelical states above the crossover magnetic field.}\n\\label{fig:schema}\n\\end{figure}\n\n\\section{Model and Methods}\n\nWe consider Dirac electrons in proximitized graphene, with sizeable (on the meV scale) spin-orbit interactions of the valley-Zeeman and Rashba types. Such interactions are induced by the proximity effects with TMDCs or TIs, as discussed above. To investigate finite systems, zigzag ribbons and flakes, we implement the following tight-binding Hamiltonian, \\cite{Gmitra2016,Kochan2017,Gmitra2013}\n\\begin{align}\n \\hat{\\mathcal{H}}&=\\sum_{\\langle i,j \\rangle}t c_{is}^\\dagger c_{js}+\\sum_{i}\\Delta\\xi_i c_{is}^\\dagger c_{is} \\nonumber \\\\\n &+\\frac{2i}{3}\\sum_{\\langle i,j \\rangle}\\lambda_R c_{is}^\\dagger c_{js^\\prime}\\left[\\left(\\hat{\\mathbf{s}}\\times \\mathbf{d}_{ij}\\right)_{z}\\right]_{ss^\\prime} \\nonumber \\\\\n &+\\frac{i}{3}\\sum_{\\langle\\langle i,j \\rangle\\rangle}\\frac{\\lambda_{I}^{i}}{\\sqrt{3}}c_{is}^\\dagger c_{js} \\left[\\nu_{ij}\\hat{\\mathbf{s}}_z\\right]~, \\label{eq:tb-ham}\n\\end{align}\nwhere $c_{is}^\\dagger$ and $c_{is}$ are the creation and annihilation operators for site $i$ and spin $s$, $\\langle i,j \\rangle$ denotes the nearest, $\\langle\\langle i,j \\rangle\\rangle$ the next nearest neighbors.\nThe Hamiltonian of Eq.~\\eqref{eq:tb-ham} has four terms: First, the nearest-neighbour hopping with amplitude $t$ occurs between sites $i$ and $j$ with spin preservation. Second, the proximity effects induces the staggered potential $\\Delta$ with signs $\\xi_i$ equal to $+1$ or $-1$ for A and B sublattices, respectively. The third term describes\nRashba spin-orbit coupling \\cite{Rashba2009,Tsaran2014} with amplitude $\\lambda_{R}$, which breaks horizontal reflection symmetry and mixes states of opposite spins and sublattices. Symbols $\\mathbf{d}_{ij}$ and $\\hat{\\mathbf{s}}$ denote the unit vector from site $j$ to site $i$ and the vector of spin Pauli matrices, respectively. Finally, the fourth term models the valley-Zeeman spin-orbit coupling \\cite{Gmitra2015, Wang2015, Yang2016}. This term preserves the spin, but the intra-sublattice hopping is different for clockwise ($\\nu_{ij} = -1$) and counterclockwise ($\\nu_{ij} = +1$) paths along a hexagonal ring from site $j$ to $i$. The intrinsic spin-orbit coupling $\\lambda_{I}^{i}$ is written here in a more general way, allowing for different strengths at $A$ and $B$ sublattices. The orbital effects of a perpendicular magnetic field are modelled by Peierl's phase \\cite{Peierls1933,PhysRev.115.1460}. We do not consider\nthe Zeeman effects of the field, as they are negligible for the fields we consider, on the millitesla scale. \n\nIn the following, when we present numerical results, we use parameters from first-principles\ncalculations for graphene\/WSe$_2$, see Ref. \\cite{Gmitra2016}: the nearest-neighbor hopping $t=-2.507$~eV, staggered potential $\\Delta=0.56$~meV, Rashba SOC parameter $\\lambda_R=0.54$~meV, and intrinsic SOC parameters $\\lambda_I^A=1.22$~meV and $\\lambda_I^B=-1.16$~meV. \n\nWe wish to investigate micron-size systems, in which magnetic orbital effects are not quenched, and in which the intravalley edge states are not gapped out. Since the spin-orbit parameters are on the meV scales, the considered systems need to be large enough to resolve such energies in the subband structure. To reduce computational efforts and avoid dealing with intractably large structures, we employ a scaling trick ~\\cite{Liu2015}, which allows us to consider smaller structures but with rescaled parameters. \nFinite-size level spacings in graphene ribbons are on the order of $\\Delta E\\approx\\pi\\hbar v_F \/ w$~\\cite{Lin2008}, which can be reduced by changing the Fermi velocity $v_F\\rightarrow v_F\/r= \\sqrt{3}a t \/2\\hbar r$, by rescaling the nearest-neighbor hopping $t$. Such a rescaling affects the interpretation of the lattice constant $a$, which is mapped to $ra$, in order to keep the energy spectrum invariant. The cyclotron energy in graphene is given by $\\hbar\\omega_c = \\sqrt{2e\\hbar B} v_F$. To preserve this energy scale, we rescale the external perpendicular magnetic field by $B\\rightarrow Br^2$. The rescaling does not influence the underlying physics as long as we consider low-energy states for which the linear dispersion holds\\cite{Liu2015}. \n\nThe scaling trick permits us to keep the cyclotron energy scale at the level of SOC parameters, while decreasing the computational burden to simulate large-scale systems. The main criterion for choosing the rescaling parameter is $r\\ll3t\\pi\/E_{\\text{max}}$, where $E_{\\text{max}}$ is the maximal energy of interest~\\cite{Liu2015}. We wish to resolve energies up to $E_{\\text{max}}=2$ meV, which safely covers the spin-orbit gap region in which the edge states form, the scaling parameter $r\\ll3t\\pi\/E_{\\text{max}}$=11809. In our work we we choose $r=400$. With that, the finite-size effects are decreased to $0.05$~meV for a width of 400 unit cells. The magnetic field strength equivalent to this energy scale is about $10^{-5}$~T, well below the value for the crossover magnetic field (see the next chapter) for gap closing.\n\nTo calculate the band structure of zigzag nanoribbons and the electronic states of a graphene flake with proximity induced spin-orbit interaction, we implement the above model using the Python-based numerical package KWANT~\\cite{Groth2014}.\n\n\\section{Results}\n\n\\subsection{Bulk results}\n\nThe starting point for analyzing the behavior of edge states of proximitized graphene in the presence of an external perpendicular magnetic field is the study of Landau levels in the bulk system. A detailed derivation of the Hamiltonian of the Landau levels for bulk proximitized graphene is presented in Ref.~\\cite{Frank2020}. Graphene with spin-orbit coupling in magnetic fields was studied from different perspectives by other groups as well~\\cite{Cysne2018, DeMartino2011}.\n\nWe construct the Landau fan diagram, shown in Fig.~\\ref{fig:bulk_landau}. Comparing to the Kane-Mele model \\cite{Kane2005,Shevtsov2012} for graphene, a staggered intrinsic SOC does not preserve the gap in the presence of an external magnetic field. The bulk band gap mainly forms between nonzero Landau levels from the $K$ and $K^{\\prime}$ valley, which can be shifted by the magnetic field. On the other hand, in the Kane-Mele model, a bulk band gap forms between the zero Landau levels, which is stable under an external magnetic field. In the systems with valley-Zeeman spin-orbit coupling, the bulk band gap closes and reopens at a crossover value of the external magnetic field equal to \\cite{Frank2020}\n\\begin{align}\n B_{c}=\n \\frac{\n \\left[\\lambda_I^A-\\lambda_I^B-2\\Delta\\right]\n \\left[\\left(\\Delta+\\lambda_I^A\\right)\\left(\\Delta-\\lambda_I^B\\right)+4\\lambda_R^2\\right]\n }\n {\n 2e\\hbar v_F^2\\left[2\\Delta+\\lambda_I^A-\\lambda_I^B\\right]\n }.\n\\end{align}\nFor the given parameters set from Ref.~\\cite{Gmitra2015,Frank2018} of our Hamiltonian\\eqref{eq:tb-ham}, the crossover value of the magnetic field will be equal to 1.942~mT for the graphene\/WSe$_2$ heterostructure. This crossover point distinguishes two regimes, below the crossover field, a bulk band gap formed by the different nonzero Landau levels, while above the crossover field bounded by the first Landau levels.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{2.png}\n\\end{center}\n\\label{fig:bulk_landau}\n\\caption{Calculated evolution of the bulk Landau levels in graphene\/WSe$_2$ with increasing of the external magnetic field. The crossover magnetic field is indicated as a dashed line. The color code corresponds to the $s_z$ expectation value in the left column and $\\kappa_z$ (valley) expectation value in the right column, see Ref.~\\onlinecite{Frank2020}.}\n\\end{figure}\n\n\n\\subsection{Zigzag ribbon results}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{3.png}\n\\end{center}\n\\label{fig:band}\n\\caption{Calculated band structures of a proximitized graphene zigzag ribbon in the presence of a perpendicular magnetic field B=0~mT (a), B=1.5~mT, and B=3~mT. The color code denotes spin polarization in the left column and edge polarization in the right column. Thin dashed lines indicate bulk Landau levels.}\n\\end{figure}\n\nThe band structure of a zigzag graphene ribbon is calculated for different values of the external perpendicular magnetic field. We consider zigzag ribbons\nwith the width of 4.1~$\\mu$m. With the scaling factor $r=400$, the simulation size of the ribbon is reduced to 72 carbon-carbon bonds. The results for the zigzag ribbon band structure for magnetic fields of 0~mT, 1.5~mT, and 3.0~mT are shown in Fig.~\\ref{fig:band}. At zero magnetic field, we can distinguish two types of edge states: spin unpolarized intravalley states and pseudohelical intervalley states with strong spin polarization~\\cite{Frank2018, Gmitra2016}. The intravalley edge states are more delocalized towards the bulk than intervalley states, due to their spectral proximity to the bulk states. \n\nTo compare with the bulk~\\cite{Frank2020}, we also plot bulk Landau levels at $K$ and $K^{\\prime}$. For small magnetic fields (1.5~mT), the Landau levels start to form and bands at the $K$ and $K^{\\prime}$ valleys lose their dispersion. In general, the Landau levels are rather bulk-like, indicated by the absence of edge polarization. They coincide with the analytic prediction\\cite{Frank2020}. \n\nAt the borders of the bulk continuum, edge states form, which are responsible for the quantum Hall effect in graphene. With the increase of the external magnetic field, intervalley edge states shift in $k$-space: states with the same spin polarization shift in the same direction, while states with the opposite spin polarization shift in the opposite direction. Generally, intervalley edge states are not much affected by the perpendicular magnetic field. However, the intravalley states get strongly modified, merging with the bulk bands; this happens already at magnetic fields less than 1.5~mT. This is similar to what is predicted for 2D topological insulators in a magnetic field~\\cite{Bottcher2019}. The bulk gap is openning at the crossover magnetic field $B_c \\approx 1.942$~mT. As \na consequence, the intravalley edge states disappear, leaving behind the lone pairs of pseudohelical edge states, ona pair at each zigzag edge. \n\n\\subsection{Flake results}\nTo analyze the nature of the edge states, we have calculated low energy states of differently sized proximitized graphene flakes. At zero magnetic field the pseudohelical states which form at the zigzag edges reflect back to the intravalley states, which generates standing waves spread out through the flake. In nanoribbons, with sizes less than a micron, the intravalley states are gapped out and the pseudohelical states tunnel through the armchair edges, allowing them to fully propagate through the flake edges. \n\nAs we showed above, for $B > B_c$ the intravalley states in wide ribbons are gapped out as well. Can then the pseudohelical states follow the same scenario as in nanoribbons? The answer is no. While at zero magnetic field the pseudohelical states cannot reflect back at the armchair edges due to time-reversal symmetry (the pseudohelical pair is formed by time-reversal partners), in the presence of a magnetic field such a reflection is possible. In fact, as we demonstrate below, this reflection is perfect, leading to the localization of the pseudohelical pair at zigzag edges and formation of standing waves that carry pure spin current. \nThe qualitative difference between propagating pseudohelical edge states and pure spin current non-propagating states is depicted in Fig.\\ref{fig:states}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{2_.pdf}\n\\end{center}\n\\caption{(a) Schematic representation of pseudohelical states in a proximitized graphene flake. The states form at zigzag edges, propagate towards the armchair edges, along which they tunnel and undergo spin flip due to Rashba coupling. The colors represent the spin which is perpendicular to the sheet. At zero magnetic field the tunneling along the armchair edges is perfect. (b) In large flakes, if $B > B_c$, the counterpropagating pseudohelical edge states are perfectly reflected at the armchair edges. The pair then forms a standing, nonpropagating wave, which carries net spin current, but no charge current. The resulting state is spin unpolarized. See also Fig.~\\ref{fig:schema}(c).}\n\\label{fig:states}\n\\end{figure}\n\nWhile the formation of the pseudohelical states can be explained by perfect tunnelling of the intervalley states across the armchair edge, the formation of the pure spin current states corresponds to perfect reflection off the armchair edge. Such a difference in the mechanisms of the formation of states is possible due to the exponential decay of the probability of tunnelling between zigzag edges as the distance between them increases. It is not possible to continuously go from one limit to another due to the periodic subband gap opening and closing~\\cite{Frank2018}. However, we can clearly distinguish pseudohelical states for a narrow graphene flake and pure spin current states for a wide graphene flake, because these states represent two different physical limits. \nWe estimated the distance between zigzag edges at which perfect tunneling still occurs as the typical spin-flip length of the Rashba spin-orbit interaction\n\\begin{align}\n l_{R}\\approx3a\\frac{t}{\\lambda_R}.\n\\end{align}\nFor our parameters (graphene\/WSe$_2$) $l_{R}$ is about 3.4~$\\mu$m, which is less than the flake width we use in this work. \n\nPseudohelical states and pure spin current states are formed by combinations of intervalley states. They both consist of states (1) and (2) at the left edge and of (3) and (4) at the right edge, see Fig.\\ref{fig:states}(c). Both types of states, pseudohelical states and pure spin current states, are formed using the same edge states and the only difference is the mechanism of their combination.\n\nWe have specifically investigated pure spin current (i.e., spin current without any charge current) states for a flake whose size is equal to 16.3584~$\\mu$m $\\times$ 4.0896~$\\mu$m, taking into account the scaling factor $r=400$, which corresponds to an auxiliary 288 $\\times$ 72 carbon-carbon bond sized flake. The width of this flake is greater than the Rashba spin-flip length, so the perfect reflection mechanism is dominant in this case. We trace the evolution of the highest occupied state with incremental steps in the external perpendicular magnetic field. \n\nThe spin and charge currents and densities through a cut in the middle of the flake, orthogonal to the armchair direction are shown in Fig.\\ref{fig:cut}. The data indicates that with the increase of the magnetic field, the charge current and spin densities disappear and complete edge localization of the state occurs. Also, the spatial spin current oscillations vanish and the highest occupied states become pure spin current states with negligible spin density. A similar situation occurs with the lowest unoccupied state. \n\nThe evolution of the spin currents of the highest occupied and the lowest unoccupied states are shown in Fig.\\ref{fig:spin}. Both states exhibit similar behavior of their spin currents, located on different zigzag edges of the flake. Pure spin currents arise in the presence of an external magnetic above the crossover value $B_c$. The evolution of the charge density is similar to the previous case, depicted in Fig.~\\ref{fig:charge}. With an external magnetic field below the crossover value, the states closest to the Fermi level are a mixture of strongly localized intervalley states and weakly localized intravalley states; they form a standing wave at the zigzag boundary, as tunneling through the armchair edge is forbidden (this can explicitly be seen in gapped ribbon armchair band structures~\\cite{Frank2018}).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{4.pdf}\n\\end{center}\n\\caption{Calculated spin (a) and charge (b) currents of the highest occupied flake states (see text for flake parameters) through the cut in the middle of the flake and in the presence of a\nperpendicular magnetic field. Spin (c) and charge (d) densities of the highest occupied state at the cut in the middle of the flake and in the presence of a magnetic field.}\n\\label{fig:cut}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{5_new.png}\n\\end{center}\n\\caption{Evolution of the spin current of the highest occupied flake state in the left column and of the lowest unoccupied flake state in the right column.}\n\\label{fig:spin}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{6_new.png}\n\\end{center}\n\\caption{Evolution of the charge density of the highest occupied flake state in the left column and of the lowest unoccupied flake state in the right column.}\n\\label{fig:charge}\n\\end{figure}\n\n\nHowever, with a magnetic field above the crossover value $B_c$, we find states with pronounced pure spin current, localized strongly at a single edge. From the zigzag ribbon band-structure analysis, we can conclude that the pure spin current states correspond to the combination of the intervalley edge states on the same edge, which is stable under the application of a magnetic field. Similar to the pseudohelical states, pure spin current states are stable against scattering on defects at the zigzag edge of the graphene flake, see Appendix A.\n\n\\section{Conclusion}\nWe investigated the behavior of the electronic structure of proximitized graphene ribbons and flakes of microscopic sized in the presence of a perpendicular magnetic field. We have chosen \ngraphene\/WSe$_2$ heterostructure for the numerical parameters, as in such structures pseudohelical states were predicted to exist. We found that under the influence of a magnetic field, the pseudohelical (intervalley) edge states are preserved, while the intravalley edge states disappear at the crossover value of the magnetic field $B_c \\approx 1.9~mT$. This value corresponds to the closing\/reopening of the gap between the bulk Landau levels. \n\nWe also studied a finite flake of micron sizes, \nfinding that instead of perfect tunneling of the pseudohelical states at the armchair edges at zero magnetic field, for $B > B_c$ the states perfectly reflect to their counterpropagating partners and form non-propagating, spin unpolarized pure spin current states which are stable under scattering off zigzag edge defects. \n\nPure spin current states should be observable in wide flakes of at least a few microns (for graphene\/TMDC heterostructures) already at rather weak (a few millitesla) fields. Lone pseudohelical pairs at a given edge should be observable in wide ribbons for $B > B_c$ of in nanosized flakes where perfect tunneling through armchair edges is allowed and intravalley states are gapped even at zero magnetic field. \n\n\n\\begin{acknowledgments}\nWe would like to thank D. Kochan for useful discussions. This work was supported by the DFG SFB Grant No. 1277 (A09 and B07), DFG SPP 2244 (Project-ID 443416183), the International Doctorate Program Topological Insulators of the Elite Network of Bavaria. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de).\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro} \n\n\nThe Kronecker-Weierstrass theory of $m \\times n$ matrix pencils provides a complete classification in terms of \n$\\GL_{m,n}(\\mathbb{F})$-orbits, which are equivalence classes under the action of $\\GL_{m}(\\mathbb{F}) \\times \\GL_{n}(\\mathbb{F})$:\n\\begin{equation*}\n \\left( P, \\, U \\right) \\cdot \\left( \\mu A + \\lambda B \\right) = \\mu P A U^\\T + \\lambda P B U^\\T,\n\\end{equation*}\nwhere the pencil $(A,B)$ is expressed in homogeneous coordinates. \nHere, $\\mathbb{F}$ denotes a field, usually $\\mathbb{R}$ or $\\mathbb{C}$.\nThese orbits are represented by Kronecker canonical forms, which are characterized by unique minimal indices describing the singular \npart of the pencil along with elementary divisors associated with its regular part \\cite{Gantmacher1960}. \nIt follows then that these elementary divisors are $\\GL_{m}(\\mathbb{F}) \\times \\GL_{n}(\\mathbb{F})$-invariant.\n\n\nThis theory has been extended by Ja'Ja' and Atkinson \\cite{JaJa1979b, Atkinson1991}, who have characterized the orbits of the larger \ngroup of tensor equivalence transformations $\\GL_{m}(\\mathbb{F}) \\times \\GL_{n}(\\mathbb{F}) \\times \\GL_2 (\\mathbb{F})$, acting on \npencils via\n\\begin{equation*}\n \\left( P, \\, U, \\, T \\right) \\cdot \\left( \\mu A + \\lambda B \\right) \n = \\mu P (t_{11} A + t_{12} B) U^\\T + \\lambda P (t_{21} A + t_{22} B) U^\\T,\n\\end{equation*} \nwhere \n\\begin{equation*}\n T = \\begin{pmatrix}\n t_{11} & t_{12} \\\\ t_{21} & t_{22}\n \\end{pmatrix}\n \\in \\GL_2 (\\mathbb{F}).\n\\end{equation*} \nFor simplicity of notation, we use the shorthands $\\GL_{m,n,2}(\\mathbb{F}) \\triangleq \\GL_{m}(\\mathbb{F}) \\times \n\\GL_{n}(\\mathbb{F}) \\times \\GL_2 (\\mathbb{F})$ and $\\GL_{m,n}(\\mathbb{F}) \\triangleq \\GL_{m}(\\mathbb{F}) \\times \\GL_{n}(\\mathbb{F})$. \nJa'Ja' \\cite{JaJa1979b} has shown that the Kronecker minimal indices of a pencil are $\\GL_{m,n,2}(\\mathbb{F})$-invariant, and so the \nsingular part of a pencil is preserved by $\\GL_{m,n,2}(\\mathbb{F})$ as well. \nHowever, the elementary divisors of its regular part are not. Nevertheless, their powers still remain the same, which motivates the \nterminology ``invariant powers,'' used by Ja'Ja' \\cite{JaJa1979b}. Atkinson \\cite{Atkinson1991} went on to prove that, for an \nalgebraically closed field $\\mathbb{F}$, the equivalence classes of regular pencils are characterized by those powers and also by \ncertain ratios which completely describe the elementary divisors. Specifically, recalling that over such a field all elementary \ndivisors are powers of linear factors of the form\n\\begin{equation*}\n \\phi_i(\\mu, \\lambda) = \\alpha_i \\mu + \\beta_i \\lambda \\quad \\text{for some } \\alpha_i, \\beta_i \\in \\mathbb{F},\n\\end{equation*}\nthese ratios are defined as $\\gamma_i \\triangleq \\alpha_i \/ \\beta_i \\in \\mathbb{F} \\cup \\{\\infty\\}$.\n\n\nWhen viewed as a tensor, the (tensor) rank\\footnote{This is not to be confused with the \\emph{normal rank} of the pencil \n$\\mu A + \\lambda B$, simply defined as $\\rank(\\mu A + \\lambda B)$ \\cite{Gantmacher1960, Ikramov1993}.} of a pencil $\\mu A + \\lambda B$ \nis defined as the minimal number $r$ of rank-one matrices $U_1, \\ldots, U_r \\in \\mathbb{F}^{m \\times n}$ such that $A, B \\in \\sspan \n\\{U_1, \\ldots, U_r\\}$ \\cite{JaJa1979}. Equivalently, it is given by the \nminimal number $r$ such that one can find vectors $u_i \\in \\mathbb{F}^m$, $v_i \\in \\mathbb{F}^n$ and $w_i \\in \\mathbb{F}^2$ satisfying\n\\begin{equation*}\n A \\otimes e_1 + B \\otimes e_2 = \\sum_{i=1}^r u_i \\otimes v_i \\otimes w_i,\n\\end{equation*} \nwhere $e_i$ denotes the canonical basis vector of $\\mathbb{F}^2$. Under the action of $(P,U,T) \\in \\GL_{m,n,2}(\\mathbb{F})$, this \nexpression is transformed into the tensor\n\\begin{equation*}\n \\left( P, U, T \\right) \\cdot \\left( A \\otimes e_1 + B \\otimes e_2 \\right) \n = \\sum_{i=1}^r \\left( P \\, u_i \\right) \\otimes \\left( U \\, v_i \\right) \\otimes \\left( T \\, w_i \\right),\n\\end{equation*} \nfrom which it is visible that the tensor rank is $\\GL_{m,n,2}(\\mathbb{F})$-invariant. The rank of $m \\times n \\times 2$ tensors \ncan thus be studied by considering $\\GL_{m,n,2}(\\mathbb{F})$-orbits and associated representatives (see, e.g., the classification of \n$\\GL_{2,2,2}(\\mathbb{R})$-orbits undertaken by De Silva and Lim \\cite{deSilva2008}).\n\n\nOne application of the study of $\\GL_{m,n,2}(\\mathbb{F})$-orbits is in algebraic complexity theory, since the tensor rank \nof $\\mu A + \\lambda B$ quantifies the minimal number of multiplications needed to simultaneously evaluate a pair of \nbilinear forms $g_1(x,y) = \\langle x, A y \\rangle$ and $g_2(x,y) = \\langle x, B y \\rangle$ \\cite{JaJa1979}. In the case where $\\mathbb{F}$ is \nalgebraically closed, Ja'Ja' \\cite{JaJa1979} has derived results which allow determining the tensor rank of any pencil based on its \nKronecker canonical form. Sumi et al.~\\cite{Sumi2009} have extended these results to pencils over any field $\\mathbb{F}$.\n\nTensor equivalence transformations can also be employed to avoid so-called infinite elementary divisors of regular \npencils \\cite{Ikramov1993}. These arise when matrix $B$ is singular (note that both $A$ and $B$ can be singular but still \nsatisfy $\\det(\\mu A + \\lambda B) \\not\\equiv 0$). In non-homogeneous coordinates, the polynomial $\\det (A + \\lambda B)$ of an $n \n\\times n$ pencil $A + \\lambda B$ has degree $s = \\rank B$, and its characteristic polynomial is said to have infinite \nelementary divisors of combined degree $n - s$. In this case,\nthe tensor equivalence transformation $A + \\lambda B \\mapsto B + \\lambda (A + \\alpha B)$ can be employed for any $\\alpha \\in \n\\mathbb{F}\\setminus\\{0\\}$ such that $\\rank (A + \\alpha B) = n$, yielding a pencil having only finite elementary divisors, \nincluding some of the form $\\lambda^q$ induced by the infinite elementary divisors of $A + \\lambda B$.\nThe existence of such an $\\alpha$ is guaranteed by definition, since $A + \\lambda B$ is regular. In other words, every regular pencil \n$\\mu A + \\lambda B$ is $\\GL_{m,n,2}(\\mathbb{F})$-equivalent to another pencil $\\mu A' + \\lambda B'$ such that $B'$ is \nnonsingular. In fact, it is always $\\GL_{m,n,2}(\\mathbb{F})$-equivalent to some $\\mu A' + \\lambda B'$ with nonsingular \nmatrices $A'$ and $B'$.\n\nOn the other hand, not every regular pencil $\\mu A + \\lambda B$ constituted by nonsingular matrices $A$ and $B$ is \n$\\GL_{n,n,2}(\\mathbb{F})$-equivalent to some other pencil $\\mu A' + \\lambda B'$ such that either $A'$ or $B'$ are \nsingular (or both). Take, for instance, $\\mathbb{F} = \\mathbb{R}$ and\n\\begin{equation*}\n A = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}, \\quad B = \\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix}.\n\\end{equation*} \nNo tensor equivalence transformation in $\\GL_{m,n,2}(\\mathbb{R})$ of this pencil can yield $\\mu A' + \\lambda B'$ such that \neither $A'$ or $B'$ is singular. Obviously, this property is $\\GL_{m,n,2}(\\mathbb{R})$-invariant.\nIt turns out that each $\\GL_{m,n,2}(\\mathbb{F})$-orbit $\\mathcal{O}$ of a matrix pencil space can be classified on the basis \nof its associated \\emph{minimal ranks} $r$ and $s$, with $r \\ge s$, which are such that every pencil $\\mu A + \\lambda B \\in \n\\mathcal{O}$ with $\\rank A \\ge \\rank B$ satisfies $\\rank A \\ge r$ and $\\rank B \\ge s$.\nThis notion of intrinsic complexity of a matrix pencil is complementary to its tensor rank, in the sense that pencils of same \ntensor rank do not necessarily have the same minimal ranks and vice-versa. For simplicity, we will compactly denote the minimal ranks \nof a pencil by $ \\rho(A,B) = (r,s)$.\n\nIt turns out that there is a direct connection between the minimal ranks of a pencil and the decomposition of its associated \nthird-order tensor in block terms consisting of matrix-vector outer products, as introduced by De Lathauwer \n\\cite{deLathauwer2008b}. Namely, the components of $\\rho(A,B)$ are the minimal numbers $r$ and $s$ satisfying\n\\begin{equation*}\n A \\otimes e_1 + B \\otimes e_2 = \\left( \\sum_{i=1}^{r} u_i \\otimes v_i \\right) \\otimes w +\n \\left( \\sum_{i=1}^{s} x_i \\otimes y_i \\right) \\otimes z,\n\\end{equation*} \nwhere $r \\ge s$ and $\\{w, z\\}$ forms a basis for $\\mathbb{F}^2$. The theoretical properties of such block-term decompositions \n(henceforth abbreviated as BTD) of $m \\times n \\times 2$ tensors are therefore related to properties of matrix pencils via this notion \nof minimal ranks. \n\nIn this paper, we will more generally define the minimal ranks of $m \\times n$ matrix polynomials $\\sum_{k=1}^{d} \\lambda^{k-1} \nA_k$, which include matrix pencils as a special case. This property of matrix polynomials is directly related to the BTD of $m \n\\times n \\times d$ tensors. In particular, similarly to the tensor rank, it induces a hierarchy of matrix polynomials, albeit a more \ninvolved one. \nWe derive results which determine the minimal ranks of any matrix pencil in Kronecker canonical form.\nA classification of $\\GL_{m,n,2}(\\mathbb{R})$-orbits of real $m \\times n$ pencils in terms of their minimal ranks is then\ncarried out for $m, n \\le 4$.\nOn the basis of these results, we proceed to show that:\n\\begin{enumerate}\n \\item The set of real $2k \\times 2k$ pencils which are $\\GL_{2k,2k,2}(\\mathbb{R})$-equivalent to some $\\mu A + \\lambda B$\nwith $\\rank A \\le 2k-1$ and $\\rank B \\le 2k-1$ is not closed in the norm topology for any positive integer $k$.\n \\item No real $2k \\times 2k$ pencil having minimal ranks $(2k,2k)$ admits a best approximation in the set above described. \n\\end{enumerate}\nThe first above result is analogous to the fact that a set of tensors having rank bounded by some number $r>1$ is generally not \nclosed. \nSimilarly, the second one parallels the fact that no element of certain sets of real rank-$r$ tensors admits a best approximation of \na certain rank\\footnote{For instance, no real $2 \\times 2 \\times 2$ tensor of rank $3$ admits a best rank-$2$ approximation.} $r' < r$ \nin the norm topology \\cite{Stegeman2006, deSilva2008}. \nThis second result is of consequence to applications relying on the BTD, since the set of real pencils having minimal ranks $(2k,2k)$ \nis open in the norm topology, thus having positive volume. For complex-valued pencils, the results of \\cite{Qi2017} imply that such a \nnon-existence phenomenon can only happen over sets of zero volume. We shall give a template of examples of (possibly complex) pencils \nhaving no best approximation on a given set of pencils with strictly lower minimal ranks.\n\nIt should be noted that the fact that a tensor might not admit an approximate BTD with a certain prescribed structure (referring \nto the number of blocks and their multilinear ranks \\cite{deLathauwer2008b}) is already known. Specifically, De Lathauwer \n\\cite{deLathauwer2008c} has provided an example relying on a construction similar to that of De Silva and Lim \\cite{deSilva2008} \nconcerning the case of low-rank tensor approximation. Our example given in \\Cref{sec:ill-posed-examples} is in the same spirit. \nNonetheless, to our knowledge ours is the first work showing the existence of a positive-volume set of tensors having no approximate \nBTD of a given structure, a phenomenon which is known to happen for low-rank tensor approximation \\cite{Stegeman2006, deSilva2008}.\n\n\n\n \n\\section{Minimal ranks of pencils}\n\\label{sec:pencils}\n\nFor brevity, we will henceforth express matrix pencils only in non-homogeneous coordinates.\n\n\\subsection{Definition and basic results}\n\\label{sec:min-ranks}\n\nGiven its prominent role in what follows, the $\\GL_{m,n,2}(\\mathbb{F})$-orbit of a pencil deserves a special notation:\n\\begin{equation*}\n \\mathcal{O}(A,B) \\triangleq \n \\{(P,U,T) \\cdot (A + \\lambda B) \\ | \\ (P,U,T) \\in \\GL_{m,n,2}(\\mathbb{F}) \\}.\n\\end{equation*}\nIt will also be helpful to introduce the sets\n\\begin{equation}\n\\label{Bset}\n \\mathcal{B}_{r,s} \\triangleq \n \\left\\{ A + \\lambda B \\; | \\;\n \\exists \\, A' + \\lambda B' \\in \\mathcal{O}(A,B) \n \\mbox{ such that }\n \\rank A' \\le r, \\, \\rank B' \\le s \\right\\}.\n\\end{equation} \nClearly, $\\mathcal{B}_{r,s} = \\mathcal{B}_{s,r}$, and thus we shall assume that $r \\ge s$ without loss of generality.\nFor simplicity, when $r \\ge s = 0$ we can also write $\\mathcal{B}_r$ instead of $\\mathcal{B}_{r,0}$. According to this convention and \nto definition \\eqref{Bset}, we have, for instance, $\\mathcal{B}_{r} \\subseteq \\mathcal{B}_{r,s} \\subseteq \\mathcal{B}_{r+1,s}$. \nFurthermore, $\\mathcal{B}_{r,s}$ is by definition $\\GL_{m,n,2}(\\mathbb{F})$-invariant, because the relation $(A,B) \\sim \n(A',B')$ defined as $A' + \\lambda B' \\in \\mathcal{O}(A,B)$ is reflexive and transitive, i.e., it defines an equivalence class. Hence:\n\n\\begin{lem}\n $A + \\lambda B \\in \\mathcal{B}_{r,s}$ if and only if $\\mathcal{O}(A,B) \\subseteq \\mathcal{B}_{r,s}$.\n\\end{lem}\t\n\nAs far as the question of whether $A + \\lambda B$ is in $\\mathcal{B}_{r,s}$ for some $(r,s)$ is concerned, all that matters is the \naction of $\\GL_2(\\mathbb{F})$. Indeed, if $A' + \\lambda B' = (P,U,T) \\cdot (A + \\lambda B)$ are such that $\\rank A' = r$ and $\\rank B' \n= s$, then\\footnote{Note that we denote the identity of $\\GL_m(\\mathbb{F})$ by $E_m$ or, when no ambiguity arises, simply by $E$.} $A'' \n+ \\lambda B'' = (P^{-1},U^{-1},E) \\cdot (A' + \\lambda B') = (E,E,T) \\cdot (A + \\lambda B)$ satisfies $\\rank A'' = \\rank P^{-1} A' \n U^{-\\T} = \\rank A'$ and $\\rank B'' = \\rank P^{-1} B' U^{-\\T} = \\rank B'$. We have shown the following.\n\n\\begin{lem}\n\\label{lem:GL2}\n$A + \\lambda B \\in \\mathcal{B}_{r,s}$ if and only if there exists $T \\in \\GL_2(\\mathbb{F})$ such that $A' + \\lambda B' = (E, E, T) \n\\cdot (A + \\lambda B)$ satisfies $\\rank A' \\le r$ and $\\rank B' \\le s$. In other words, $\\mathcal{B}_{r,s}$ can be equivalently \ndefined as the set of all pencils which are $\\GL_2(\\mathbb{F})$-equivalent to some $A' + \\lambda B'$ satisfying $\\rank A' \\le r$ and \n$\\rank B' \\le s$.\n\\end{lem}\n\n\nLet us now formally define the minimal ranks in terms of the introduced notation.\n\n\\begin{defn}\n\\label{def:min-ranks}\nLet $A + \\lambda B$ be an $m \\times n$ pencil over $\\mathbb{F}$. The \\emph{minimal ranks} of $A + \\lambda B$, denoted as \n$\\rho(A,B) = (r,s)$, are defined as\n\\begin{align}\n \\label{s-def}\ns \\triangleq & \\ \\min_{(t,u) \\neq 0} \\rank (t A + u B), \\\\\nr \\triangleq & \\ \\min_{(t',u') \\notin \\sspan\\{(t^\\star,u^\\star)\\}} \\rank(t' A + u' B), \n\\label{r-def}\n\\end{align} \nwhere $(t^\\star,u^\\star)$ is any minimizer\\footnote{Observe that such a minimizer is not unique. Besides the obvious family of \nminimizers of the form $(c t^\\star, c u^\\star)$, there may be also multiple non-collinear minimizers. For instance, for the pencil $E_4 \n+ \\lambda (a E_2 \\oplus b E_2)$, with $a \\neq b$, there are two non-collinear minimizers: $(-a,1)$ and $(-b,1)$.} of \\eqref{s-def}. We \nobviously have $r\\ge s$.\nWhen denoting a pencil as $P(\\lambda) = A + \\lambda B$, we shall also use the notation $\\rho(P) = \\rho(A,B)$.\n\\end{defn}\n\nThe first thing to note is that $r$ is well-defined, i.e., its value is always the same regardless of the minimizer $(t^\\star,u^\\star)$\npicked in the definition \\eqref{r-def}. For different collinear minimizers of \\eqref{s-def}, this is immediately clear. Now if two \nnon-collinear minimizers $(t^\\star,u^\\star)$ and $(t^{\\star\\star}, u^{\\star\\star})$ exist for \\eqref{s-def}, then $r = s$ must hold.\nIt is also clear from \\eqref{s-def} and \\eqref{r-def} that the minimal ranks of a pencil $A + \\lambda B$ are the ranks of \nmatrices $A'$ and $B'$ of some pencil $A' + \\lambda B'$ in the $\\GL_{2}(\\mathbb{F})$-orbit of $A + \\lambda B$. Indeed, \ntaking $B' = t_{21} A + t_{22} B$ and $A' = t_{11} A + t_{12} B$, where $(t_{21},t_{22})$ and $(t_{11},t_{12})$ are the minimizers of \n\\eqref{s-def} and \\eqref{r-def}, respectively, then $A + \\lambda B = (E,E,T^{-1}) \\cdot (A' + \\lambda B')$, with $T = (t_{ij}) \\in \n\\GL_2(\\mathbb{F})$. Moreover, the minima in \\eqref{s-def} and \\eqref{r-def} are unchanged under a transformation from \n$\\GL_{m,n}(\\mathbb{F})$, implying that the value of $\\rho$ is an invariant of $\\GL_{m,n,2}$ action. Summarizing, we have:\n\n\\begin{prop}\n \\label{prop:rho-orbit}\n Every $A' + \\lambda B' \\in \\mathcal{O}(A,B)$ satisfies $\\rho(A,B) = \\rho(A',B')$. In particular, if $A' + \\lambda B' \\in \n\\mathcal{O}(A,B)$ satisfies $(\\rank A', \\rank B') = \\rho(A,B)$, we say that $A' + \\lambda B'$ attains the minimal ranks of $A + \\lambda \nB$.\n\\end{prop}\n\nFrom the above discussion, $\\rho(A,B) = (r,s)$ implies $A + \\lambda B \\in \\mathcal{B}_{r,s}$. However, the converse is not true. For \ninstance, $E_n + \\lambda E_n \\in \\mathcal{B}_{n,n}$ but $\\rho(E_n, E_n) = (n,0)$. In general, if $A = c B$ or $B = c A$ for \nsome $c \\in \\mathbb{F}$ (including the possibility $c = 0$), then $\\rho(A,B) = (r,0)$ with $r = \\max\\{\\rank A, \\rank B\\}$. \nConversely, $s = 0$ only if $A$ and $B$ are proportional. We thus have the following result. \n\n\\begin{lem}\n \\label{lem:rho-r-0}\nA pencil $A + \\lambda B$ satisfies $\\rho(A,B) = (r,0)$ if and only if $A$ and $B$ are proportional. Furthermore, $r = \n\\max\\{\\rank A, \\rank B\\}$.\n\\end{lem}\n\n\nThe terminology ``minimal ranks'' is motivated by the fact that, if $\\rho(A,B) = (r,s)$ and $A + \\lambda B \\in \n\\mathcal{B}_{r',s'}$ for \nsome $r' \\ge s'$, then \\emph{both} $r' \\ge r$ and $s' \\ge s$ must hold. The definitions in \\eqref{s-def} and \\eqref{r-def} clearly \nimply $A + \\lambda B \\notin \\mathcal{B}_{r',s'}$ for any pair $r'$ and $s' < s$ or any pair $r' < r$ and $s' = s$.\nIt remains to show that $A + \\lambda B \\notin \\mathcal{B}_{r',s'}$ also for $r > r' \\ge s' > s$. \nSuppose on the contrary that $A + \\lambda B \\in \\mathcal{B}_{r',s'}$ with $r > r' \\ge s' > s$. \nThis implies there exists $T \\in \\GL_2\\left( \\mathbb{F} \\right)$ such that \n\\begin{equation*}\n \\rank (t_{11} A + t_{12} B) \\le r' \\quad \\text{and} \\quad \\rank (t_{21} A + t_{22} B) \\le s', \\quad \n \\text{with} \\quad r > r' \\ge s' > s.\n\\end{equation*}\nBut then, $r' > s$ implies $(t_{11},t_{12}) \\notin \\sspan\\{(t^\\star,u^\\star)\\}$, where $(t^\\star,u^\\star)$ is a minimizer of \n\\eqref{s-def}. This contradicts the definition of $r$ given by \\eqref{r-def}. \n\n\n\\begin{prop}\n\\label{prop:r-s}\nIf $\\rho(A,B) = (r,s)$ and $r' \\ge s'$, then $A + \\lambda B \\in \\mathcal{B}_{r',s'}$ if and only if $r' \\ge r$ and $s' \\ge s$.\n\\end{prop}\n\n\nWe consider now some examples.\n\n\\begin{exmp}\nA regular $n \\times n$ pencil $A + \\lambda B$ can only belong to $\\mathcal{B}_r$ if $r = n$. Indeed, $A + \\lambda B \\in \\mathcal{B}_r$ \nimplies $A$ and $B$ are proportional, say $B = \\alpha A$, and $\\rank(t A + u B) = \\rank((t + \\alpha u) A) \\le r$ for any \n$(t,u) \\in \\mathbb{F}^2$. As a concrete example, $E + \\lambda E$ is clearly in $\\mathcal{B}_n$ but not in any $\\mathcal{B}_{r'}$ with \n$r' < n$.\n\\end{exmp}\n\n\\begin{exmp}\nRegular $n \\times n$ pencils can also be in $\\mathcal{B}_{r,s}$ for some $n > r \\ge s > 0$. For example, the regular $3 \\times 3$ \npencil\\footnote{The dimensions of the zero blocks in that expression should be clear from the context.} $E_2 \\oplus 0 + \\lambda (0 \n\\oplus E_1)$ is in $\\mathcal{B}_{2,1}$. Yet, the constraint $r + s \\ge n$ must be satisfied. Indeed, $A + \\lambda B \\in \n\\mathcal{B}_{r,s}$ means $A + \\lambda B$ is $\\GL_{m\\times n\\times 2}(\\mathbb{F})$-equivalent to some $A' + \\lambda B'$, with $\\rank A' \n\\le r$ and $\\rank B' \\le s$. If $r + s < n$, then clearly $\\det(A' + \\lambda B') \\equiv 0$, implying neither $A' + \\lambda B'$ nor $A \n+ \\lambda B$ is regular.\n\\end{exmp}\n\nThe next two examples underline how the elementary divisors of a regular pencil determine its minimal ranks. A general result \nestablishing this connection will be presented ahead.\n\n\\begin{exmp}\n The pencil \n \\begin{equation*}\n Q + \\lambda E =\n \\begin{pmatrix} a & b \\\\ -b & a \\end{pmatrix}\n + \\lambda \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} \n \\end{equation*} \n with $b \\neq 0$ has minimal ranks $\\rho(Q,E) = (2,2)$ in $\\mathbb{R}$, because\n \\begin{equation*}\n \\rank \\begin{pmatrix} u + t a & t b \\\\ -t b & u + t a \\end{pmatrix} = 2\n \\end{equation*} \nfor any $(t,u) \\in \\mathbb{R}^2 \\setminus \\{0\\}$. However, this is not true in $\\mathbb{C}$, because the $\\GL_2(\\mathbb{C})$ \ntransformation $(Q,E) \\mapsto ((-a + bi) E + Q, E )$ yields $B + \\lambda E$, where\n\\begin{equation*}\n B = \\begin{pmatrix} bi & b \\\\ -b & bi \\end{pmatrix} = \n \\begin{pmatrix} bi & bi \\\\ -b & -b \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\\\ 0 & -i \\end{pmatrix} \n\\end{equation*} \nhas rank 1. This difference comes from the fact that $Q + \\lambda E$ has a single elementary divisor over $\\mathbb{R}$, namely \n$\\lambda^2 + 2a\\lambda + (a^2 + b^2)$, which cannot be factored into powers of linear forms since its roots are complex. \nIn fact, $Q$ is diagonalizable over $\\mathbb{C}$, since it is similar to\n\\begin{equation*}\n B' = \\begin{pmatrix} a + bi & 0 \\\\ 0 & a - bi \\end{pmatrix}\n\\end{equation*}\nFrom \\Cref{prop:rho-orbit}, we have $\\rho(Q,E) = \\rho(B',E)$, and it is not hard to see that $\\rho(B',E) = (1,1)$. \n\\end{exmp}\n\n\\begin{exmp}\nDefining\n\\begin{equation*}\n H = \\begin{pmatrix} a & 1 & 0 \\\\ 0 & a & 1 \\\\ 0 & 0 & a \\end{pmatrix}, \\quad\n H' = \\begin{pmatrix} a & 1 & 0 \\\\ 0 & a & 0 \\\\ 0 & 0 & a \\end{pmatrix}\n \\quad \\text{and} \\quad \n H'' = \\begin{pmatrix} a & 1 & 0 \\\\ 0 & a & 0 \\\\ 0 & 0 & b \\end{pmatrix}\n\\end{equation*} \nwith $a \\neq b$, we have $\\rho(H,E) = (3,2)$, $\\rho(H',E) = (3,1)$ and $\\rho(H'',E) = (2,2)$. Note that the three \nconsidered pencils are regular and, in particular, the eigenvalues of the first two are the same but their elementary divisors are not. \nIn fact, their invariant polynomials are $\\{(a+\\lambda)^3,1,1\\}$ for $H + \\lambda E$, $\\{(a+\\lambda)^2,a+\\lambda,1\\}$ for $H' \n+ \\lambda E$, and $\\{(a+\\lambda)^2,b+\\lambda,1\\}$ for $H'' + \\lambda E$. \n\\end{exmp}\n\n\n\\subsection{Induced hierarchy of matrix pencils}\n\\label{sec:hierarchy}\n\nThe tensor rank induces a straightforward hierarchy in any tensor space, namely, $\\mathcal{S}_0 \\subset \\mathcal{S}_1 \\subset \n\\mathcal{S}_2 \\subset \\dots$, where $\\mathcal{S}_r$ contains all tensors of rank up to $r$. Our definition of minimal ranks also \ninduces a hierarchy which can be expressed by using the definition of the sets $\\mathcal{B}_{r,s}$ given in \\eqref{Bset}. However, \nsuch a hierarchy is more intricate, as now we have, for instance, $\\mathcal{B}_{r,s} \\subset \n\\mathcal{B}_{r+1,s}$ and $\\mathcal{B}_{r,s} \\subset \\mathcal{B}_{r,s+1}$ but $\\mathcal{B}_{r+1,s} \\not\\subset \n\\mathcal{B}_{r,s+1}$ and $\\mathcal{B}_{r,s+1} \\not\\subset \\mathcal{B}_{r+1,s}$.\n\n\n\\Cref{fig:hierarchy} contains a diagram depicting the hierarchy of $\\mathcal{B}_{r,s}$ sets in the space of $m \\times n$ real pencils, \ndenoted by $\\mathcal{P}_{m,n}(\\mathbb{R})$. \nWe assume $m \\le n$ without loss of generality, since $\\mathcal{P}_{m,n}(\\mathbb{R})$ and $\\mathcal{P}_{n,m}(\\mathbb{R})$ have \nidentical structures. \nFor even $m$, the set $\\mathcal{B}_{m,m} \\subset \\mathcal{P}_{m,n}(\\mathbb{R})$ is always non-empty, but for odd $m$ the set \n$\\mathcal{B}_{m,m} \\subseteq \\mathcal{P}_{m,n}(\\mathbb{R})$ is non-empty if and only if $m < n$.\nThis is because an $m \\times m$ pencil has full minimal ranks if and only if it is regular and its elementary divisors cannot be \nwritten as powers of linear forms, as we shall prove in the next section. In $\\mathcal{P}_{m,m}(\\mathbb{R} )$, this means that a \npencil $A + \\lambda B$ satisfies $\\rho(A,B) = (m,m)$ if and only if it is $\\GL_{m,m,2}(\\mathbb{R})$-equivalent to some other pencil \n$Q + \\lambda E$ where no eigenvalue of $Q$ is in $\\mathbb{R}$. Since complex-valued eigenvalues of a real matrix necessarily \narise in pairs, this can evidently only happen for even values of $m$.\nFor concreteness, three examples concerning $\\mathcal{P}_{2,2}(\\mathbb{R})$, $\\mathcal{P}_{2,3}(\\mathbb{R})$ and \n$\\mathcal{P}_{3,3}(\\mathbb{R})$ are shown in \\Cref{fig:hierarchy}.\n\n\n\\begin{figure}[h!]\n\\centering\n \\includegraphics[width=0.8\\textwidth]{hierarchy.eps}\n\\caption{Hierarchy of sets of real pencils in $\\mathcal{P}_{m,n}(\\mathbb{R})$ according to their minimal ranks: illustration of \nthe general form (left) and concrete examples for three pencil spaces (right). The notation $\\mathcal{B}_{r,s} \\relbar\\joinrel\\mathrel\\RHD \n\\mathcal{B}_{r',s'}$ stands for $\\mathcal{B}_{r,s} \\subset \\mathcal{B}_{r',s'}$. See \\eqref{Bset} for a definition of the sets \n$\\mathcal{B}_{r,s}$.}\n \\label{fig:hierarchy}\n\\end{figure}\n\n\\subsection{Minimal ranks of Kronecker canonical forms}\n\\label{sec:Kronecker}\n\nWe now show how the minimal ranks of a pencil can be determined from its Kronecker canonical form. The notation $J_m(a) \n\\triangleq a E_m + H_m$, where $H_m \\triangleq \\sum_{l=1}^{m-1} e_l \\otimes e_{l+1}$, will be used for a \nJordan block of size $m$ associated with the finite elementary divisor $(a + \\lambda)^m$. In this definition, the vectors $e_l$ \ndenote as usual the canonical basis vectors of their corresponding spaces. \nA canonical $v \\times v$ block associated with an infinite elementary divisor will be expressed as $N_v(\\lambda) \\triangleq E_m \n+ \\lambda H_v$.\nLet us first consider regular pencils.\n\n\n\\begin{lem}\n\\label{lem:rho-regular} \n Let $A + \\lambda B$ be a regular $n \\times n$ pencil and let \n$$ \n A' + \\lambda B' = \\left[ N_{v_1}(\\lambda) \\oplus \\dots \\oplus N_{v_j}(\\lambda) \\right] \\oplus \n \\left[ (A_1 \\oplus \\dots \\oplus A_k \\oplus Q) + \\lambda E \\right]\n$$ \nbe its Kronecker canonical form, where the elementary divisors of $ Q + \\lambda E_q$ cannot be factored into powers of linear forms\nand $A_l = J_{m_l}(a_l)$ for some $a_l \\in \\mathbb{F}$, $l \\in \\{1,\\ldots,k\\}$. Let $k_s$ be the largest number of blocks $A_l$ whose \nelementary divisors share a common factor $a^\\star + \\lambda$ and $k_r$ be the second largest number of blocks $A_l$ whose elementary \ndivisors share a common factor $a^{\\star\\star} + \\lambda$ (with $a^{\\star\\star} \\neq a^{\\star}$). Then \n\\begin{equation}\n\\label{rho-reg}\n \\rho(A,B) = \\left( n - k'_r, n - k'_s \\right),\n\\end{equation}\nwhere $k'_s$ and $k'_r$ are the first and second largest components of $(j,k_s,k_r)$, respectively. \n\\end{lem}\n\n\\begin{proof}\nBy virtue of \\Cref{prop:rho-orbit}, we have $\\rho(A, B) = \\rho(A',B')$. It thus suffices to show that $\\rho(A',B') = (n - k'_r, n - \nk'_s)$. The steps are as follows.\n\\begin{enumerate}\n\n\\item First, we claim that $t Q + u E_q$ is nonsingular for any $(t,u) \\neq 0$. \nThis claim is trivially true if $t=0$ and $u \\neq 0$. For $t \\neq 0$ (and $u$ possibly null), the argument is as follows. Suppose for \na contradiction that $t Q + u E_q$ is singular, with $t \\neq 0$. Without loss of generality, we may take $t=1$. Then, there \nexists $P \\in \\GL_q(\\mathbb{F})$ such that \n\\begin{equation*}\n P \\left( Q + u E_q \\right) P^{-1} = P Q P^{-1} + u E_q = F \\oplus J_p(0),\n\\end{equation*} \nwhere $0 < p \\le q$. But then, $P Q P^{-1} = (F - u E_{q-p}) \\oplus J_p(-u)$, implying $Q + \\lambda E_q$ has an elementary \ndivisor of the form $(-u + \\lambda)^p$, which contradicts the hypothesis that the elementary divisors of $Q$ cannot be written as\npowers of linear forms. As a consequence, $\\rho(Q,E_q) = (q,q)$. In particular, if $Q = A'$ (i.e., $n = q$), then \\eqref{rho-reg} \nyields $\\rho(A',B') = (n,n)$ (because $j = k_s = k_r = 0$), as required.\n\n\\item Now, note that $\\rank\\, (t J_{m}(a) + u E_m) < m$ for $(t,u) \\neq 0$ if and only if $(t,u) = c(1,-a)$ for some $c \\neq 0$, in \nwhich case $\\rank\\, (t J_{m}(a) + u E_m) = \\rank J_{m}(0) = m-1$. Moreover, $\\rank\\, (t E_v + u H_v) < v$ for $(t,u) \\neq 0$ if and \nonly if $(t,u) = (0,c)$ for some $c \\neq 0$, implying $\\rank\\, (t E_v + u H_v) = v-1$.\nHence, since by definition $k_s \\ge k_r$, we have three cases:\n\\begin{enumerate}[(i)]\n\n \\item If $k_s \\ge j \\ge k_r$, then we can apply the following transformation\n \\begin{equation*}\n (A',B') \\mapsto (B', A' -a^\\star B') \n \\end{equation*} \nto obtain an $\\GL_{n,n,2}(\\mathbb{F})$-equivalent pencil attaining the minimal ranks of $(A',B')$. Indeed, $\\rank \\, (A' -a^\\star B') \n= n - k_s$ is minimal among all linear combinations $t A' + u B'$ with $(t,u) \\neq 0$. Hence, \\eqref{s-def} must equal $n - k_s = n - \nk'_s$. Furthermore, $\\rank B' = n - j$ is minimal among all linear combinations $t A' + u B'$ with $(t,u) \\neq c (1, -a^\\star)$. So, \n\\eqref{r-def} must equal $r' = n - j = n - k'_r$.\n\n \\item If $k_s \\ge k_r \\ge j$, then using a similar argument we deduce that the $\\GL_{n,n,2}(\\mathbb{F})$-equivalent pencil $(A' \n-a^{\\star\\star} B') + \\lambda( A' -a^\\star B' )$ attains the minimal ranks of $(A',B')$, showing that $\\rho(A',B') = (n-k_r, n-k_s) \n= (n-k'_r, n-k'_s)$.\n\n \\item Finally, if $j \\ge k_s \\ge k_r$, then following the same line of thought we have that $(A' -a^{\\star} B') + \\lambda B'$ \nattains the minimal ranks of $(A',B')$, that is, $\\rho(A',B') = (n-k_r, n-j) = (n-k'_r, n-k'_s)$.\n\n\\end{enumerate}\n\n\\end{enumerate}\n\\end{proof}\n\n\nFor a singular $m \\times n$ canonical pencil $A + \\lambda B$ having no regular part, computing the minimal ranks is \nstraightforward, because $a_{i,j}$ and $b_{i,j}$ cannot be both nonzero for any given pair of indices $(i,j)$. Indeed, the \ncanonical block $L_k(\\lambda)$ related to a minimal index $k$ associated with the columns is the $k \\times (k+1)$ pencil of \nthe form\n\\begin{equation*}\nL_k(\\lambda) \\triangleq\n \\begin{pmatrix}\n \\lambda & 1 & & & \\\\\n & \\lambda & 1 & & \\\\\n & & \\ddots & \\ddots & \\\\\n & & & \\lambda & 1\n \\end{pmatrix}.\n\\end{equation*} \nAny $\\GL_2(\\mathbb{F})$ transformation applied to $L_k(\\lambda)$ yields some pencil $A' + \\lambda B'$ such that $\\rank \nA' = \\rank B' = k$. In other words, $L_k(\\lambda)$ has minimal ranks $(k,k)$. By the same argument, the canonical block $R_l(\\lambda)$ \nrelated to a minimal index $l$ associated with the rows, which is an $(l+1) \\times l$ pencil defined as $R_l(\\lambda) \\triangleq \nL^\\T_l(\\lambda)$, has minimal ranks $(l,l)$. The special case $k = 0$ (or $l = 0$) also adheres to that \nrule, as its minimal ranks are $(0,0)$.\nNow, adjoining blocks having these forms yields a singular pencil \n$L_{k_1}(\\lambda) \\oplus \\dots \\oplus L_{k_p}(\\lambda) \\oplus R_{l_1}(\\lambda) \\oplus \\dots \\oplus R_{l_q}(\\lambda)$ whose minimal \nranks are clearly the sum of the minimal ranks of the blocks. Note that this is true even for the zero minimal indices $k_1 = \\dots = \nk_{p'} = l_1 = \\dots = l_{q'} = 0$, since they correspond to $p'$ null columns and $q'$ null rows, and so the minimal ranks must be \nbounded by $\\min\\{m - q', n-p'\\}$. We have arrived at the following result. \n\n\\begin{lem}\n\\label{lem:rho-singular}\nLet $A + \\lambda B$ be a singular $m \\times n$ pencil having the form $A + \\lambda B = L_{k_1}(\\lambda) \\oplus \\dots \\oplus \nL_{k_p}(\\lambda) \\oplus R_{l_1}(\\lambda) \\oplus \\dots \\oplus \nR_{l_q}(\\lambda)$, where $k_1, \\ldots, k_p$ are the minimal indices associated with its columns (henceforth called minimal column \nindices) and $l_1,\\ldots,l_q$ are the minimal indices associated with its rows (minimal row indices). \nThen, \\begin{equation*}\n \\rho(A,B) = (\\bar{s},\\bar{s}), \\quad \\text{where } \\bar{s} = k_1 + \\dots + k_p + l_1 + \\dots + l_q.\n\\end{equation*}\n\\end{lem}\n\nFor an arbitrary $m \\times n$ pencil $A + \\lambda B$, the block diagonal structure of its Kronecker canonical form allows a direct \ncombination of the previous results, yielding the main theorem of this section.\n\n\\begin{theo}\n\\label{theo:rho-canonical}\nLet $A + \\lambda B$ be an arbitrary $m \\times n$ pencil with Kronecker canonical form $S(\\lambda) \\oplus (A' + \\lambda B')$, where \n$S(\\lambda)$ is its singular part and $A' + \\lambda B'$ is regular. Suppose $S(\\lambda)$ has minimal column indices $k_1, \\ldots, k_p$ \nand minimal row indices $l_1, \\ldots, l_q$. Define $\\bar{s} = k_1 + \\dots + k_p + l_1 + \\dots + l_q$. Then, its minimal ranks are given \nby\n\\begin{equation}\n \\label{rho-canonical}\n \\rho(A, B) = (r' + \\bar{s}, s' + \\bar{s}), \n\\end{equation} \nwhere $(r', s') = \\rho(A',B')$, whose components are given by \\Cref{lem:rho-regular}.\n\\end{theo}\n\nThe above result implies that both minimal ranks of an $n \\times n$ singular pencil must be strictly smaller than $n$. This is because \nthe sum of the minimal indices of its singular part (which equals $\\bar{s}$ in \\eqref{rho-canonical}) can never attain the largest \ndimension of that part. Hence, if an $n \\times n$ pencil has minimal ranks $(n,n)$ or $(n,n-1)$, then it is necessarily regular. \nIn particular, $n \\times n$ pencils with full minimal ranks can be characterized as follows.\n\n\\begin{corol}\n\\label{cor:full-min-rank}\nAn $n \\times n$ pencil $A + \\lambda B$ satisfies $\\rho(A,B) = (n,n)$ if and only if it is regular and its elementary divisors cannot be \nwritten as powers of linear forms.\n\\end{corol}\n\n\n\n\n\\subsection{Classification of $\\GL_{m,n,2}(\\mathbb{R})$-orbits for $m,n \\le 4$}\n\\label{sec:classif}\n\nUsing \\Cref{theo:rho-canonical}, a complete classification of all Kronecker canonical forms of $m \\times n$ pencils over \n$\\mathbb{R}$ is provided in \\Cref{tab:1,tab:2,tab:3,tab:4} for $m, n \\in \\{1,\\ldots,4\\}$. Each such form is associated with a \nfamily of $\\GL_{m,n,2}(\\mathbb{R})$-orbits. We denote canonical blocks whose elementary divisors are powers of second-order \nirreducible polynomials by\n\\begin{equation*}\n Q_{2k}(a,b) = E_k \\kron \\begin{pmatrix} a & b \\\\ -b & a \\end{pmatrix} + J_2(0) \\kron E_k \\quad \\text{with} \\quad b \\neq 0,\n\\end{equation*}\nwhere $\\kron$ denotes the Kronecker product. Observe that, because we consider orbits of $\\GL_{m,n,2}(\\mathbb{R})$ action, we \ncan represent each family of orbits by a canonical form having no infinite elementary divisors (which can always be avoided by \nemploying a $\\GL_{2}(\\mathbb{R})$ transformation).\n\nIt can be checked that each described family with $m, n \\le 3$ corresponds to a single orbit,\\footnote{Atkinson \n\\cite{Atkinson1991} had already pointed out that, over an algebraically closed field $\\mathbb{F}$, there are only finitely many \n$\\GL_{3,n,2}(\\mathbb{F})$-orbits for any $n$. Thus, over $\\mathbb{R}$ this must be true of orbits whose elementary divisors are \npowers of linear forms.} except for $\\mathscrc{R}'_{3,2}$, which encompasses an infinite number of non-equivalent orbits. All families \nhaving dimensions $m = 3$ and $n = 4$ (or $m = 4$ and $n = 3$) also contain only one orbit each. These properties can be \nverified by inspecting the equivalent pencils of \\Cref{tab:equiv} shown ahead in \\Cref{sec:equiv}: for $\\min\\{m,n\\} \\le 3$, the only \nfamily whose given canonical form depends on a parameter is that of $\\mathscrc{R}'_{3,2}$.\nFor $m = n = 4$, infinitely many non-equivalent orbits are contained by each family in general. \nTo avoid redundancies, families of orbits having zero minimal indices are omitted in the tables, since each such family corresponds \nto some other one of lower dimensions. For instance for $m=n=4$, if a singular pencil $A + \\lambda B$ has minimal indices \n$k_1 = k_2 = l_1 = l_2 = 0$, then it can be expressed in the form $0 \\oplus (A' + \\lambda B')$, where both blocks in this \ndecomposition have size $2 \\times 2$. So, the canonical form of $A' + \\lambda B'$ can be inspected to determine the properties \nof $A + \\lambda B$. Similarly, not all combinations of canonical blocks are included for the singular part, because the shown \nproperties remain the same if we transpose these blocks. To exemplify, it can be checked that $L_1 \\oplus R_2$ and $R_1 \\oplus L_2$ \nhave the same dimensions, tensor rank, multilinear rank and minimal ranks, because the roles played by column and row minimal indices \nare essentially the same, up to a transposition.\n\nA family is denoted with the letter $\\mathscrc{R}$ or the letter $\\mathscrc{S}$ if it encompasses regular or singular pencils, \nrespectively. The subscript indices of each family indicate its minimal ranks, and primes are used to distinguish among otherwise \nidentically labeled families. The tensor rank of each canonical form was determined using Corollary 2.4.1 of Ja'Ja' \\cite{JaJa1979} \nand Theorem 4.6 of Sumi et al.~\\cite{Sumi2009}, which requires taking into account the minimal indices of the pencil and also its \nelementary divisors. The values given in the column ``multilinear rank'' were determined by inspection; for a definition see \n\\cite{deSilva2008}. Specifically, for an $m \\times n$ pencil $A + \\lambda B$ viewed as an $m \\times n \\times 2$ tensor $A \n\\otimes e_1 + B \\otimes e_2$, the multilinear rank is the triple $(r_1,r_2,r_3)$ satisfying\n\\begin{equation*}\n r_1 = \\rank \\begin{pmatrix} A & B \\end{pmatrix}, \\quad \n r_2 = \\rank \\begin{pmatrix} A^\\T & B^\\T \\end{pmatrix}, \\quad \n r_3 = \\dim \\sspan \\{A, B\\}.\n\\end{equation*} \nIt should be noted that, in the above equation, $\\sspan \\{A, B\\}$ denotes the subspace of $\\mathbb{R}^{m \\times n}$ spanned by \nthe matrices $A$ and $B$, whose dimension is at most two. Finally, in order to determine the minimal ranks (column labeled ``$\\rho$'') \nof each family, \\Cref{theo:rho-canonical} was applied.\n\n\nWe point out that another classification of pencil orbits is given by Pervouchine \n\\cite{Pervouchine2004}, but his study is concerned with closures of orbits and pencil bundles, not with tensor rank or minimal ranks. \nOur list is therefore a complement to the one he provides. Furthermore, the hierarchy of closures of pencil bundles he has \npresented bears no direct connection with the hierarchy of sets $\\mathcal{B}_{r,s}$ we present in \\Cref{sec:hierarchy}, which is \neasily determined by the numbers $r,s$ associated with each such set.\n\n\\begin{table}[t!]\n\\begin{center}\n\\def1.6{1.6}\n\\begin{tabular}{c | c |c|c|c|c}\n\\hline \n\\multirow{2}{*}{Family} & \\multirow{2}{*}{Canonical form} & $m \\times n$ & tensor & multilinear & \\multirow{2}{*}{$\\rho$} \\\\[-8pt]\n& & & rank & rank & \\\\\n\\hline\n$\\mathscrc{R}_{1,0}$ & $a + \\lambda\\, E_1$ & $1 \\times 1$ & 1 & $(1,1,1)$ & $(1,0)$ \\\\[2pt] \n\\hline \n$\\mathscrc{S}_{1,1}$ & $L_1(\\lambda)$ & $1 \\times 2$ & 2 & $(1,2,2)$ & $(1,1)$ \\\\[2pt] \n\\hline\n\\end{tabular}\n\\end{center}\n \\caption{Families of canonical forms of $1 \\times 1$ and $1 \\times 2$ real pencils having no zero minimal indices, with $a_i \n\\neq a_j$ for $i \\neq j$.}\n \\label{tab:1}\n\\end{table}\n\n\\begin{table}[t!]\n\\begin{center}\n\\def1.6{1.6}\n\\begin{tabular}{c | c |c|c|c|c}\n\\hline \n\\multirow{2}{*}{Family} & \\multirow{2}{*}{Canonical form} & $m \\times n$ & tensor & multilinear & \\multirow{2}{*}{$\\rho$} \\\\[-8pt] \n& & & rank & rank & \\\\\n\\hline\n$\\mathscrc{R}_{1,1}$ & $a_1 \\oplus a_2 + \\lambda\\, E_2$ & $2 \\times 2$ & 2 & $(2,2,2)$ & $(1,1)$ \\\\ \n$\\mathscrc{R}_{2,0}$ & $a \\oplus a + \\lambda\\, E_2$ & $2 \\times 2$ & 2 & $(2,2,1)$ & $(2,0)$ \\\\[2pt]\n\\hline \n$\\mathscrc{R}_{2,1}$ & $J_2(a) + \\lambda\\, E_2$ & $2 \\times 2$ & 3 & $(2,2,2)$ & $(2,1)$ \\\\[2pt] \n\\hline \n$\\mathscrc{R}_{2,2}$ & $Q_2(a,b) + \\lambda\\, E_2$ & $2 \\times 2$ & 3 & $(2,2,2)$ & $(2,2)$ \\\\[2pt] \n\\hline\n$\\mathscrc{S}_{2,2}$ & $L_2(\\lambda)$ & $2 \\times 3$ & 3 & $(2,3,2)$ & $(2,2)$ \\\\[2pt] \n$\\mathscrc{S}_{2,1}$ & $L_1(\\lambda) \\oplus (a + \\lambda\\, E_1)$ & $2 \\times 3$ & 3 & $(2,3,2)$ & $(2,1)$ \\\\[2pt] \n\\hline\n$\\mathscrc{S}'_{2,2}$ & $L_1(\\lambda) \\oplus L_1(\\lambda)$ & $2 \\times 4$ & 4 & $(2,4,2)$ & $(2,2)$ \\\\[2pt] \n\\hline\n\\end{tabular}\n\\end{center}\n \\caption{Families of canonical forms of $2 \\times 2$ and $2 \\times 3$ real pencils having no zero minimal indices, with $a_i \n\\neq a_j$ for $i \\neq j$ and $b \\neq 0$.}\n \\label{tab:2}\n\\end{table}\n\n\n\\begin{table}[t!]\n\\begin{center}\n\\def1.6{1.6}\n\\begin{tabular}{c | c |c|c|c|c}\n\\hline \n\\multirow{2}{*}{Family} & \\multirow{2}{*}{Canonical form} & $m \\times n$ & tensor & multilinear & \\multirow{2}{*}{$\\rho$} \\\\[-8pt] \n& & & rank & rank & \\\\\n\\hline \n$\\mathscrc{S}''_{2,2}$ & $L_1(\\lambda) \\oplus R_1(\\lambda)$ & $3 \\times 3$ & 4 & $(3,3,2)$ & $(2,2)$ \\\\[2pt] \n\\hline \n$\\mathscrc{R}'_{2,2}$ & $a_1 \\oplus a_2 \\oplus a_3 + \\lambda\\, E_3$ & $3 \\times 3$ & 3 & $(3,3,2)$ & $(2,2)$ \\\\ \n$\\mathscrc{R}'_{2,1}$ & $a_1 \\oplus a_2 \\oplus a_2 + \\lambda\\, E_3$ & $3 \\times 3$ & 3 & $(3,3,2)$ & $(2,1)$ \\\\ \n$\\mathscrc{R}_{3,0}$ & $a \\oplus a \\oplus a + \\lambda\\, E_3$ & $3 \\times 3$ & 3 & $(3,3,1)$ & $(3,0)$ \\\\[2pt] \n\\hline \n$\\mathscrc{R}_{3,2}$ & $J_3(a) + \\lambda\\, E_3$ & $3 \\times 3$ & 4 & $(3,3,2)$ & $(3,2)$ \\\\ \n$\\mathscrc{R}_{3,1}$ & $a \\oplus J_2(a) + \\lambda\\, E_3$ & $3 \\times 3$ & 4 & $(3,3,2)$ & $(3,1)$ \\\\ \n$\\mathscrc{R}''_{2,2}$ & $a_1 \\oplus J_2(a_2) + \\lambda\\, E_3$ & $3 \\times 3$ & 4 & $(3,3,2)$ & $(2,2)$ \\\\[2pt]\n\\hline \n$\\mathscrc{R}'_{3,2}$ & $a \\oplus Q_2(c,d) + \\lambda\\, E_3$ & $3 \\times 3$ & 4 & $(3,3,2)$ & $(3,2)$ \\\\[2pt] \n\\hline\n$\\mathscrc{S}'''_{2,2}$ & $L_1(\\lambda) \\oplus (a_1 \\oplus a_2 + \\lambda\\, E_2)$ & $3 \\times 4$ & 4 & $(3,4,2)$ & $(2,2)$ \\\\ \n$\\mathscrc{S}_{3,1}$ & $L_1(\\lambda) \\oplus (a \\oplus a + \\lambda\\, E_2)$ & $3 \\times 4$ & 4 & $(3,4,2)$ & $(3,1)$ \\\\ \n$\\mathscrc{S}_{3,2}$ & $L_1(\\lambda) \\oplus (J_2(a) + \\lambda\\, E_2)$ & $3 \\times 4$ & 5 & $(3,4,2)$ & $(3,2)$ \\\\ \n$\\mathscrc{S}_{3,3}$ & $L_1(\\lambda) \\oplus (Q_2(a,b) + \\lambda\\, E_2)$ & $3 \\times 4$ & 5 & $(3,4,2)$ & $(3,3)$ \\\\\n$\\mathscrc{S}'_{3,2}$ & $L_2(\\lambda) \\oplus (a + \\lambda\\, E_1)$ & $3 \\times 4$ & 4 & $(3,4,2)$ & $(3,2)$ \\\\ \n$\\mathscrc{S}'_{3,3}$ & $L_3(\\lambda)$ & $3 \\times 4$ & 4 & $(3,4,2)$ & $(3,3)$ \\\\[2pt] \n\\hline\n\\end{tabular}\n\\end{center}\n \\caption{Families of canonical forms of $3 \\times 3$ and $3 \\times 4$ real pencils having no zero minimal indices, with $a_i \n\\neq a_j$ for $i \\neq j$, $b \\neq 0$ and $d \\neq 0$.}\n \\label{tab:3}\n\\end{table}\n\n\n\\begin{table}[t!]\n\\begin{center}\n\\def1.6{1.6}\n\\begin{tabular}{c | c |c|c|c|c}\n\\hline \n\\multirow{2}{*}{Family} & \\multirow{2}{*}{Canonical form} & $m \\times n$ & tensor & multilinear & \\multirow{2}{*}{$\\rho$} \\\\[-8pt]\n& & & rank & rank & \\\\\n\\hline \n$\\mathscrc{S}''_{3,2}$ & $L_1(\\lambda) \\oplus R_1(\\lambda) \\oplus (a + \\lambda\\, E_1)$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(3,2)$ \n\\\\[2pt] \n$\\mathscrc{S}''_{3,3}$ & $L_2(\\lambda) \\oplus R_1(\\lambda) $ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(3,3)$ \\\\[2pt] \n\\hline\n$\\mathscrc{R}_{4,0}$ & $a \\oplus a \\oplus a \\oplus a + \\lambda\\, E_4$ & $4 \\times 4$ & 4 & $(4,4,1)$ & $(4,0)$ \\\\ \n$\\mathscrc{R}'''_{2,2}$ & $a_1 \\oplus a_1 \\oplus a_2 \\oplus a_2 + \\lambda\\, E_4$ & $4 \\times 4$ & 4 & $(4,4,2)$ & $(2,2)$ \\\\ \n$\\mathscrc{R}'_{3,1}$ & $a_1 \\oplus a_2 \\oplus a_2 \\oplus a_2 + \\lambda\\, E_4$ & $4 \\times 4$ & 4 & $(4,4,2)$ & $(3,1)$ \\\\ \n$\\mathscrc{R}''_{3,2}$ & $a_1 \\oplus a_2 \\oplus a_3 \\oplus a_3 + \\lambda\\, E_4$ & $4 \\times 4$ & 4 & $(4,4,2)$ & $(3,2)$ \\\\ \n$\\mathscrc{R}_{3,3}$ & $a_1 \\oplus a_2 \\oplus a_3 \\oplus a_4 + \\lambda\\, E_4$ & $4 \\times 4$ & 4 & $(4,4,2)$ & $(3,3)$ \\\\[2pt] \n\\hline\n$\\mathscrc{R}'''_{3,2}$ & $a_1 \\oplus a_2 \\oplus J_2(a_2) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(3,2)$ \\\\ \n$\\mathscrc{R}''''_{3,2}$ & $a_1 \\oplus a_1 \\oplus J_2(a_2) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(3,2)$ \\\\\n$\\mathscrc{R}_{4,1}$ & $a \\oplus a \\oplus J_2(a) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(4,1)$ \\\\ \n$\\mathscrc{R}'_{3,3}$ & $J_2(a_1) \\oplus J_2(a_2) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(3,3)$ \\\\ \n$\\mathscrc{R}''_{3,3}$ & $a_1 \\oplus a_2 \\oplus J_2(a_3) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(3,3)$ \\\\ \n$\\mathscrc{R}'''_{3,3}$ & $a_1 \\oplus J_3(a_2) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(3,3)$ \\\\ \n$\\mathscrc{R}_{4,2}$ & $a \\oplus J_3(a) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(4,2)$ \\\\ \n$\\mathscrc{R}'_{4,2}$ & $J_2(a) \\oplus J_2(a) + \\lambda\\, E_4$ & $4 \\times 4$ & 6 & $(4,4,2)$ & $(4,2)$ \\\\ \n$\\mathscrc{R}_{4,3}$ & $J_4(a) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(4,3)$ \\\\[2pt] \n\\hline\n$\\mathscrc{R}''''_{3,3}$ & $a_1 \\oplus a_2 \\oplus Q_2(c,d) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(3,3)$ \\\\ \n$\\mathscrc{R}''_{4,2}$ & $a \\oplus a \\oplus Q_2(c,d) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(4,2)$ \\\\\n$\\mathscrc{R}'_{4,3}$ & $J_2(a) \\oplus Q_2(c,d) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(4,3)$ \\\\[2pt] \n\\hline\n$\\mathscrc{R}_{4,4}$ & $Q_2(a,b) \\oplus Q_2(c,d) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(4,4)$ \\\\ \n$\\mathscrc{R}'_{4,4}$ & $Q_4(a,b) + \\lambda\\, E_4$ & $4 \\times 4$ & 5 & $(4,4,2)$ & $(4,4)$ \\\\ \n$\\mathscrc{R}''_{4,4}$ & $Q_2(a,b) \\oplus Q_2(a,b) + \\lambda\\, E_4$ & $4 \\times 4$ & 6 & $(4,4,2)$ & $(4,4)$ \\\\[2pt] \n\\hline\n\\end{tabular}\n\\end{center}\n \\caption{Families of canonical forms of $4 \\times 4$ real pencils having no zero minimal indices, with $a_i \n\\neq a_j$ for $i \\neq j$, $b \\neq 0$ and $d \\neq 0$.}\n \\label{tab:4}\n\\end{table}\n\n\\subsection{Minimal ranks of matrix polynomials}\n\\label{sec:matrix-poly}\n\n \nOne can generalize \\Cref{def:min-ranks} to matrix polynomials of finite degree as follows. The minimal ranks of a \ndegree-$(d-1)$ polynomial $P(\\lambda) = \\sum_{k=1}^{d} \\lambda^{k-1} A_k$ should correspond to the minimal values $r_1 \\ge \\dots \\ge \nr_d$ such that we can find \na transformation $T \\in \\GL_d(\\mathbb{F})$ for which $(E,E,T) \\cdot P(\\lambda) = \\sum_{k=1}^{d} \\lambda^{k-1} A'_k$ where $\\rank A'_k \n= r_k$ for $k=1,\\ldots,d$. One can thus introduce the \\emph{rank-minimizing subspaces} $\\mathcal{T}_1, \\ldots, \\mathcal{T}_l$ of \n$\\mathbb{F}^d$ with respect to $P(\\lambda)$, where $1 \\le l \\le d$, which are inductively defined in the following manner. First, \nlet $\\mathcal{T}_1$ be the subspace of $\\mathbb{F}^d$ spanned by all solutions of \n\\begin{equation*}\n \\min_{ (t_{1}, \\ldots, t_{d}) \\neq 0 } \\rank \\left( \\sum_{k=1}^{d} t_{k} A_k \\right) = \\bar{r}_1.\n\\end{equation*} \nIf $\\dim \\mathcal{T}_1 = d$, then $\\mathcal{T}_1$ is the only rank-minimizing subspace, that is, $l=1$. Otherwise, we define next \n$\\mathcal{T}_2$ as the subspace of $\\mathbb{F}^d$ spanned by all solutions of \n\\begin{equation*}\n \\min_{ (t_{1}, \\ldots, t_{d}) \\notin \\mathcal{T}_1 } \\rank \\left( \\sum_{k=1}^{d} t_{k} A_k \\right) = \\bar{r}_2.\n\\end{equation*} \nIf $\\dim \\mathcal{T}_1 + \\dim \\mathcal{T}_2 = d$, then we have $\\mathcal{T}_1 \\oplus \\mathcal{T}_2 = \\mathbb{F}^2$ and $l=2$. \nOtherwise, one continues in this fashion until $\\dim \\mathcal{T}_1 + \\dots + \\dim \\mathcal{T}_l = d$ for some $l$, which must happen \nafter finitely many steps. So, each $\\mathcal{T}_p$ is defined as the subspace spanned by the solutions of \n\\begin{equation}\n \\label{ms-Tp}\n \\min_{ (t_{1}, \\ldots, t_{d}) \\notin \\bar{\\mathcal{T}} } \n \\rank \\left( \\sum_{k=1}^{d} t_{k} A_k \\right) = \\bar{r}_p,\n \\quad \\text{where} \\quad\n \\bar{\\mathcal{T}} = \n \\begin{cases}\n \\{ 0 \\}, & p = 1, \\\\\n \\mathcal{T}_1 \\oplus \\dots \\oplus \\mathcal{T}_{p-1}, & 1 < p \\le l.\n \\end{cases}\n\\end{equation} \nIt is clear that $\\mathcal{T}_1 \\oplus \\dots \\oplus \\mathcal{T}_l = \\mathbb{F}^d$. The minimal ranks are then associated with this \ndecomposition, as defined below.\n\n\\begin{defn}\n\\label{def:min-ranks-poly}\nLet $ P(\\lambda) = \\sum_{k=1}^{d} \\lambda^{k-1} A_k$ be an $m \\times n$ matrix polynomial over $\\mathbb{F}$ of degree (at most) $d-1$, \nand let $\\mathcal{T}_1, \\ldots, \\mathcal{T}_l$ be the rank-minimizing subspaces of $\\mathbb{F}^d$ associated with $P(\\lambda)$, with \n$\\dim \\mathcal{T}_p = d_p$.\nThe minimal ranks of $P(\\lambda)$, denoted by $\\rho(P)$, are defined as the components of the $d$-tuple \n\\begin{equation*}\n \\rho(P) = (r_1,\\ldots,r_d) \\triangleq ( \n \\underbrace{\\bar{r}_l,\\ldots,\\bar{r}_l}_{d_l \\text{ times}},\n \\underbrace{\\bar{r}_{l-1},\\ldots,\\bar{r}_{l-1}}_{d_{l-1} \\text{ times}},\n \\ldots,\n \\underbrace{\\bar{r}_2,\\ldots,\\bar{r}_2}_{d_2 \\text{ times}} \n \\underbrace{\\bar{r}_1,\\ldots,\\bar{r}_1}_{d_1 \\text{ times}} \n ),\n\\end{equation*}\nwhere the numbers $\\bar{r}_p$ are given by \\eqref{ms-Tp} and satisfy $\\bar{r}_l > \\bar{r}_{l-1} > \\dots > \\bar{r}_1$.\nWe say that the components of $\\rho(P)$ which equal $\\bar{r}_p$ are associated with $\\mathcal{T}_p$.\n\\end{defn}\n\nFor $d=2$, \\Cref{def:min-ranks-poly} is equivalent to \\Cref{def:min-ranks}. In particular, $\\rho(A,B) = (r,s)$ satisfies $r = s$ if \nand only if $\\mathbb{F}^2$ has a single associated rank-minimizing subspace $\\mathcal{T}_1 = \\mathbb{F}^2$ with respect to $A + \n\\lambda B$. An analogue of \\Cref{lem:GL2} also clearly holds for matrix polynomials.\n\nLet us introduce \n\\begin{equation*}\n\\label{Bset-poly}\n \\mathcal{B}_{r_1,\\ldots,r_d} \\triangleq \n \\left\\{ P(\\lambda) \n \\; \\Bigg| \\;\n \\exists \\, \\sum_{k=1}^{d} \\lambda^{k-1} A'_k \\in \\mathcal{O}(P) \n \\mbox{ such that }\n \\rank A_k' \\le r_k \\mbox{ for } k=1,\\ldots,d \\right\\},\n\\end{equation*}\nwhere $\\mathcal{O}(P)$ stands for the $\\GL_{m,n,d}$-orbit of $P(\\lambda)$.\n$\\mathcal{B}_{r_1,\\ldots,r_d}$ is clearly invariant with respect to a permutation of $r_1,\\ldots,r_d$, and thus we shall assume \n$r_1 \\ge \\dots \\ge r_d$. \nAssuming $\\rho(P) = (r_1,\\ldots,r_d)$, the construction of the rank-minimizing subspaces shows there is a transformation $T \\in \n\\GL_d(\\mathbb{F})$ yielding $(E,E,T) \\cdot P(\\lambda) = \\sum_{k=1}^{d} \\lambda^{k-1} A'_k$ such that $\\rank A'_k = r_k$,\nand thus by definition $P(\\lambda) \\in \\mathcal{B}_{r_1,\\ldots,r_d}$. \nNow, since the rows of any other $T' \\in \\GL_d(\\mathbb{F})$ must span $\\mathbb{F}^d$, it is easy to see \nthat $P(\\lambda)$ cannot belong to any $\\mathcal{B}_{r_1,\\ldots,r_{q-1},r'_q,r_{q+1},\\ldots,r_d}$ such that $r_q > r'_q \\ge \nr_{q+1}$. Indeed, this would contradict the construction of the subspaces $\\mathcal{T}_p$ as being spanned by \\emph{all} solutions of \nthe rank minimization problem \\eqref{ms-Tp}.\nIn fact, if $s_1 \\ge \\dots \\ge s_d$ and $P(\\lambda) \\in \\mathcal{B}_{s_1,\\ldots,s_d}$, then we must have $s_q \\ge r_q$ for all $q \\in \n\\{1,\\ldots,d\\}$, because the assumption $r_q > s_q \\ge s_{q+1} \\ge \\dots \\ge s_d$ is inconsistent with the above construction \nof the rank-minimizing subspaces $\\mathcal{T}_1,\\ldots,\\mathcal{T}_l$ of $\\mathbb{F}^d$ with respect to $P(\\lambda)$. This is the \ncentral argument of the following generalization of \\Cref{prop:r-s}.\n\n\n\\begin{prop}\nLet $ P(\\lambda) = \\sum_{k=1}^{d} \\lambda^{k-1} A_k$ be an $m \\times n$ matrix polynomial over $\\mathbb{F}$.\nIf $\\rho(P) = (r_1,\\ldots,r_d)$ and $s_1 \\ge \\dots \\ge s_d$, then $P(\\lambda) \\in \\mathcal{B}_{s_1,\\ldots,s_d}$ if and only if \n$s_k \\ge r_k$ for all $k \\in \\{1,\\ldots,d\\}$.\n\\end{prop}\n\n\n\nIn view of these extensions, it makes sense to consider the hierarchy of $m \\times n$ matrix polynomials of any finite degree \nin terms of their minimal ranks, leading to a diagram such as that of \\Cref{fig:hierarchy}.\n \n\n\\subsection{Decomposing third-order tensors into matrix-vector tensor products of minimal ranks}\n\\label{sec:MVD}\n\n\nThe connection between $m \\times n$ matrix pencils and $m \\times n \\times 2$ tensors has been exploited time and \nagain to derive many results, such as the characterization of $\\GL_{m,n,2}(\\mathbb{R})$-orbits in terms of their tensor ranks carried \nout by Ja'Ja' and Sumi et al.~\\cite{JaJa1979, Sumi2009}. This correspondence is also well-suited to study block-term \ndecompositions of $m \\times n \\times 2$ tensors composed by matrix-vector products, as we can directly associate the ranks of these \nblocks with the ranks of the matrices constituting the pencil.\n\nMore generally, a natural connection exists between a block-term decomposition of an $m \\times n \\times d$ tensor and $m \\times n$ \npolynomial of degree $d-1$. Given any third order tensor $X \\in \\mathbb{F}^{m} \\otimes \\mathbb{F}^{n} \\otimes \\mathbb{F}^{d}$, it can \nalways be written in the form\n\\begin{equation*}\n X = \\sum_{k=1}^d \\left( \\sum_{j=1}^{r_k} u^{(k)}_j \\otimes v^{(k)}_j \\right) \\otimes w_k,\n\\end{equation*} \nwhere the vectors $w_1, \\ldots, w_d$ are linearly independent. Choosing coordinates for these spaces, this expression can be \nidentified with\n\\begin{equation}\n \\label{BTD}\n X = \\sum_{k=1}^d A_k \\otimes w_k \\quad \\in \\mathbb{F}^{m \\times n \\times d},\n\\end{equation} \nwhere $A_k$ is a matrix of rank at most $r_k$. By suitably defining an isomorphism $\\mathbb{F}^d \\simeq \n\\mathbb{F}_{d-1}[\\lambda]$, where $\\mathbb{F}_{d-1}[\\lambda]$ denotes the space of degree-$(d-1)$ polynomials, the tensor in \n\\eqref{BTD} can be associated with the matrix polynomial\n\\begin{equation*}\n P(\\lambda) = \\sum_{k=1}^d \\lambda^{k-1} A_k.\n\\end{equation*}\n\n\nFrom this link, it becomes evident that the minimal ranks of $P(\\lambda)$ characterize the intrinsic complexity of the tensor $X$ in \n\\eqref{BTD} in terms of the ranks of the matrices appearing in the sum. It thus quantifies the dimensions of the smooth manifolds \nwhose join set contains $X$; see the recent work by Breiding, and Vannieuwenhoven \\cite{Breiding2018} for a discussion on this \ninterpretation of \\eqref{BTD}. In the case of a matrix pencil $P(\\lambda) = A + \\lambda B$ ($d=2$), our results in \n\\Cref{sec:Kronecker} allow one to compute $\\rho(A,B)$ from the Kronecker canonical form of the pencil. Similar characterizations for \nhigher values of $d$ would be surely valuable for the study of the BTD and its properties.\n\n\n\n\\section{A pencil may have no best approximation with strictly lower minimal ranks}\n\\label{sec:ill-posed}\n\nThis section investigates the existence of optimal approximations of a pencil $A + \\lambda B \\in \\mathcal{P}_{m,n}(\\mathbb{F})$ on \n$\\mathcal{B}_{r,s}$, for a given pair $(r,s)$ satisfying $r \\ge s$, for $\\mathbb{F} = \\mathbb{R}$ or $\\mathbb{F} = \n\\mathbb{C}$. \nDefining such approximations requires a topology for $\\mathcal{P}_{m,n}(\\mathbb{F})$. We shall pick the norm topology, which is \nthe same regardless of the chosen norm, since $\\mathcal{P}_{m,n}(\\mathbb{F})$ has a finite dimension. It can be introduced by \nconsidering the inner product\n\\begin{equation}\n \\label{inner}\n \\langle A + \\lambda B, C + \\lambda D \\rangle \\triangleq \\langle A , C \\rangle + \\langle B, D \\rangle, \n\\end{equation}\nwhere the inner product appearing in the right-hand side is the standard Euclidian one given by\n\\begin{equation*}\n \\langle A , C \\rangle \\triangleq \\trace A^* C, \n\\end{equation*}\nwith $A^*$ denoting the adjoint of $A$. This leads to the Euclidian norm\n\\begin{equation*}\n \\| A + \\lambda B \\| \\triangleq \\sqrt{ \\langle A + \\lambda B , A + \\lambda B \\rangle }\n = \\sqrt{ \\|A\\|^2 + \\|B\\|^2},\n\\end{equation*}\nwhich induces the topology. In the above expression, $\\|A\\| \\triangleq \\sqrt{\\langle A, A \\rangle}$ is the Frobenius norm of $A$.\nWith this definition, the approximation problem can be formulated as\n\\begin{equation}\n \\label{best-app}\n \\inf_{ A' + \\lambda B' \\in \\mathcal{B}_{r,s} } \\| A + \\lambda B - (A' + \\lambda B' ) \\|.\n\\end{equation}\nWe are thus interested in determining whether this infimum is attained for a given pencil $A + \\lambda B$ and some choice of ranks $(r,s)$. \nEvidently, this question is only of interest at all when $A + \\lambda B \\not\\in \\mathcal{B}_{r,s}$. To check whether this holds, we \nshall rely on the concept of minimal ranks and its associated results developed in \\Cref{sec:pencils}.\n\nIn view of the isomorphism between $\\mathcal{P}_{m,n}(\\mathbb{F})$ and $\\mathbb{F}^{m \\times n \\times 2}$ discussed in \\Cref{sec:MVD}, \nthe results of the present section apply also to the approximation of tensors from $\\mathbb{F}^{m \\times n \\times 2}$ by a sum \nof two matrix-vector tensor products, in the tensor norm topology.\nTo define this topology, one can consider the Frobenius norm \n\\begin{equation*}\n \\| A \\otimes e_1 + B \\otimes e_2 \\| \\triangleq \\sqrt{\\langle A \\otimes e_1 + B \\otimes e_2 , A \\otimes e_1 + B \\otimes e_2 \\rangle }\n\\end{equation*}\nwhere the scalar product is defined for rank-one tensors as\n\\begin{equation*}\n \\langle u \\otimes v \\otimes w, \\, x \\otimes y \\otimes z \\rangle \\triangleq \\langle u, x \\rangle \\cdot \\langle v, y \\rangle \\cdot \\langle w, z \\rangle\n\\end{equation*}\nand extends bi-linearly to tensors of arbitrary rank. Therefore, \n$$ \\| A \\otimes e_1 + B \\otimes e_2 \\| = \\sqrt{\\|A\\|^2 + \\|B\\|^2} = \\| A + \\lambda B \\|.$$\n\n\n\\subsection{A template for ill-posed pencil approximation problems}\n\\label{sec:ill-posed-examples}\n\nExamples of tensors having no best rank-$r$ approximation in the norm topology have been known for quite a while; see \n\\cite{Comon2014, deSilva2008} and references therein. De Lathauwer \\cite{deLathauwer2008c} has employed the same kind of \nconstruction to provide an example of a tensor having no best approximate block-term decomposition constituted by two blocks of \nmultilinear rank $(2,2,2)$. In the next proposition, we resort to a similar expedient to derive a template of ill-posed instances \nof problem \\eqref{best-app} of a certain kind.\n\n\\begin{prop}\n\\label{prop:exist-PL}\n Let $A, B$ be $m \\times s$ matrices and $C, D$ be $n \\times s$ matrices and suppose \n\\begin{equation}\n\\label{cond-ex}\n\\min\\left\\{ \\, \\rank \\begin{pmatrix} A & B \\end{pmatrix}, \\, \n \\rank \\begin{pmatrix} C & D \\end{pmatrix} \\,\\right\\} > r = \\frac{3s}{2} . \n\\end{equation} \n Then, the pencil \n \\begin{equation}\n P(\\lambda) = \\left( A C^\\T + B D^\\T \\right) + \\lambda \\, B C^\\T \\quad \\in \\mathcal{P}_{m,n}(\\mathbb{F})\n \\end{equation}\nhas no best approximation in $\\mathcal{B}_{s,s}$.\n\\end{prop}\n\n\\begin{proof}\nFirst, let us show that for any transformation $T \\in \\GL_2(\\mathbb{F})$, at least one of the matrices $t_{11} (A C^\\T + B D^\\T) + \nt_{12} B C^\\T$ and $t_{21} (A C^\\T + B D^\\T) + t_{22} B C^\\T$ has rank strictly larger than $s$. \nNote that we can write $t_{i1} (A C^\\T + B D^\\T) + t_{i2} B C^\\T = F_i \\begin{pmatrix} C & D \\end{pmatrix}^\\T$, where\n$$ F_i = \\begin{pmatrix} A & B \\end{pmatrix} \\begin{pmatrix} t_{i1} E_s & 0 \\\\ t_{i2} E_s & t_{i1} E_s \\end{pmatrix}. $$\nWe have $\\rank F_i = \\rank \\begin{pmatrix} A & B \\end{pmatrix} > r$ for $t_{i1} \\neq 0$, and so Sylvester's inequality implies that \nthe product $F_i \\begin{pmatrix} C & D \\end{pmatrix}^\\T$ has rank strictly greater than $2(r - s) = s$.\nBecause $t_{11}$ and $t_{21}$ obviously cannot be both zero, the statement is true. Hence, we \nconclude that $P(\\lambda) \\notin \\mathcal{B}_{s,s}$.\nNext, let\n \\begin{align}\n \\label{Pn}\nP_n(\\lambda) \\triangleq & \\ n \n \\left[ \\left( B + \\frac{1}{n} A \\right)\\left( C + \\frac{1}{n} D \\right)^\\T \\right] \n \\left( 1 + \\frac{1}{n} \\lambda \\right) - n \\left( B C^\\T \\right) \\\\\n = & \\ P(\\lambda) + \\frac{1}{n}\\left[ \\left( A C^\\T \\right) \\lambda +\n \\left( B D^\\T \\right) \\lambda +\n \\left( A D^\\T \\right) \\right] \n + \\frac{1}{n^2} \\left( A D^\\T \\right) \\lambda. \n \\label{Pn2}\n \\end{align}\nBy construction, $P_n(\\lambda) \\in \\mathcal{B}_{s,s}$, while \\eqref{Pn2} reveals that $P_n(\\lambda) \\rightarrow P(\\lambda) $ as \n$n \\rightarrow \\infty$. Hence, since $P(\\lambda) \\notin \\mathcal{B}_{s,s}$, it holds that \n\\begin{equation*}\n \\inf_{A' + \\lambda B' \\in \\mathcal{B}_{s,s}} \\, \n \\| P(\\lambda) - (A' + \\lambda B') \\| = 0\n\\end{equation*} \nis not attained.\n\\end{proof}\n\n\\begin{rem}\n The condition \\eqref{cond-ex} is tight in the sense that one can find matrices $A$, $B$, $C$ and $D$ satisfying $\\rank \n\\begin{pmatrix} A & B\\end{pmatrix} \\le r$ and $\\rank \\begin{pmatrix} C & D \\end{pmatrix} \\le r$ such that \n $P(\\lambda) = \\left( A C^\\T + B D^\\T \\right) + \\lambda \\, B C^\\T \\in \\mathcal{B}_{s,s}$. Take, for instance, $m, n \\ge 6$, $s=4$ and\n\\begin{align*}\n A = \\begin{pmatrix} a_1 & a_2 & a_3 & a_4 \\end{pmatrix}, \\quad & \\quad\n B = \\begin{pmatrix} a_1 & a_2 & b_3 & b_4 \\end{pmatrix}, \\\\\n C = \\begin{pmatrix} c_1 & c_2 & c_3 & c_4 \\end{pmatrix}, \\quad & \\quad\n D = \\begin{pmatrix} d_1 & d_2 & c_3 & c_4 \\end{pmatrix},\n\\end{align*}\nwhere $a_1, a_2, a_3, a_4, b_3, b_4$ are linearly independent, and the same applies to $c_1, c_2, c_3, c_4, d_1, d_2$. We have then\n$\\rank \\begin{pmatrix} A & B \\end{pmatrix} = \\rank \\begin{pmatrix} C & D \\end{pmatrix} = 6 = \\frac{3s}{2} = r$. Choosing $t_{i1} = 1$ \nand $t_{i2} = -1$ yields $F_i \\begin{pmatrix} C & D \\end{pmatrix}^\\T = (a_3 - b_3) c_3^\\T + (a_4 - b_4) c_4^\\T + B D^\\T $, which \nclearly cannot have rank larger than $s = 4$.\n\\end{rem}\n\n\n\n\\subsection{Ill-posedness over a positive-volume set of real pencils}\n\\label{sec:positive-volume}\n\nIn this section, we will prove that no (regular) $2k \\times 2k$ pencil having only complex-valued eigenvalues admits a best \napproximation in $\\mathcal{B}_{2k-1,2k-1}$ in the norm topology, for any positive integer $k$. The set containing all such pencils is \ndefined as \n\\begin{align}\n \\label{C-set}\n \\mathcal{C} \\triangleq & \\ \\{ A + \\lambda B \\in \\mathcal{P}_{2k,2k}(\\mathbb{R}) \\ | \\ \\mathcal{O}(A,B) \\text{ contains } Q + \\lambda \nE, \n \\text{ where } Q \\text{ has no real eigenvalues }\\} \\\\\n = & \\ \\{ A + \\lambda B \\in \\mathcal{P}_{2k,2k}(\\mathbb{R}) \\ | \\ \\rho(A,B) = (2k,2k)\\}, \\nonumber\n\\end{align} \nwhere the equality is due to \\Cref{cor:full-min-rank}. \nFor instance, in the case $k=2$ this set is constituted by all orbits of the families $\\mathscrc{R}_{4,4}$, $\\mathscrc{R}'_{4,4}$ and \n$\\mathscrc{R}''_{4,4}$ of \\Cref{tab:4}. We start by showing $\\mathcal{C}$ is open, and therefore has positive volume (since \nit is always nonempty).\n\n\\begin{lem}\n\\label{lem:C-open}\nThe set $\\mathcal{C} \\subset \\mathcal{P}_{2k,2k}(\\mathbb{R})$ defined by \\eqref{C-set} is open in the norm topology.\n\\end{lem}\n\n\\begin{proof}\nTake an arbitrary pencil $A + \\lambda B \\in \\mathcal{C}$. By definition, it can be written as $(P,U,T) \\cdot (Q + \\lambda E)$,\nwhere $(P,U,T) \\in \\GL_{2k,2k,2}(\\mathbb{R})$ and \n$Q = Q_{2 k_1}(a_1,b_1) \\oplus Q_{2 k_2}(a_2,b_2) \\oplus \\dots \\oplus Q_{2 k_l}(a_l,b_l)$, with $k_1 + \\dots + k_l = k$ and \n$b_i \\neq 0$ for $i=1,\\ldots,l$.\nConsider any other pencil $A' + \\lambda B'$ lying in an open ball of radius $\\sqrt{\\epsilon}$ centered on $A + \\lambda B$. We \nhave\n\\begin{equation*}\n \\left\\| (P,U,T) \\cdot (Q + \\lambda E) - (A' + \\lambda B') \\right\\|^2\n= \\left\\| (P,U,T) \\cdot \\left(Q - A'' + \\lambda(E - B'') \\right) \\right\\|^2 < \\epsilon,\n\\end{equation*} \nwhere $A'' + \\lambda B'' = (P^{-1}, U^{-1}, T^{-1}) \\cdot (A' + \\lambda B') \\not\\equiv Q + \\lambda E$. But, since $(P,U,T) \\in \n\\GL_{2k,2k,2}(\\mathbb{R})$, then we have\n$\n \\| (P,U,T) \\cdot (C + \\lambda D) \\| \\ge \\sigma \\|C + \\lambda D\\|\n$ \nfor any $C + \\lambda D \\in \\mathcal{P}_{2k,2k}(\\mathbb{R})$, where $\\sigma > 0$ is the smallest singular value of the linear \noperator $(P,U,T) : \\mathcal{P}_{2k,2k}(\\mathbb{R}) \\rightarrow \\mathcal{P}_{2k,2k}(\\mathbb{R})$. So, \n\\begin{equation*}\n \\left\\| Q - A'' \\right\\|^2 + \\left\\| E - B'' \\right\\|^2 \n\\le \\sigma^{-2} \\left\\| (P,U,T) \\cdot \\left(Q - A'' + \\lambda(E - B'') \\right) \\right\\|^2\n < \\sigma^{-2} \\, \\epsilon.\n\\end{equation*}\nHence, a sufficiently small $\\epsilon$ can be chosen to guarantee that $\\|Q - A''\\| \\le \\epsilon_1$ and $\\| E - B'' \\| \\le \\epsilon_2$ \nfor any $\\epsilon_1, \\epsilon_2 > 0$. By continuity of the eigenvalues of the pencil $Q + \\lambda E$, it follows that there \nexists $\\epsilon > 0$ such that every such $A'' + \\lambda B''$ can be written as $A'' + \\lambda B'' = \\left( X, Y, Z \\right) \\cdot \n(Q' + \\lambda E)$ for some $(X,Y,Z) \\in \\GL_{2k,2k,2}(\\mathbb{R})$ and $Q'$ having $k$ pairs of complex conjugate eigenvalues \n(with possibly some identical pairs). Therefore, $A' + \\lambda B' = \\left( PX, UY, TZ \\right) \\cdot (Q' + \\lambda E) \\in \n\\mathcal{C}$. Because this applies to every $A' + \\lambda B'$ in the chosen open ball of radius $\\epsilon$, then $A + \\lambda B$ is an \ninterior point of $\\mathcal{C}$.\n\\end{proof}\n\n\\begin{corol}\n\\label{cor:Y-bestapp}\n The set $\\mathcal{B}_{2k,2k-1} \\subset \\mathcal{P}_{2k,2k}(\\mathbb{R})$ is closed in the norm topology. Consequently, \n $$\n \\inf_{A' + \\lambda B' \\in \\mathcal{B}_{2k,2k-1}} \\| A + \\lambda B - (A' + \\lambda B') \\|\n $$\n is always attained by some $A' + \\lambda B' \\in \\mathcal{B}_{2k,2k-1}$.\n\\end{corol}\n\n\\begin{proof}\nFollows from \\Cref{lem:C-open} by using the fact that $\\mathcal{B}_{2k,2k-1} = \\mathcal{P}_{2k,2k}(\\mathbb{R}) \\setminus \n\\mathcal{C}$.\n\\end{proof}\n\n\nAnother way to define $\\mathcal{C}$ is by stating that it contains all $2k \\times 2k$ pencils $A + \\lambda B$ such that $\\det(u A + t \nB) = 0$ if and only if $t=u=0$. This corresponds to the set of absolutely nonsingular $2k \\times 2k \\times 2$ tensors defined by Sumi \net al.~in their paper \\cite{Sumi2013}. It was shown in \\cite[Theorem 2.5]{Sumi2013} that, for any positive integer $k$ and $d > 1$, \nthe set of absolutely nonsingular $2k \\times 2k \\times d$ tensors is open. \\Cref{lem:C-open} therefore provides an alternative proof \nof that fact for the case $d = 2$.\n\n\n\\begin{theo}\n\\label{theo:closure}\n The set $\\mathcal{B}_{2k-1,2k-1} \\subset \\mathcal{P}_{2k,2k}(\\mathbb{R})$ is not closed in the norm topology. Furthermore, its \nclosure is given by $ \\overline{\\mathcal{B}_{2k-1,2k-1}} = \\mathcal{B}_{2k,2k-1} \\subset \\mathcal{P}_{2k,2k}(\\mathbb{R})$.\n\\end{theo}\n\n\\begin{proof}\nTo prove this claim, it is sufficient\\footnote{Indeed, if $\\mathcal{X} \\subset \\mathcal{Y}$, $\\mathcal{Y} = \n\\overline{\\mathcal{Y}}$ and $\\mathcal{Y} \\subseteq \\overline{\\mathcal{X}}$, then $\\overline{\\mathcal{X}} \\subseteq \n\\overline{\\mathcal{Y}}$ and thus $\\mathcal{Y} \\subseteq \\overline{\\mathcal{X}} \\subseteq \\overline{\\mathcal{Y}} = \\mathcal{Y}$, \nimplying $\\overline{\\mathcal{X}} = \\mathcal{Y}$.} to show that for every pencil $A + \\lambda B \\in \\mathcal{B}_{2k,2k-1} \\subset \n\\mathcal{P}_{2k,2k}(\\mathbb{R})$ having minimal ranks $\\rho(A,B) = (2k,2k-1)$, we can find a sequence of pencils in \n$\\mathcal{B}_{2k-1,2k-1}$ which converges to $A + \\lambda B$. \nFirst, recall that every pencil $A + \\lambda B$ with $\\rho(A,B) = (2k,2k-1)$ must be regular.\nFrom \\Cref{lem:rho-regular}, $\\rho(A,B) = (2k,2k-1)$ holds precisely when the canonical form of $A + \\lambda B$ comprises \neither a single Jordan block $J_{m_1}(b)$ for some $b \\in \\mathbb{R}$, or a single block $N_{m_1}(\\lambda)$ having an infinite \nelementary divisor, with even dimension $m_1 > 0$. Let us first focus on the case with no infinite elementary divisors, where the \ncanonical form is either of the form $J_{2k}(a) + \\lambda E$ (with $m_1 = 2k$) or of the form $J_{m_1}(a) \\oplus Q + \\lambda \nE$, where the elementary divisors of $Q + \\lambda E$ cannot be factored into powers of linear forms. The latter possibility \nexists of course only when $k > 1$. Let us consider each case separately:\n\\begin{enumerate}[(i)]\n\n \\item In the former case, define $Z_p(\\lambda) \\triangleq W_p + \\lambda E$, where \n \\begin{equation*}\nW_p \\triangleq J_{2k-1}(a) \\oplus \\left(a + \\frac{1}{p}\\right) + e_{2k-1} \\otimes e_{2k} = \n\\begin{pmatrix}\na & 1 & & & & \\\\\n & a & 1 & & & \\\\\n & & \\ddots & \\ddots & & \\\\\n & & & a & 1 & \\\\\n & & & & a & 1 \\\\ \n & & & & & a + 1\/p \\\\\n\\end{pmatrix},\n\\end{equation*}\nwhere $p$ is a positive integer. It is not hard to see that $Z_p(\\lambda)$ is $\\GL_{2k,2k,2}(\\mathbb{R})$-equivalent to \n$J_{2k-1}(a) \\oplus (a + 1\/p) + \\lambda E$, since their elementary divisors coincide.\nHence, by \\Cref{lem:rho-regular}, $Z_p(\\lambda)$ has minimal ranks $(2k-1,2k-1)$, and thus belongs to $\\mathcal{B}_{2k-1,2k-1}$. \nOn the other hand, $A + \\lambda B = \\left( P, U, T \\right) \\cdot (J_{2k}(a) + \\lambda E)$ for some $\\left( P, U, T \\right) \n\\in \\GL_{2k,2k,2}(\\mathbb{R})$, and so we have\n\\begin{equation*}\n \\lim_{p \\rightarrow \\infty}\n \\left\\| A + \\lambda B - \\left( P, U, T \\right) \\cdot Z_p(\\lambda) \\right\\|\n = \\lim_{p \\rightarrow \\infty}\n \\left\\| \\left( P, U, T \\right) \\cdot \\left( J_{2k}(a) - W_p \\right) \\right\\| = 0.\n\\end{equation*} \nIt follows that the sequence of pencils $\\{ \\left( P, U, T \\right) \\cdot Z_p(\\lambda)\\}_{p \\in \\mathbb{N}}$ converges to $A + \\lambda \nB$ as $p \\rightarrow \\infty$.\n\n\\item The same argument can be employed in the second case, now with $Z'_p(\\lambda) \\triangleq W'_p + \\lambda E$, where\n$$ W'_p \\triangleq J_{m_1 - 1}(a) \\oplus (a + 1\/p) \\oplus Q + e_{m_1 - 1} \\otimes e_{m_1}.$$\nFrom this definition, $Z_p(\\lambda)$ is $\\GL_{2k,2k,2}(\\mathbb{R})$-equivalent to $J_{m_1-1}(a) \\oplus (a + 1\/p) \\oplus \nQ + \\lambda E$, which is also in $\\mathcal{B}_{2k-1,2k-1}$. Writing again $A + \\lambda B = \\left( P, U, T \\right) \\cdot (J_{m_1}(a) \n\\oplus Q + \\lambda E)$, it follows that\n\\begin{equation*}\n \\lim_{p \\rightarrow \\infty}\n \\left\\| A + \\lambda B - \\left( P, U, T \\right) \\cdot Z'_p(\\lambda) \\right\\|\n = \\lim_{p \\rightarrow \\infty}\n \\left\\| \\left( P, U, T \\right) \\cdot \\left( J_{m_1}(a) \\oplus Q - W'_p \\right) \\right\\| = 0.\n\\end{equation*}\n\\end{enumerate}\nFinally, note that the above argument extends easily to the case where $A + \\lambda B$ has one infinite divisor, because there \nstill exists $(P,U,T) \\in \\GL_{m,n,2}(\\mathbb{R})$ such that either $A + \\lambda B = \\left( P, U, T \\right) \\cdot (J_{2k}(a) + \\lambda \nE)$ or $A + \\lambda B = \\left( P, U, T \\right) \\cdot (J_{m_1}(a) \\oplus Q + \\lambda E)$, since infinite elementary divisors can always \nbe avoided with a $\\GL_{2}(\\mathbb{R})$ transformation.\n\\end{proof}\n\nMotivated by the fact that pencils with minimal ranks $(2k,2k-1)$ can be arbitrarily well approximated by pencils having minimal ranks \n$(2k-1,2k-1)$, one may define the ``border minimal ranks'' of a pencil in the same fashion as the border rank is defined for tensors \n(see, e.g., \\cite{deSilva2008}). Specifically, \\Cref{theo:closure} shows that every real $2k \\times 2k$ pencil with minimal ranks \n$(2k,2k-1)$ has ``border minimal ranks'' $(2k-1,2k-1)$.\nAs a consequence, no pencil with minimal ranks $(2k,2k-1)$ has a best approximation $\\mathcal{B}_{2k-1,2k-1}$ in the norm topology, as \nstated next.\n\n\n\\begin{corol}\n\\label{corol:approx-43} \n If $A + \\lambda B \\in \\mathcal{P}_{2k,2k}(\\mathbb{R})$ satisfies $\\rho(A,B) = (2k,2k-1)$, then \n \\begin{equation*}\n \\inf_{A' + \\lambda B' \\in \\mathcal{B}_{2k-1,2k-1}} \\| A + \\lambda B - (A' + \\lambda B') \\| = 0\n \\end{equation*}\nis not attained by any $A' + \\lambda B' \\in \\mathcal{B}_{2k-1,2k-1}$.\n\\end{corol}\n\nThe next result establishes that the best approximation of any pencil $A + \\lambda B \\in \\mathcal{C}$ on $\\mathcal{B}_{2k,2k-1}$ \nmust have minimal ranks $(2k,2k-1)$, otherwise it is not optimal. It is thus in the same spirit of Lemma 8.2 of De Silva and Lim \n\\cite{deSilva2008}, which states that for positive integers $r,s$ such that $r \\ge s$, the best approximation of a rank-$r$ tensor \nhaving rank up to $s$ always has rank $s$. \n\n\\begin{lem}\n\\label{lem:approx-43}\n Let $A + \\lambda B \\in \\mathcal{C}$. Then, in the norm topology we have \n \\begin{equation*}\n \\arg\\min_{A' + \\lambda B' \\in \\mathcal{B}_{2k,2k-1}} \\| A + \\lambda B - (A' + \\lambda B') \\|\n \\quad \\subset \\quad \\mathcal{B}_{2k,2k-1} \\setminus \\left( \\mathcal{B}_{2k,2k-2} \\cup \\mathcal{B}_{2k-1,2k-1} \\right).\n \\end{equation*} \n In other words, every best approximation $A' + \\lambda B'$ of $A + \\lambda B$ on $\\mathcal{B}_{2k,2k-1}$ is such that \n$\\rho(A',B') = (2k,2k-1)$. \n\\end{lem}\n\n\\begin{proof}\nTake any $A' + \\lambda B' \\in \\mathcal{B}_{2k,2k-2} \\cup \\mathcal{B}_{2k-1,2k-1}$. By definition, it can be written as\n$(t_{11} U + t_{12} V) + \\lambda (t_{21} U + t_{22} V)$, with $T = (t_{ij}) \\in \\GL_2(\\mathbb{R})$ and either $(\\rank U, \\rank V) \\le \n(2k-1,2k-1)$ or $(\\rank U, \\rank V) \\le (2k,2k-2)$, where the inequality is meant entry-wise. Let us define now $Z_1(\\lambda) \n\\triangleq \nt_{11}(A- A') + \\lambda t_{21}(B- B')$ and $Z_2(\\lambda) \\triangleq t_{12}(A- A') + \\lambda t_{22}(B- B')$. Since $A' + \\lambda B' \n\\not\\equiv A + \\lambda B$, then $Z_i(\\lambda) \\not\\equiv 0$ for at least one $i \\in \\{1,2\\}$. Thus, there exists a rank-one matrix $W$ \nsuch that $\\langle Z_i(\\lambda), W + \\lambda W \\rangle \\neq 0$ for a certain $i \\in \\{1,2\\}$. Without loss of generality, we can \nassume that $\\|t_{1i} W + \\lambda t_{2i} W \\|^2 = (t_{1i}^2 + t_{2i}^2) \\|W\\|^2 = 1$. For any $c \\in \\mathbb{R}$, we have\n\\begin{multline*}\n \\| A + \\lambda B - (A' + \\lambda B' + c \\, (t_{1i} W + \\lambda t_{2i} W) ) \\|^2 \\\\\n = \\| A + \\lambda B - (A' + \\lambda B') \\|^2 - 2c \\langle Z_i(\\lambda) , W + \\lambda W \\rangle + c^2.\n\\end{multline*} \nNow, if we choose $c = \\langle Z_i(\\lambda) , W + \\lambda W \\rangle \\neq 0$, then clearly \n\\begin{equation*}\n \\| A + \\lambda B - (A' + \\lambda B' + c \\, (t_{1i} W + \\lambda t_{2i} W) ) \\|^2\n = \\| A + \\lambda B - (A' + \\lambda B') \\|^2 - c^2 < \\| A + \\lambda B - (A' + \\lambda B') \\|^2.\n\\end{equation*}\nThis means that $A' + \\lambda B' + c \\, (t_{1i} W + \\lambda t_{2i} W)$ is closer to $A + \\lambda B$ than $A' + \\lambda B'$. Using now \nthe expressions given for $A'$ and $B'$, we find that\n\\begin{equation*}\nA'' + \\lambda B'' \\triangleq\nA' + \\lambda B' + c \\, (t_{1i} W + \\lambda t_{2i} W) =\n (t_{11} U + t_{12} V + c t_{1i} W) + \\lambda (t_{21} U + t_{22} V + c t_{2i} W).\n\\end{equation*}\nIf $i=1$, we have $ (E,E,T^{-1}) \\cdot (A'' + \\lambda B'') = (U + c W) + \\lambda V $, otherwise \n$ (E,E,T^{-1}) \\cdot (A'' + \\lambda B'') = U + \\lambda (V + c W) $. Either way, since $\\rank W = 1$, then $A'' + \\lambda B'' \n\\in \\mathcal{B}_{2k,2k-1}$. This shows that for any $A' + \\lambda B' \\in \\mathcal{B}_{2k-1,2k-1} \\cup \\mathcal{B}_{2k,2k-2}$, we can \nalways find some other pencil $A'' + \\lambda B''$ in $\\mathcal{B}_{2k,2k-1}$ such that $\\| A + \\lambda B - (A'' + \\lambda B'') \n\\| < \\| A + \\lambda B - (A' + \\lambda B') \\|$. Because a best approximation of $A + \\lambda B \\in \\mathcal{B}_{2k,2k-1}$ must exist \ndue to \\Cref{cor:Y-bestapp}, we conclude that it can only belong to $\\mathcal{B}_{2k,2k-1} \\setminus \\left( \\mathcal{B}_{2k-1,2k-1} \n\\cup \\mathcal{B}_{2k,2k-2}\\right)$, i.e., it necessarily has minimal ranks $(2k,2k-1)$.\n\\end{proof}\n\nWe are now in a position to prove the main result of this section.\n\n\\begin{theo}\nIn the norm topology, if $A + \\lambda B \\in \\mathcal{C}$ then \n \\begin{equation*}\n \\inf_{A' + \\lambda B' \\in \\mathcal{B}_{2k-1,2k-1}} \\| A + \\lambda B - (A' + \\lambda B') \\| \n \\end{equation*}\nis not attained by any $A' + \\lambda B' \\in \\mathcal{B}_{2k-1,2k-1}$. In other words, the problem above stated has no solution \nwhen $\\rho(A,B)=(2k,2k)$. \n\\end{theo}\n\n\\begin{proof}\n \\Cref{cor:Y-bestapp} and \\Cref{lem:approx-43} imply that for every pencil $A + \\lambda B \\in \\mathcal{C}$ there exists another \npencil $A' + \\lambda B'$ such that $\\rho(A',B') = (2k,2k-1)$ and\n\\begin{equation}\n\\label{B43-B33}\n \\left\\| A + \\lambda B - (A' + \\lambda B') \\right\\| < \\left\\| A + \\lambda B - (C + \\lambda D) \\right\\|, \n \\quad \\forall \\, C + \\lambda D \\in \\mathcal{B}_{2k-1,2k-1} \\subset \\mathcal{B}_{2k,2k-1}. \n\\end{equation}\nBut, it follows from \\Cref{theo:closure} that any such $A' + \\lambda B'$ can be arbitrarily well approximated by pencils from \n$\\mathcal{B}_{2k-1,2k-1}$, that is, \n\\begin{equation}\n \\inf_{C + \\lambda D \\in \\mathcal{B}_{2k-1,2k-1}} \\| A' + \\lambda B' - (C + \\lambda D) \\| = 0, \n\\end{equation}\nwhilst no $C + \\lambda D \\in \\mathcal{B}_{2k-1,2k-1}$ can attain that infimum because $A' + \\lambda B' \\notin \\mathcal{B}_{2k-1,2k-1}$.\nCombining the above facts, we conclude that\n\\begin{equation}\n \\inf_{C + \\lambda D \\in \\mathcal{B}_{2k-1,2k-1}} \\| A + \\lambda B - (C + \\lambda D) \\| = \\| A + \\lambda B - (A' + \\lambda B') \\|\n\\end{equation} \ncannot be attained by any $C + \\lambda D \\in \\mathcal{B}_{2k-1,2k-1}$.\n\\end{proof}\n\n\n \n \\section{Conclusion}\n\\label{sec:concl}\n\nThis work defines and studies a fundamental property of a matrix pencil, which we have called its minimal ranks. \nThe structure of a space of pencils can be better understood on the basis of this notion and its properties.\nIn particular, endowing $\\mathcal{P}_{m,n}( \\mathbb{F} )$ with a norm, we have studied the problem of approximation of a \npencil by another one having strictly lower minimal ranks in the induced norm topology. An optimal approximation may not exist, and \nour results show that this is true for every pencil of the set $\\mathcal{C} \\subset \\mathcal{P}_{2k,2k}( \\mathbb{R} )$ if an \napproximation is sought over $\\mathcal{B}_{2k-1,2k-1}$ for any positive integer $k$. $\\mathcal{C}$ is open, which shows that, \ncontrarily to the complex-valued case, this phenomenon can happen for pencils forming a positive volume set. Translated to a tensor \nviewpoint, our result states that certain $2k \\times 2k \\times 2$ real tensors forming a positive-volume set have no best approximate \nblock-term decomposition with two rank-$(2k-1)$ blocks. \n\nAs we have shown, the definition and essential properties of the minimal ranks can be readily extended to matrix polynomials, which \nare associated with more general $m \\times n \\times d$ tensors. We believe this should provide a useful element for the study of \nthird-order block-term decompositions composed by matrix-vector tensor products. In particular, results enabling the computation \nof the minimal ranks of a general matrix polynomial would certainly be helpful in this regard.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}