data_all_eng_slimpj/shuffled/split2/finalzzacqm

{"text":"\\section{Introduction}\n\nLet $\\mathbb{F}$ be an arbitrary field. Cayley algebras (or algebras of octonions) over $\\mathbb{F}$ constitute a well-known class of nonassociative algebras (see, e.g. \\cite[Chapter VIII]{KMRT} and references therein). They are unital nonassociative algebras $\\mathcal{C}$ of dimension eight over $\\mathbb{F}$, endowed with a nonsingualr quadratic multiplicative form (the \\emph{norm}) $q:\\mathcal{C}\\rightarrow \\mathbb{F}$. Hence $q(xy)=q(x)q(y)$ for any $x,y\\in\\mathcal{C}$, and the polar form $b_q(x,y):= q(x+y)-q(x)-q(y)$ is a nondegenerate bilinear form.\n\nAny element in a Cayley algebra $\\mathcal{C}$ satisfies the degree $2$ equation:\n\\begin{equation}\\label{eq:CayleyHamilton}\nx^2-b_q(x,1)x+q(x)1=0.\n\\end{equation}\nBesides, the map $x\\mapsto \\bar x:= b_q(x,1)1-x$ is an involution (i.e., an antiautomorphism of order $2$) and the \\emph{trace} $t(x):= b_q(x,1)$ and norm $q(x)$ are given by $t(x)1=x+\\bar x$, $q(x)1=x\\bar x=\\bar x x$ for any $x \\in \\mathcal{C}$. Two Cayley algebras $\\mathcal{C}_1$ and $\\mathcal{C}_2$, with respective norms $q_1$ and $q_2$, are isomorphic if and only if the norms $q_1$ and $q_2$ are isometric.\n\nIf the characteristic of $\\mathbb{F}$ is not $2$, then $\\mathcal{C}=\\mathbb{F} 1\\oplus \\mathcal{C}^0$, where $\\mathcal{C}^0$ is the subspace of trace zero elements (i.e., the subspace orthogonal to $\\mathbb{F} 1$ relative to $b_q$). For $x,y\\in \\mathcal{C}^0$,  \\eqref{eq:CayleyHamilton} shows that $xy+yx=-b_q(x,y)1$, while $t([x,y])=[x,y]+\\overline{[x,y]}=[x,y]+[\\bar y,\\bar x]=0$, so $[x,y]:= xy-yx\\in\\mathcal{C}^0$. In particular,\n\\begin{equation}\\label{eq:xyC0}\nxy=-\\frac{1}{2}b_q(x,y)1+ \\frac{1}{2}[x,y],\n\\end{equation}\nso the projection of $xy$ in $\\mathcal{C}^0$ is $\\frac{1}{2}[x,y]$. Moreover, the following relation holds (see, e.g. \\cite[Theorem 4.23]{EKmon}):\n\\begin{equation}\\label{eq:xyy}\n[[x,y],y]=2b_q(x,y)y-2b_q(y,y)x,\n\\end{equation}\nso the multiplication in $\\mathcal{C}$ and its norm are determined by the bracket in $\\mathcal{C}^0$.\n\nThe Lie algebra of derivations of a Cayley algebra $\\mathcal{C}$ is defined by\n\\[ \\Der(\\mathcal{C}) := \\{ d\\in {\\mathfrak{gl}}(\\mathcal{C}): d(xy)=d(x)y+xd(y)\\ \\forall x,y\\in\\mathcal{C}\\}. \\]\nIn general, if $M$ is a module for a Lie algebra $\\mathcal{L}$, we say that a bilinear product $\\cdot : M \\times M \\rightarrow M$ is \\emph{$\\mathcal{L}$-invariant} if $\\mathcal{L}$ acts on $(M, \\cdot)$ by derivations: $x(u \\cdot v) = x(u)\\cdot v + u \\cdot x(u)$, for any $x \\in \\mathcal{L}$, $u,v \\in M$.\n\nIf the norm $q$ of a Cayley algebra $\\mathcal{C}$ is isotropic (i.e., there exists $0\\ne x\\in \\mathcal{C}$ with $q(x)=0$), then $\\mathcal{C}$ is unique up to isomorphism. In this case, the Cayley algebra $\\mathcal{C}$ is said to be \\emph{split}, and it has a \\emph{good basis} $\\{ p_1, p_2,u_1,u_2,u_3,v_1,v_2,v_3\\}$ with multiplication given in Table \\ref{ta:good_basis}. We denote by $\\mathcal{C}_s$ the split Cayley algebra.\n\n\\begin{table}[!h]\\label{ta:good_basis}\n\\[\n\\vcenter{\\offinterlineskip\n\\halign{\\hfil$#$\\enspace\\hfil&#\\vrule height 12pt width1pt depth 4pt\n &\\hfil\\enspace$#$\\enspace\\hfil\n &\\hfil\\enspace$#$\\enspace\\hfil&#\\vrule width .5pt\n &\\hfil\\enspace$#$\\enspace\\hfil\n &\\hfil\\enspace$#$\\enspace\\hfil\n &\\hfil\\enspace$#$\\enspace\\hfil&#\\vrule width .5pt\n &\\hfil\\enspace$#$\\enspace\\hfil\n &\\hfil\\enspace$#$\\enspace\\hfil\n &\\hfil\\enspace$#$\\enspace\\hfil&#\\vrule height 12pt width1pt depth 4pt\\cr\n &\\omit\\hfil\\vrule width 1pt depth 4pt height 10pt\n   &p_1&p_2&\\omit&u_1&u_2&u_3&\\omit&v_1&v_2&v_3&\\cr\n \\noalign{\\hrule height1pt}\n p_1&& p_1&0&&u_1&u_2&u_3&&0&0&0&\\cr\n p_2&&0&p_2&&0&0&0&&v_1&v_2&v_3&\\cr\n &\\multispan{12}{\\leaders\\hregleta\\hfill}\\cr\n u_1&&0&u_1&&0&v_3&-v_2&&-p_1&0&0&\\cr\n u_2&&0&u_2&&-v_3&0&v_1&&0&-p_1&0&\\cr\n u_3&&0&u_3&&v_2&-v_1&0&&0&0&-p_1&\\cr\n &\\multispan{12}{\\leaders\\hregleta\\hfill}\\cr\n v_1&&v_1&0&&-p_2&0&0&&0&u_3&-u_2&\\cr\n v_2&&v_2&0&&0&-p_2&0&&-u_3&0&u_1&\\cr\n v_3&&v_3&0&&0&0&-p_2&&u_2&-u_1&0&\\cr\n \\noalign{\\hrule height1pt}}}\n\\]\n\\caption{{\\vrule width 0pt height 15pt}Multiplication table in a good basis of the split Cayley algebra.}\n\\end{table}\n\nGiven a finite-dimensional simple Lie algebra $\\frg$ of type $X_r$ over the complex numbers, and a Chevalley basis $\\mathcal{B}$, let $\\frg_\\mathbb{Z}$ be the $\\mathbb{Z}$-span of $\\mathcal{B}$ (a Lie algebra over $\\mathbb{Z}$). The Lie algebra $\\frg_\\mathbb{F}:=\\frg_\\mathbb{Z}\\otimes_\\mathbb{Z} \\mathbb{F}$ is the \\emph{Chevalley algebra} of type $X_r$. In particular, the Chevalley algebra of type $G_2$ is isomorphic to $\\Der(\\mathcal{C}_s)$ (see, e.g. \\cite[\\S 4.4]{EKmon}). For any Cayley algebra $\\mathcal{C}$, the Lie algebra $\\Der(\\mathcal{C})$ is a twisted form of the Chevalley algebra $\\Der(\\mathcal{C}_s)$. (Recall that, if $\\cA$ and $\\mathcal{B}$ are algebras over $\\mathbb{F}$, then $\\cA$ is a \\emph{twisted form} of $\\mathcal{B}$ whenever $\\cA\\otimes_\\mathbb{F} \\mathbb{F}_{\\text{alg}}\\cong \\mathcal{B}\\otimes_\\mathbb{F}\\FF_{\\text{alg}}$, for an algebraic closure $\\mathbb{F}_{\\text{alg}}$ of $\\mathbb{F}$.)\n\nIf the characteristic of $\\mathbb{F}$ is neither $2$ nor $3$, then the Chevalley algebra of type $G_2$ is simple; this is the split simple Lie algebra of type $G_2$. Moreover, two Cayley algebras $\\mathcal{C}_1$ and $\\mathcal{C}_2$ are isomorphic if and only if their Lie algebras of derivations are isomorphic (see \\cite[Theorem IV.4.1]{Seligman} or \\cite[Theorem 4.35]{EKmon}).\n\nThe goal of this paper is to show some surprising features of Cayley algebras over fields of characteristic $7$, $3$ and $2$. \n\nSection \\ref{se:char7} studies the case of characteristic $7$ and is divided in three subsections. In Section \\ref{subsec1}, we review a construction of $\\mathcal{C}_s$ due to Dixmier \\cite{Dixmier} in terms of transvectants which was originally done in characteristic $0$, but it is valid in any characteristic $p \\geq 7$. In this construction, $\\mathcal{C}_s$ appears as the direct sum of the trivial one-dimensional module and the restricted irreducible seven-dimensional module $V_6$ for the simple Lie algebra ${\\mathfrak{sl}}_2(\\mathbb{F})$, which embeds into $\\Der(\\mathcal{C}_s)$ as its principal ${\\mathfrak{sl}}_2$ subalgebra. When the characteristic is $7$, this action of ${\\mathfrak{sl}}_2(\\mathbb{F})$ by derivations on $\\mathcal{C}_s$ may be naturally extended to an action by derivations of the Witt algebra $W_1:= \\Der\\left(\\mathbb{F}[X]\/(X^7)\\right)$, explaining the fact, first proved in \\cite[Lemma 13]{Premet} (see also \\cite{Herpel_Stewart}), that $W_1$ embeds into the split simple Lie algebra of type $G_2$. In Section \\ref{subsec2}, we show that, when $\\mathbb{F}$ is algebraically closed, $V_6$ is the unique non-trivial non-adjoint restricted irreducible module for $W_1$ with a nonzero invariant product. Then, in Section \\ref{subsec3} we prove that, in characteristic $7$ and even when the ground field is not algebraically closed, all the twisted forms of the Witt algebra embed into $\\Der(\\mathcal{C}_s)$, and any two embeddings of the same twisted form are conjugate by an automorphism.\n \nSection \\ref{se:char3} is devoted to the case of characteristic $3$. In this situation, it is known that the Chevalley algebra of type $G_2$ is not simple, but it contains an ideal isomorphic to the projective special linear Lie algebra ${\\mathfrak{psl}}_3(\\mathbb{F})$. We review this situation and prove that it is still valid that two Cayley algebras are isomorphic if and only if their Lie algebras of derivations are isomorphic.\n\nFinally, in Section \\ref{se:char2}, we prove that the Lie algebra of derivations $\\Der(\\mathcal{C})$ of any Cayley algebra $\\mathcal{C}$ over a field $\\mathbb{F}$ of characteristic $2$ is always isomorphic to the projective special linear Lie algebra ${\\mathfrak{psl}}_4(\\mathbb{F})$. A proof of this fact when $\\mathcal{C} = \\mathcal{C}_s$ appears in \\cite[Corollary 4.32]{EKmon}. Hence, in this case, it is plainly false that two Cayley algebras are isomorphic if and only if their Lie algebras of derivations are isomorphic. We show that the isomorphism classes of twisted forms of ${\\mathfrak{psl}}_4(\\mathbb{F})$, which is here the Chevalley algebra of type $G_2$, are in bijection with the isomorphism classes of central simple associative algebras of degree $6$ endowed with a symplectic involution.\n\n\\medskip\n\n\n\\section{Characteristic $7$}\\label{se:char7}\n\n\n\\subsection{Dixmier's construction} \\label{subsec1}\n\nLet $\\mathbb{F}[x,y]$ be the polynomial algebra in two variables over a field $\\mathbb{F}$ of characteristic $0$. The general linear Lie algebra ${\\mathfrak{gl}}_2(\\mathbb{F})$ acts by derivations on $\\mathbb{F}[x,y]$ preserving the degree of each polynomial. For any $n\\geq 0$, denote by $V_n$ the subspace of homogeneous polynomials of degree $n$ in $\\mathbb{F}[x,y]$. For any $i,j,q\\geq 0$ with $q\\leq i,j$, consider the $q$-\\emph{transvectant} $V_i\\times V_j\\rightarrow V_{i+j-2q}$ given by\n\\begin{multline*}\n(f,g)_q:= \\frac{(i-q)!}{i!}\\frac{(j-q)!}{j!}\n\\left(\\frac{\\partial^qf}{\\partial x^q}\\frac{\\partial^qg}{\\partial y^q}\n-\\binom{q}{1}\n\\frac{\\partial^qf}{\\partial x^{q-1}\\partial y}\\frac{\\partial^qg}{\\partial x\\partial y^{q-1}}\\right.\\\\\n\\left. +\\binom{q}{2}\n\\frac{\\partial^qf}{\\partial x^{q-2}\\partial y^2}\\frac{\\partial^qg}{\\partial x^2\\partial y^{q-2}}\n-+\\cdots\\right),\n\\end{multline*}\nfor $f\\in V_i$ and $g\\in V_j$.\n\nIt turns out that the split Cayley algebra $\\mathcal{C}_s$ is isomorphic to the algebra defined on $\\mathbb{F} 1\\oplus V_6$ with multiplication given by\n\\begin{equation}\\label{eq:CsDixmier}\n(\\alpha 1+f)(\\beta 1+g):= \\bigl(\\alpha\\beta-\\frac{1}{20}(f,g)_6\\bigr)1+\n\\bigl(\\alpha g+\\beta f+(f,g)_3\\bigr),\n\\end{equation}\nfor any $\\alpha,\\beta\\in\\mathbb{F}$ and $f,g\\in V_6$ (see \\cite[3.6 Proposition]{Dixmier}); equipped with this product, the subspace $V_6$ becomes the subspace of trace zero elements in $\\mathcal{C}_s$. The existence of this isomorphism is based on the following identity given in \\cite[3.5 Lemme]{Dixmier}:\n\\begin{equation}\\label{eq:fgg}\n((f,g)_3,g)_3=\\frac{1}{20}\\Bigl((f,g)_6g-(g,g)_6f\\Bigr),\n\\end{equation}\nfor any $f,g\\in V_6$.\n\nConsider the following endomorphisms of $V_6$:\n\\[ e_{-1}:=\\left.x\\frac{\\partial\\ }{\\partial y}\\right|_{V_6},\\quad\ne_0:=-\\frac{1}{2}\\left.\\left(x\\frac{\\partial\\ }{\\partial x}-y\\frac{\\partial\\ }{\\partial y}\\right)\\right|_{V_6},\\quad e_1:=-\\left.y\\frac{\\partial\\ }{\\partial x}\\right|_{V_6}.\\]\nA direct calculation gives $[e_i,e_j]=(j-i)e_{i+j}$, for $i,j\\in\\{-1,0,1\\}$, so these endomorphisms span a subalgebra of ${\\mathfrak{gl}}(V_6)$ isomorphic to ${\\mathfrak{sl}}_2(\\mathbb{F})$ that acts by derivations on $V_6$. This construction also works when the characteristic of $\\mathbb{F}$ is $p\\geq 7$, and, moreover, the map  $f\\otimes g\\mapsto (f,g)_3$ gives the only, up to scalars, linear map $V_6\\otimes V_6\\rightarrow V_6$ invariant under the action of ${\\mathfrak{sl}}_2(\\mathbb{F})$.\n\nNow assume that the characteristic of $\\mathbb{F}$ is $7$. First, we will find a simpler formula describing the $3$-transvectant $(\\; , \\;)_3 : V_6 \\times V_6 \\rightarrow V_6$. For $0\\leq i\\leq 6$, denote\n\\[ m_i:= x^{6-i}y^i \\in V_6.\\]\nTaking the indices modulo $7$, the action of ${\\mathfrak{sl}}_2(\\mathbb{F})$ on $V_6$ is given by the following formulas:\n\\begin{equation}\\label{eq:e01-1mi}\n\\begin{split}\ne_{-1}(m_i)&=im_{i-1} \\text{ for } 1 \\leq i \\leq 6, \\text{ and } e_{-1}(m_0) = 0, \\\\\ne_0(m_i)&=(i+4)m_i \\text{ for any } 0 \\leq i \\leq 6, \\\\\ne_1(m_i)&=(i+1)m_{i+1} \\text{ for } 0 \\leq i \\leq 5, \\text{ and } e_1(m_6) = 0.\n\\end{split}\n\\end{equation}\n\nFor $i,j$ in the prime subfield $\\mathbb{F}_7$ of $\\mathbb{F}$, define the element $c(i,j)$ by\n\\begin{equation}\\label{eq:cij}\nc(i,j)=2(j-i)(4i+j-1)(4j+i-1)\\ \\bigl(\\in\\mathbb{F}_7\\subseteq \\mathbb{F}\\bigr).\n\\end{equation}\nIt is clear that $c(i,j)=-c(j,i)$ for any $i,j \\in \\mathbb{F}_7$, and $c(0,6)=1$. A straightforward computation gives, for any $i,j,k\\in\\mathbb{F}_7$:\n\\begin{equation}\\label{eq:cij_recurs}\n(i+j+4k+1)c(i,j)=(i+4k+4)c(i+k,j)+(j+4k+4)c(i,j+k).\n\\end{equation}\nTherefore, defining \n\\begin{equation}\\label{eq:mimj}\nm_i\\cdot m_j :=c(i,j)m_{i+j-3}\n\\end{equation}\n(with indices modulo $7$), we have\n\\[\ne_k(m_i\\cdot m_j)=e_k(m_i)\\cdot m_j+m_i\\cdot e_k(m_j),\n\\]\nfor any $k \\in \\{-1,0,1\\}$, $0\\leq i,j\\leq 6$. This means that the product in \\eqref{eq:mimj} is ${\\mathfrak{sl}}_2(\\mathbb{F})$-invariant; hence, by uniqueness and since $(m_0,m_6)_3=m_3=m_0\\cdot m_6$, we conclude that\n\\begin{equation}\\label{eq:mimj_3}\nm_i\\cdot m_j=(m_i,m_j)_3\n\\end{equation}\nfor any $0 \\leq i,j \\leq 6$. The multiplication table of $V_6$ with this product is given in Table \\ref{ta:multiplication_mis}.\n\n\\begin{table}[!h]\n\\setlength{\\tabcolsep}{6pt}\n\\renewcommand{\\arraystretch}{1.3}\n\\centering\n\\begin{tabular}{c|ccccccc|}\n$\\cdot$ & $m_0$ & $m_1$ & $m_2$ & $m_3$ & $m_4$ & $m_5$ & $m_6$ \\\\ \\hline \n$m_0$ & $0$ & $0$ & $0$ & $-m_0$ & $3m_1$ & $-3m_2$ & $m_3$   \\\\\n$m_1$ & $0$ & $0$ & $3m_0$ & $m_1$ & $0$ & $-m_3$ & $-3m_4$ \\\\\n$m_2$ & $0$ & $-3m_0$ & $0$ & $m_2$ & $-m_3$ & $0$ & $3m_5$\\\\\n$m_3$ & $m_0$ & $-m_1$ & $-m_2$ & $0$ & $m_4$ & $m_5$ & $-m_6$ \\\\\n$m_4$ & $-3m_1$ & $0$ & $m_3$ & $-m_4$ & $0$ & $3m_6$ & $0$ \\\\\n$m_5$ & $3m_2$ & $m_3$ & $0$ & $-m_5$ & $-3m_6$ & $0$ & $0$\\\\\n$m_6$ & $-m_3$ & $3m_4$ & $-3m_5$ & $m_6$ & $0$ & $0$ & $0$ \\\\ \\hline\n\\end{tabular}\n\\caption{{\\vrule width 0pt height 15pt}Multiplication table of $(V_6,\\cdot)$.}\n\\label{ta:multiplication_mis}\n\\end{table} \n\n\\begin{remark}\\label{re:explicit_iso}\nThe above arguments show that the split Cayley algebra $\\mathcal{C}_s$ is isomorphic to the algebra defined on $\\mathbb{F} 1\\oplus V_6$ with multiplication given by \\eqref{eq:CsDixmier}, or equivalently, by\n\\[\n(\\alpha 1+f)(\\beta 1+g):= \\bigl(\\alpha\\beta+(f,g)_6\\bigr)1+\n\\bigl(\\alpha g+\\beta f+f\\cdot g\\bigr).\n\\]\nAn explicit isomorphism between $\\mathbb{F} 1\\oplus V_6$ and $\\mathcal{C}_s$, in terms of a good basis of  $\\mathcal{C}_s$ as in Table \\ref{ta:good_basis}, is given by:\n\\[\n\\left( \\begin{matrix}\n\\ 1 \\ & \\ m_0\\ &\\ m_1\\ &\\ m_2\\ &\\ m_3\\ &\\ m_4\\ &\\ m_5\\ &\\ m_6\\ \\\\\n\\downarrow&\\downarrow&\\downarrow&\\downarrow&\\downarrow&\\downarrow&\\downarrow&\\downarrow\\\\\np_1 + p_2 & -3v_3&3u_2&3u_1&-p_1+p_2&-3v_1&-3v_2&3u_3\n\\end{matrix} \\right).\n\\]\n\\end{remark}\n\nNow, for any $k \\in \\{ 2,\\ldots,5 \\}$, define the endomorphism $e_k$ of $V_6$ by\n\\begin{equation}\\label{eq:eks}\ne_k(m_i)=\\begin{cases} (i+4k+4)m_{i+k}&\\text{if $i+k \\leq 6$,}\\\\\n   0&\\text{otherwise.}\n   \\end{cases}\n\\end{equation}\nWe will show that $e_k$ is a derivation of $(V_6,\\cdot)$ for any $k=\\{-1,0,\\ldots,5\\}$. Take $0\\leq i,j \\leq 6$. If $i+j+k-3\\leq 6$, $i+k\\leq 6$, and $j+k\\leq 6$, then\n\\begin{equation}\\label{eq:ekmimj}\n\\begin{split}\ne_k (m_i \\cdot m_j) &= (i+j + 4k + 1) c(i,j)  m_{i+j+k-3} \\\\\n &= \\Bigl((i+4k+4)c(i+k,j) + (j+4k+4)c(i,j+k)\\Bigr) m_{i+j+k-3} \\\\\n &= e_k(m_i) \\cdot m_j + m_i \\cdot e_k(m_j).\n\\end{split}\n\\end{equation}\nIf $i+j+k-3>6$, then $e_k(m_i\\cdot m_j)=0=e_k(m_i)\\cdot m_j=m_i\\cdot e_k(m_j)$. Finally, if $i+j+k-3\\leq 6$ and $i+k>6$ (the same happens if $j+k>6$), then $e_k(m_i)=0$ and $c(i+k,j)$ is one of $c(0,0)$, $c(0,1)$, $c(0,2)$ or $c(1,1)$, but all these are equal to $0$, so \\eqref{eq:ekmimj} applies.\n\nBecause of \\eqref{eq:CsDixmier} and \\eqref{eq:fgg}, we may extend the action of $e_k$ to $ \\mathbb{F} 1\\oplus V_6 \\cong \\mathcal{C}_s$ by means of $e_k(1)=0$, obtaining that $e_k$ is a derivation of the split Cayley algebra $\\mathcal{C}_s$. Furthermore, one checks at once that\n\\begin{equation}\\label{eq:Witt_basis}\n[e_i,e_j]=(j-i)e_{i+j}\n\\end{equation}\nfor any $i,j \\in \\{ -1,0,\\ldots,5 \\}$, with $e_i=0$ when $i$ is outside $\\{ -1,0,\\ldots,5 \\}$. Thus, the span of $\\{ e_i:-1\\leq i\\leq 5 \\}$ in $\\Der(V_6,\\cdot)$ is isomorphic to the Witt algebra $W_1:=\\Der\\left(\\mathbb{F}[X]\/(X^7)\\right)$; the map $e_i\\leftrightarrow x^{i+1}\\frac{\\partial }{\\partial x}$ gives an explicit isomorphism of these Lie algebras, where $x$ denotes the class of $X$ modulo $(X^7)$.\n\nSince $\\Der(\\mathcal{C}_s)$ is the split simple Lie algebra of type $G_2$, the above arguments provide an elementary proof of the next result.\n\n\\begin{theorem}[\\cite{Premet}]\\label{th:W1_char7}\nIf the characteristic of the ground field $\\mathbb{F}$ is $7$, then the Witt algebra $W_1$ embeds as a subalgebra of the simple split Lie algebra of type $G_2$.\n\\end{theorem}\n\n\\begin{remark}\nA specific embedding of $W_1$ into $\\Der(\\mathcal{C}_s)$ may be given in terms of a good basis of $\\mathcal{C}_s$. Given $x,y\\in\\mathcal{C}_s$, the derivation of $\\mathcal{C}_s$ defined by $D_{x,y}:=\\mathrm{ad}_{[x,y]}+[\\mathrm{ad}_x,\\mathrm{ad}_y]$ (where $\\mathrm{ad}_x(y):=[x,y]$) is called the \\emph{inner derivation induced by $x$ and $y$}. Then, the assignment\n\\begin{align*}\ne_{-1} &\\mapsto D_{p_1 - p_2,u_1} + D_{u_2, v_1},& e_3 &\\mapsto -D_{v_1, v_2} = 4D_{p_1 - p_2,u_3}, \\\\\ne_0 &\\mapsto 2D_{u_3, v_3} + 3D_{u_2,v_2},  &\\quad e_4 &\\mapsto 3D_{v_1, u_3}, \\\\\ne_1 &\\mapsto D_{p_1 - p_2,v_1} + D_{u_1,v_2}, & e_5 &\\mapsto 5D_{v_2,u_3},  \\\\\ne_2 &\\mapsto 3D_{u_1, u_3} = 2D_{p_1 - p_2,v_2}, &&\n\\end{align*} \ngives an explicit embedding of $W_1$ into $\\Der(\\mathcal{C}_s)$.\n\\end{remark}\n\n\\smallskip\n\n\n\\subsection{Invariant bilinear products on modules for $W_1$} \\label{subsec2}\n\nIt turns out that $V_6$ is a very special module for the Witt algebra in characteristic $7$ (see Theorem \\ref{th:main} below).\n\nAssume for this section that $\\mathbb{F}$ is an algebraically closed field of characteristic $p \\geq 5$. Let $\\mathcal{L}$ be a finite-dimensional restricted Lie algebra over $\\mathbb{F}$ with $p$-mapping denoted by $[p]$. Given any character $\\chi \\in \\mathcal{L}^*$, there is a finite-dimensional algebra $u(\\mathcal{L},\\chi)$, called the \\emph{reduced enveloping algebra of $\\mathcal{L}$ associated with $\\chi$}, that is a quotient of the universal enveloping algebra of $\\mathcal{L}$ and whose irreducible modules coincide precisely with the irreducible modules for $\\mathcal{L}$ with character $\\chi$. We say that $V$ is a \\emph{restricted module} for $\\mathcal{L}$ if there is a representation $\\rho : \\mathcal{L} \\rightarrow {\\mathfrak{gl}}(V)$ that is a morphism of restricted Lie algebras, i.e., $\\rho(x^{[p]}) = \\rho(x)^{p}$, for any $x \\in \\mathcal{L}$. When $\\chi=0$, it turns out that the irreducible modules for $u(\\mathcal{L}, \\chi)$ coincide precisely with the restricted irreducible modules for $\\mathcal{L}$.\n\nLet $W=W_1$ be the Witt algebra over $\\mathbb{F}$ with basis $\\{e_i:-1\\leq i\\leq p-2\\}$ and Lie bracket as in \\eqref{eq:Witt_basis}. It is well known that $W$ has a $p$-mapping given by $e_i^{[p]} := \\delta_{i,0}e_i$.\n\nFor each $i \\in \\{-1,0,...,p-2\\}$, define $W_{(i)} = \\langle e_i, ..., e_{p-2} \\rangle$. Consider the $p$-dimensional Verma modules $L(\\lambda) = u(W,0) \\otimes_{u(W_{(0)},0)} k_\\lambda$, where $\\lambda \\in \\{0,1,...,p-1\\}$, and $k_\\lambda$ is the one-dimensional module for $W_{(0)}$ on which $W_{(1)}$ acts trivially and $e_0$ acts by multiplication by $\\lambda$. By \\cite[Lemma 2.2.1]{N92}), $L(\\lambda)$ has a basis $\\left\\{ m_{0},m_{1},...,m_{p-1}\\right\\}$ on which the action of $W$ is given by\n\\[ e_{k}(m_{j}) = ( j + ( \\lambda +1 ) ( k+1 )) m_{k+j}, \\]\nwhere $m_{j}=0$ for $j$ outside $\\{ 0,...,p-1\\}$.\n\nIt was established in \\cite{C41} (see also \\cite{S77,FN98}) that any restricted irreducible module for $W$ is isomorphic to one of the following: \n\\begin{enumerate}\n\\item The trivial one-dimensional module.\n\\item The ($p-1$)-dimensional quotient $L(p-1)\/\\langle m_0 \\rangle$.\n\\item The $p$-dimensional Verma module $L(\\lambda)$, with $\\lambda \\in \\{1,...,p-2 \\}$.\n\\end{enumerate}\n\nIn the next result the invariant bilinear products $L\\times L \\rightarrow L$ (not to be confused with invariant bilinear forms!) on irreducible restricted modules for the Witt algebra are determined.\n\n\\begin{theorem}\\label{th:main}\nLet $W=W_1$ be the Witt algebra over an algebraically closed field $\\mathbb{F}$ of characteristic $p \\geq 5$. Then:\n\\begin{enumerate}\n\\item If $p \\neq 7$, there is no non-trivial non-adjoint restricted irreducible module for $W$ with a nonzero invariant bilinear product.\n\\item If $p = 7$, there is a unique non-trivial non-adjoint restricted irreducible module for $W$ with a nonzero invariant bilinear product. Up to isomorphism and scaling of the product, this unique module is $V_6$ with product given in \\eqref{eq:mimj}. ($V_6$ is a module for $W$ by means of \\eqref{eq:e01-1mi} and \\eqref{eq:eks}.)   \n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nWe will use the above classification of restricted irreducible modules for $W$ and follow several steps:\n\\begin{romanenumerate}\n\\item \nLet $\\lambda \\in \\{0,1,...,p-1\\}$ and suppose that $L(\\lambda)$ is equipped with $\\cdot$, a $W$-invariant bilinear product. We claim that $m_{0}\\cdot m_{p-1}= \\mu m_{\\lambda }$, for some scalar $\\mu \\in \\mathbb{F}$ that determines the invariant product. Indeed, first observe that \n\\[\\begin{split}\ne_{0}\\left( m_{0}\\cdot m_{p-1}\\right)  &=\\left( \\lambda +1\\right) m_{0}\\cdot m_{p-1}+\\left( p-1+\\left( \\lambda +1\\right) \\right) m_{0}\\cdot m_{p-1} \\\\\n&=\\left( 2\\lambda +1\\right) m_{0}\\cdot m_{p-1}.\n\\end{split}\\]\nTherefore, $m_{0}\\cdot m_{p-1}$ is an eigenvector of the action of $e_{0}$ with eigenvalue $2\\lambda +1$. As $m_{\\lambda }$ is also an eigenvector of the action of $e_{0}$ with eigenvalue $2 \\lambda +1$, and the action of $e_{0}$ is diagonal with $p$ distinct eigenvalues, we must have that $m_{0}\\cdot m_{p-1}$ is a scalar multiple of $m_{\\lambda}$. Furthermore, $m_0\\otimes m_{p-1}$ generates the $W$-module $L(\\lambda)\\otimes L(\\lambda)$, so this scalar determines the invariant product.\n\n\\item For any $p \\geq 5$, we will show that if $\\cdot$ is a $W$-invariant bilinear product on the irreducible module $L(p-1) \/ \\langle m_0 \\rangle$, then it must be the zero product. Denote by $\\bar x$ the class of an element $x\\in L(p-1)$ modulo $\\langle m_0\\rangle$; then, \n\\[ e_0 \\left( \\overline{m}_1 \\cdot \\overline{m}_{p-1} \\right) = \\overline{m}_1 \\cdot \\overline{m}_{p-1} + (p-1) \\overline{m}_1 \\cdot \\overline{m}_{p-1} =\\overline{0}. \\]\nAs all the eigenvalues of the action of $e_0$ on $L(p-1) \/ \\langle m_0 \\rangle$ are nonzero, this implies that $\\overline{m}_1 \\cdot \\overline{m}_{p-1}= \\overline{0}$. Now, using the $W$-invariance, it is easy to show that $\\cdot$ must be the zero product.\n\n\\item As $L(p-2)$ is the adjoint module for $W$, it obviously has a $W$-invariant bilinear product. Hence, we exclude this case from further observations.  \n\n\\item Let $p\\geq 5$ and $\\lambda \\in \\left\\{ 1,...,p-3\\right\\}$. We will show that if $L\\left( \\lambda \\right)$ has a nonzero $W$-invariant bilinear product $\\cdot$, then $p=7$ and $\\lambda =3$.\n\nBy step (i), we may assume that $m_0\\cdot m_{p-1}=m_\\lambda$. For $k \\in \\{1,...,p-2 \\}$,\n\\begin{eqnarray*}\n(\\lambda +1)(k+1)m_k\\cdot m_{p-1} & = &e_k(m_0)\\cdot m_{p-1} \\\\\n & = & e_k(m_0\\cdot m_{p-1})= e_k(m_\\lambda) \\\\\n & = & (\\lambda+(\\lambda+1)(k+1))m_{\\lambda +k},\n\\end{eqnarray*}\nso\n\\begin{equation}\\label{eq:mkmp-1}\nm_{k}\\cdot m_{p-1}=\\frac{\\left( \\lambda +1\\right) \\left( k+1\\right) +\\lambda}{\\left( \\lambda +1\\right) \\left( k+1\\right) }m_{\\lambda+k },\n\\end{equation}\nfor $k\\in \\left\\{ 1,...,p-2\\right\\}$. On the other hand, $e_1(e_1(m_\\lambda))=e_1(e_1(m_0)\\cdot m_{p-1}$, and this gives\n\\begin{equation}\\label{eq:m2mp-1}\n3\\left( 3\\lambda +2\\right) \\left( \\lambda +1\\right) m_{\\lambda +2} =2\\left( \\lambda +1\\right) \\left( 2\\lambda +3\\right) m_{2}\\cdot m_{p-1}. \n\\end{equation}\n\nIf $2\\lambda +3=0$, \\eqref{eq:m2mp-1} gives $3\\lambda+2=0$, and this implies $p=5$ and $\\lambda=1$. But then, using \\eqref{eq:mkmp-1} we get $e_1(m_2\\cdot m_4)=2e_1(m_3)=4m_4$, while at the same time  \n\\[ e_{1}\\left( m_{2}\\cdot m_{4}\\right) = (e_1m_3) \\cdot m_4 + m_3 \\cdot (e_1 m_4) = m_{3}\\cdot m_{4} = -2m_{4},\\]\nwhich is a contradiction. \n\nHence, we assume for the rest of the proof $2\\lambda +3\\ne 0$. By \\eqref{eq:m2mp-1}, we have\n\\[\n m_{2}\\cdot m_{6}=\\frac{3\\left( 3\\lambda +2\\right) }{2\\left( 2\\lambda +3\\right) }m_{\\lambda +2}.\n  \\]\nComparing this with \\eqref{eq:mkmp-1} with $k=2$ we obtain\n\\[\n \\frac{3\\left( 3\\lambda +2\\right) }{2\\left( 2\\lambda +3\\right) } = \\frac{4\\lambda +3}{3\\left( \\lambda +1\\right) }, \\text{ so } \\lambda \\left( 11\\lambda +9\\right) =0.\n\\]\nIf $p=11$, the above relation implies that $\\lambda =0$, which is a contradiction with our choice of $\\lambda $, so no invariant bilinear product exists for $p=11$.\n\nFor the rest of the proof, assume that $p\\neq 11$ and hence $\n\\lambda =-\\frac{9}{11}$.\nNow, from $e_{1}\\left( e_{1}\\left( e_{1}(m_{\\lambda} )\\right) \\right) =e_{1}\\left(e_{1}\\left( e_{1}(m_{0})\\cdot m_{p-1}\\right) \\right)$, we obtain\n\\[ \n\\left( 3\\lambda +2\\right) \\left( 3\\lambda +3\\right) \\left( 3\\lambda +4\\right) m_{\\lambda +3} = \\left( 2\\lambda +2\\right)\\left( 2\\lambda +3\\right) \\left( 2\\lambda +4\\right) m_{3}\\cdot m_{p-1}.\n\\]\nThus,\n\\[ \nm_{3}\\cdot m_{p-1}=\\frac{3\\left( 3\\lambda +2\\right) \\left( 3\\lambda +4\\right) }{4\\left( 2\\lambda +3\\right) \\left( \\lambda +2\\right) }m_{\\lambda +3}.\n\\]\nComparing this with \\eqref{eq:mkmp-1} for $k=3$ we obtain\n\\[ \n\\frac{3\\left( 3\\lambda +2\\right) \\left( 3\\lambda +4\\right) }{\\left( 2\\lambda +3\\right) \\left( \\lambda +2\\right) } = \\frac{5\\lambda +4}{\\left( \\lambda +1\\right) }. \n\\]\nHence,\n\\[ \n17\\lambda ^{2}+38\\lambda +20=0. \n\\]\nSubstituting $\\lambda =-\\frac{9}{11}$ in this relation we obtain that $p\\mid 35$, so either $p=7$ or $p=5$. If $p=5$, then $\\lambda =-\\frac{9}{11}=1$, and it was shown above that no nonzero invariant bilinear product exists in this case. If $p=7$, then $\\lambda =-\\frac{9}{11}=3$. This completes the proof. \\qedhere\n\\end{romanenumerate}\n\\end{proof}\n\n\\smallskip\n\n\\begin{remark} \\label{re:abs irr}\nLet $V$ and $U$ be two irreducible modules for a Lie algebra $\\mathcal{L}$ over an arbitrary field $\\mathbb{F}$, and let $\\mathbb{F}_{\\text{alg}}$ be an algebraic closure of $\\mathbb{F}$. Suppose that $V\\otimes_\\mathbb{F}\\FF_{\\text{alg}}$ and $U\\otimes_\\mathbb{F}\\FF_{\\text{alg}}$ are isomorphic as modules for $\\mathcal{L}\\otimes_\\mathbb{F} \\mathbb{F}_{\\text{alg}}$; then, $V$ and $U$ are isomorphic as modules for $\\mathcal{L}$. Indeed, if $V\\otimes_\\mathbb{F}\\FF_{\\text{alg}}$ and $U\\otimes_\\mathbb{F}\\FF_{\\text{alg}}$ are isomorphic, then $\\Hom_{\\mathcal{L}}(V,U)\\otimes_\\mathbb{F}\\FF_{\\text{alg}}\\cong \\Hom_{\\mathcal{L}\\otimes_\\mathbb{F}\\FF_{\\text{alg}}}(V\\otimes_\\mathbb{F}\\FF_{\\text{alg}},U\\otimes_\\mathbb{F}\\FF_{\\text{alg}})\\ne 0$, so $\\Hom_\\mathcal{L}(V,U)\\ne 0$. The result follows since, by irreducibility, any nonzero $\\mathcal{L}$-module homomorphism from $V$ to $U$ is an $\\mathcal{L}$-module isomorphism. In particular, this implies that, even when the ground field is not algebraically closed, Theorem \\ref{th:main} applies to modules for the Witt algebra that are absolutely irreducible (i.e., they remain irreducible after extending scalars to an algebraic closure).\n\\end{remark}\n\n\\smallskip\n\n\n\\subsection{Embeddings of $W_1$ and its twisted forms into $G_2$} \\label{subsec3}\n\nIt is shown in \\cite{Premet,Herpel_Stewart} that, over an algebraically closed field of characteristic $7$, the simple Lie algebra of type $G_2$ contains a unique conjugacy class of subalgebras isomorphic to the Witt algebra. With our results above, a different proof may be given, valid for not necessarily algebraically closed fields.\n\n\\begin{theorem}\\label{th:conjugation}\nLet $\\mathbb{F}$ be an arbitrary field of characteristic $7$. Then any two subalgebras $S_1$ and $S_2$ of $\\Der(\\mathcal{C}_s)$ isomorphic to the Witt algebra are conjugate: there is an automorphism $\\varphi$ of $\\mathcal{C}_s$ such that $S_2=\\varphi S_1\\varphi^{-1}$.\n\\end{theorem}\n\\begin{proof}\nLet $S:=\\espan{e_i:-1\\leq i\\leq 5}$ be the subalgebra of $\\frg:=\\Der(\\mathcal{C}_s)$ isomorphic to the Witt algebra given by equations \\eqref{eq:e01-1mi} and \\eqref{eq:eks}, and let $\\tilde S$ be an arbitrary subalgebra of $\\frg$ isomorphic to the Witt algebra. Take a basis $\\{\\tilde e_i:-1\\leq i\\leq 5\\}$ of $\\tilde S$ with $[\\tilde e_i,\\tilde e_j]=(j-i)\\tilde e_{i+j}$. We will follow several steps:\n\n\\begin{romanenumerate}\n\\item $\\tilde S$ is a maximal subalgebra of $\\frg$.\n\\begin{proof}\nSince the Witt algebra does not admit nonsingular invariant bilinear forms (see, e.g. \\cite[Theorem 4.2]{F86}), the restriction of the Killing form $\\kappa$ of $\\frg$ to $\\tilde S$ is trivial; hence, by dimension count, $\\frg\/\\tilde S$ is isomorphic, as a module for $\\tilde S$, to the dual of the adjoint module for $\\tilde S$. In particular, $\\frg\/\\tilde S$ is an irreducible module for $\\tilde S$, and this shows the maximality of $\\tilde S$.\n\\end{proof}\n\n\\item The representation of $\\tilde S$ on $\\mathcal{C}_s^0$, the subspace of trace zero elements of $\\mathcal{C}_s$, is restricted and absolutely irreducible.\n\\begin{proof}\nAs $\\tilde S$ is an ideal in its $p$-closure in $\\frg$, step (i) implies that $\\tilde S$ is a restricted subalgebra of $\\frg$, and hence, the corresponding representation of $\\tilde S$ on $\\mathcal{C}_s^0$ is restricted. \n\nIn order to prove the second part, we may assume that $\\mathbb{F}$ is algebraically closed. Recall that the possible dimensions of restricted irreducible representations of the Witt algebra are $1$, $6$ and $7$. Suppose that $\\mathcal{C}_s^0$ has a one-dimensional trivial $\\tilde S$-submodule $X$. The space $X$ may be either nondegenerate or totally isotropic with respect to the symmetric bilinear form $b_q$ of $\\mathcal{C}_s$. \n\nIf $X = \\mathbb{F} x$ is nondegenerate, then $q(x) \\neq 0$ and $b_q(x,1) = 0$. By \\eqref{eq:CayleyHamilton}, $\\mathbb{F} 1 \\oplus X$ is a two-dimensional composition subalgebra of $\\mathcal{C}_s$, so it is isomorphic to $\\mathbb{F} p_1\\oplus\\mathbb{F} p_2$ (with $p_i$ as in Table \\ref{ta:good_basis}). Now, by \\cite[Corollary 1.7.3]{SV00}, the isomorphism $\\mathbb{F} 1 \\oplus \\mathbb{F} x  \\cong \\mathbb{F} p_1 \\oplus \\mathbb{F} p_2$ may be extended to an automorphism of $\\mathcal{C}_s$. As $\\tilde S$ annihilates $\\mathbb{F} 1 \\oplus \\mathbb{F} x$, and the subalgebra of the derivations that annihilate $\\mathbb{F} p_1\\oplus\\mathbb{F} p_2$ is isomorphic to ${\\mathfrak{sl}}_3(\\mathbb{F})$ (\\cite[Proposition 4.29]{EKmon}), we obtain that the Witt algebra embeds into ${\\mathfrak{sl}}_3(\\mathbb{F})$, which is impossible. \n\nIf $X$ is totally isotropic, then $X \\subseteq X^{\\perp}$, and $X^{\\perp} \/ X$ is a $5$-dimensional module for $\\tilde S$. As this cannot be irreducible, we deduce that all the composition factors of $\\mathcal{C}_s^0$ are one-dimensional. Hence, the representation $\\rho$ of $\\tilde S$ on $\\mathcal{C}_s^0$ is nilpotent. Since $\\mathrm{ad}_{(\\tilde e_0 + \\tilde e_i)}$ is diagonalizable for $i\\ne 0$, $-1 \\leq i \\leq 5$, then $\\mathrm{ad}_{(\\tilde e_0 + \\tilde e_i)}^p = \\mathrm{ad}_{(\\tilde e_0+ \\tilde e_i)}$. As the representation of $\\tilde S$ on $\\mathcal{C}_s^0$ is restricted, then $\\rho(\\tilde e_0 + \\tilde e_i)^p = \\rho(\\tilde e_0 + \\tilde e_i)$, and the nilpotency implies that $\\rho(\\tilde e_0+\\tilde e_i) =0$, for any $i\\ne 0$, $-1 \\leq i \\leq 5$. This is a contradiction.\n\nFinally, if $Y$ is a $6$-dimensional irreducible $\\tilde S$-submodule for $\\mathcal{C}_s^0$, then it must be nondegenerate with respect to $b_q$. This implies that $Y^{\\perp}$ is a one-dimensional nondegenerate submodule, and we may use the above argument with $X=Y^{\\perp}$. \n\\end{proof}\n\n\\item Note that $\\mathcal{C}_s^0$ is not the adjoint module for $\\tilde S$ because of the existence of the invariant bilinear form $b_q$ on $\\mathcal{C}_s^0$. \n\n\\item The previous steps together with Theorem \\ref{th:main} and Remark \\ref{re:abs irr} imply that, even when $\\mathbb{F}$ is not algebraically closed, there is a unique possibility, up to isomorphism, for $\\mathcal{C}_s^0$ as a module for $\\tilde S$, and a unique, up to scalars, nonzero $\\tilde S$-invariant product on $\\mathcal{C}_s^0$. Therefore, there is a basis $\\{\\tilde m_i: 0\\leq i\\leq 6\\}$ of $\\mathcal{C}_s^0$ with \n\\[\n\\tilde e_k(\\tilde m_i)=\\begin{cases} (i+4k+4)\\tilde m_{i+k}&\\text{if $i+k\\geq 6$,}\\\\\n   0&\\text{otherwise,}\n   \\end{cases}\n\\]\nand \n\\[\n\\frac{1}{2}[\\tilde m_i\\tilde m_j]=c(i,j)\\tilde m_{i+j-3}\n\\]\nfor $-1\\leq k\\leq 5$ and $0\\leq i,j\\leq 6$. Since the multiplication on a Cayley algebra is determined by the bracket of trace zero elements, the linear map $\\varphi$ that takes $1$ to $1$ and $m_i$ to $\\tilde m_i$, for $i \\in \\{ 0,\\ldots,6 \\}$, is an automorphism of $\\mathcal{C}_s$ such that $\\tilde S=\\varphi S\\varphi^{-1}$.\n\n\\end{romanenumerate}\n\\end{proof}\n\n\\smallskip\n\nTo finish this section, note that the Witt algebra $W$ over a field $\\mathbb{F}$ of characteristic $7$ is equal to the Lie algebra $\\mathbb{F}[X]\/(X^7)=\\mathbb{F}[Z]\/(Z^7-1)$, where $Z=X+1$. Denote by $z$ the class of $Z$ modulo $(Z^7-1)=((Z-1)^7)$. The elements $f_i=z^{i+1}\\frac{\\partial\\ }{\\partial z}$, for $i \\in \\{-1,0,\\ldots 5\\}$, form a basis of $W$ with\n\\begin{equation}\\label{eq:secondWitt_basis}\n[f_i,f_j]=(j-i)f_{i+j},\n\\end{equation}\nwhere, contrary to \\eqref{eq:Witt_basis}, the indices are taken modulo $7$. Now we may define an action of $W$ on $V_6$ by changing slightly the definition in \\eqref{eq:eks}:\n\\begin{equation}\\label{eq:fks}\nf_k(m_i)=(i+4k+4)m_{i+k},\n\\end{equation}\nfor $-1\\leq k\\leq 5$ and $0\\leq i\\leq 6$, with indices taken modulo $7$. This gives another representation of $W$ on $V_6$:\n\\[\n\\begin{split}\nf_r(f_s(m_i))&-f_s(f_r(m_i))\\\\\n &=\\Bigl((i+s+4r+4)(i+4s+4)-(i+r+4s+4)(i+4r+4)\\Bigr)m_{i+s+r}\\\\\n &=(s-r)(i+4(r+s)+4)m_{i+s+r}\\\\\n &=[f_r,f_s](m_i).\n\\end{split}\n\\]\nEquation \\eqref{eq:cij_recurs} proves that this is a representation by derivations, so $W$ embeds in $\\Der(V_6,\\cdot)$, and hence on $\\Der(\\mathcal{C}_s)$ as well. This embedding is different from the one obtained through \\eqref{eq:eks}, but Theorem \\ref{th:conjugation} shows that they are conjugate.\n\nWe may even go a step further. For any $0\\ne \\alpha\\in\\mathbb{F}$, consider the Lie algebra $W^\\alpha=\\Der\\left(\\mathbb{F}[Y]\/(Y^7-\\alpha)\\right)$, and denote by $y$ the class of $Y$ modulo $(Y^7-\\alpha)$. For any natural number $i$, consider the element $\\tilde f_i=y^{i+1}\\frac{\\partial\\ }{\\partial y}$ in $W^\\alpha$, so $\\{\\tilde f_i:-1\\leq i\\leq 5\\}$ is a basis of $W^\\alpha$, and $f_{i+7}=\\alpha f_i$ for any $i$. Define $m_{i+7}:= \\alpha m_i$. For $0\\leq i,j\\leq 6$, $c(i,j)=0$ if $i+j-3>6$ or $i+j-3<0$, so if we modify \\eqref{eq:fks} as follows:\n\\begin{equation}\\label{eq:ftildeks}\n\\tilde f_k(m_i)=(i+4k+4)m_{i+j-3},\n\\end{equation}\nfor any $i,k$ (but now $\\tilde f_{k+7}=\\alpha\\tilde f_k$ and $m_{i+7}=\\alpha m_i$). The same computations as above show that $W^\\alpha$ embeds in $\\Der(\\mathcal{C}_s)$.\n\nThe twisted forms of the Witt algebra are precisely the algebras $W^\\alpha$; if $0\\ne\\alpha\\in\\mathbb{F}^7$, $W^\\alpha$ is isomorphic to the Witt algebra, while if $\\alpha\\in\\mathbb{F}\\setminus\\mathbb{F}^7$, $W^\\alpha$ is the Lie algebra of derivations of the purely inseparable field extension $\\mathbb{F}[Y]\/(Y^7-\\alpha)$. Two algebras $W^\\alpha$ and $W^\\beta$ are isomorphic if and only if so are the algebras $\\mathbb{F}[Y]\/(Y^7-\\alpha)$ and $\\mathbb{F}[Y]\/(Y^7-\\beta)$. (See \\cite{Allen_Sweedler69} or \\cite{Waterhouse71}.)\n\nTherefore, Theorems \\ref{th:W1_char7} and \\ref{th:conjugation} may be extended as follows:\n\n\\begin{theorem}\\label{th:Walpha}\nOver a field of characteristic $7$, all the twisted forms of the Witt algebra embed in the Lie algebra of derivations of the split Cayley algebra. Moreover, any two embeddings of the same twisted form of the Witt algebra in $\\Der(\\mathcal{C}_s)$ are conjugate.\n\\end{theorem}\n\nThe last part of this theorem follows by the same arguments as in the proof of Theorem \\ref{th:conjugation}.\n\n\\begin{remark}\nIf $\\mathcal{C}$ is a non split Cayley algebra (and hence it is a division algebra, since $q$ is anisotropic), then $\\Der(\\mathcal{C})$ contains no nonzero nilpotent derivation. Indeed, if $d\\in\\Der(\\mathcal{C})$ is nilpotent, then $\\mathrm{ker}\\, d\\vert_{\\mathcal{C}^0}$ is a subspace of $\\mathcal{C}^0$ and, since $q$ is anisotropic, $\\mathcal{C}^0=\\mathrm{ker}\\, d\\vert_{\\mathcal{C}^0}\\oplus \\left(\\mathrm{ker}\\, d\\vert_{\\mathcal{C}^0}\\right)^\\perp$. Besides, since $q$ is invariant under the action of $d$ (because of \\eqref{eq:CayleyHamilton}), $d$ leaves $\\left(\\mathrm{ker}\\, d\\vert_{\\mathcal{C}^0}\\right)^\\perp$ invariant; this is a contradiction because the nilpotency of $d$ implies that any nonzero invariant subspace has nontrivial intersection with the kernel.\n\nTherefore, $\\Der(\\mathcal{C})$ cannot contain subalgebras isomorphic to twisted forms of the Witt algebra, because these algebras contain nilpotent elements. (Recall that the action of the Witt algebra on $\\mathcal{C}_s^0$ is restricted, and hence so is the action on $\\mathcal{C}^0$ of any subalgebra of $\\Der(\\mathcal{C})$ isomorphic to a twisted form of the Witt algebra.)\n\\end{remark}\n\n\\medskip\n\n\n\\section{Characteristic $3$}\\label{se:char3}\n\nLet $\\mathcal{C}$ be a Cayley algebra over a field $\\mathbb{F}$ of characteristic $p \\neq 2$. Then, the subspace of trace zero elements $\\mathcal{C}^0$ is closed under the commutator $[\\,, \\,]$, and it satisfies \\eqref{eq:xyy}. The anticommutative algebra $\\left( \\mathcal{C}^0, [\\, , \\, ] \\right)$ is a central simple Malcev algebra. If $p \\ne 2,3$, any central simple non Lie Malcev algebra is isomorphic to one of these. However, if $p=3$, then $\\mathcal{C}^0$ is a simple Lie algebra; more precisely, it is a twisted form of the projective special linear Lie algebra ${\\mathfrak{psl}}_3(\\mathbb{F})$, and any such twisted form is obtained, up to isomorphism, in this way. (See \\cite{AEMN} or \\cite[Theorem 4.26]{EKmon}.)\n\nDenote by $\\AAut(\\cA)$ the affine group scheme of automorphisms of a finite-dimensional algebra $\\cA$. Equations \\eqref{eq:xyC0} and \\eqref{eq:xyy} show that the restriction map\n\\begin{equation}\\label{eq:AutCAutC0}\n\\begin{split}\n\\AAut(\\mathcal{C})&\\longrightarrow \\AAut(\\mathcal{C}^0)\\\\\nf\\ &\\mapsto\\quad f\\vert_{\\mathcal{C}^0},\n\\end{split}\n\\end{equation}\ngives an isomorphism of group schemes. (The reader may consult \\cite[Chapter VI]{KMRT} for the basic facts of affine group schemes.)\n\nFor the rest of this section, assume that $\\mathcal{C}$ is a Cayley algebra over a field $\\mathbb{F}$ of characteristic $3$, and write $\\frg\\ :=\\Der(\\mathcal{C})$. Any derivation $d\\in\\frg$ satisfies $d(1)=0$ and $d(\\mathcal{C}^0)\\subseteq \\mathcal{C}^0$, and hence, because of \\eqref{eq:xyC0} and \\eqref{eq:xyy}, we may identify $\\frg$ with $\\Der(\\mathcal{C}^0)$. Since $\\mathcal{C}^0$ is a Lie algebra, $\\mathrm{ad}_{\\mathcal{C}^0}$ is an ideal of $\\frg$: the ideal of inner derivations. In fact, $\\mathrm{ad}_{\\mathcal{C}^0}$ is the only proper ideal of $\\frg$, and the quotient $\\frg\/\\mathrm{ad}_{\\mathcal{C}^0}$ is again isomorphic to $\\mathcal{C}^0\\cong \\mathrm{ad}_{\\mathcal{C}^0}$ (see \\cite{AEMN}).\n\nIn the split case, $\\mathrm{ad}_{\\mathcal{C}_s^0}$ is the ideal of the Chevalley algebra of type $G_2$ generated by the root spaces corresponding to the short roots (see \\cite[p.~156]{Steinberg}).\n\n\\begin{lemma}\\label{le:inner}\nAny derivation of $\\frg :=\\Der(\\mathcal{C})$ is inner.\n\\end{lemma}\n\\begin{proof}\nLet $d\\in \\Der(\\frg)$ and let ${\\mathfrak i}=\\mathrm{ad}_{\\mathcal{C}^0}$ be the unique proper ideal of $\\frg$. The ideal ${\\mathfrak i}$ is simple, so ${\\mathfrak i}=[{\\mathfrak i},{\\mathfrak i}]$, and hence $d({\\mathfrak i})\\subseteq [d({\\mathfrak i}),{\\mathfrak i}]\\subseteq {\\mathfrak i}$, so $d\\vert_{\\mathfrak i}$ is a derivation of ${\\mathfrak i}=\\mathrm{ad}_{\\mathcal{C}^0}\\cong\\mathcal{C}^0$. Since any derivation $\\delta$ of $\\mathcal{C}^0$ extends to a derivation of $\\mathcal{C}$ by means of $\\delta(1)=0$, it follows that there exists a $\\delta\\in\\frg$ such that $d\\vert_{\\mathfrak i}=\\mathrm{ad}_\\delta\\vert_{\\mathfrak i}$. That is, $d(f)=[\\delta,f]$ for any $f\\in{\\mathfrak i}=\\mathrm{ad}_{\\mathcal{C}^0}$, so the derivation $\\tilde d=d-\\mathrm{ad}_\\delta$ satisfies $\\tilde d({\\mathfrak i})=0$.\n\nBut if $\\tilde d$ is a derivation of $\\frg$ such that $\\tilde d({\\mathfrak i})=0$, then $[\\tilde d(\\frg),{\\mathfrak i}]\\subseteq \\tilde d\\bigl([\\frg,{\\mathfrak i}]\\bigr)+[\\frg,\\tilde d({\\mathfrak i})]=0$, so $\\tilde d(\\frg)$ is contained in the centralizer of ${\\mathfrak i}$ in $\\frg$, which is trivial. In particular, the derivation $\\tilde d=d-\\mathrm{ad}_\\delta$ above is trivial, and hence $d=\\mathrm{ad}_\\delta$ is inner.\n\\end{proof}\n\n\\begin{theorem}\\label{th:char3}\nLet $\\mathcal{C}$ be a Cayley algebra over a field $\\mathbb{F}$ of characteristic $3$, and let $\\frg :=\\Der(\\mathcal{C})$ be its Lie algebra of derivations. The the adjoint map\n\\[\n\\begin{split}\n\\mathrm{Ad}: \\AAut(\\mathcal{C})&\\longrightarrow \\AAut(\\frg)\\\\\n       f\\ &\\mapsto\\ \\varphi(f):d\\mapsto fdf^{-1},\n\\end{split}\n\\]\nis an isomorphism of affine group schemes.\n\\end{theorem}\n\\begin{proof}\nLet $\\mathbb{F}_{\\text{alg}}$ be an algebraic closure of $\\mathbb{F}$. The group homomorphism $\\mathrm{Ad}_{\\mathbb{F}_{\\text{alg}}}:\\Aut(\\mathcal{C}_{\\mathbb{F}_{\\text{alg}}})\\rightarrow \\Der(\\mathcal{C}_{\\mathbb{F}_{\\text{alg}}})$ is injective, where $\\mathcal{C}_R=\\mathcal{C}\\otimes_\\mathbb{F} R$, for any unital commutative and associative algebra $R$ over $\\mathbb{F}$. This is because $f\\mathrm{ad}_xf^{-1}=\\mathrm{ad}_{f(x)}$ for any $x\\in \\mathcal{C}^0_{\\mathbb{F}_{\\text{alg}}}$ and $f\\in\\Aut(\\mathcal{C}_{\\mathbb{F}_{\\text{alg}}})$, so $\\mathrm{Ad}_{\\mathbb{F}_{\\text{alg}}}(f)=\\id$ implies $f\\vert_{\\mathcal{C}^0_{\\mathbb{F}_{\\text{alg}}}}=\\id$; hence, $f=\\id$. \n\nAs any $\\varphi\\in\\Aut\\left(\\frg_{\\mathbb{F}_{\\text{alg}}}\\right)$ preserves the only proper ideal ${\\mathfrak i}_{\\mathbb{F}_{\\text{alg}}}=\\mathrm{ad}_{\\mathcal{C}^0_{\\mathbb{F}_{\\text{alg}}}}$, the automorphism $\\varphi$ induces an automorphism $f$ of $\\mathcal{C}^0_{\\mathbb{F}_{\\text{alg}}}$, which extends to an automorphism of $\\mathcal{C}_{\\mathbb{F}_{\\text{alg}}}$ also denoted by $f$. Then $\\varphi\\mathrm{Ad}(f)^{-1}\\vert_{{\\mathfrak i}_{\\mathbb{F}_{\\text{alg}}}}=\\id$, and, as in the proof above, a simple argument gives $\\varphi\\mathrm{Ad}(f)^{-1}=\\id$, so $\\varphi=\\mathrm{Ad}(f)$. This shows that $\\mathrm{Ad}_{\\mathbb{F}_{\\text{alg}}}$ is a bijection.\n\nBut it also shows that $\\dim \\AAut(\\frg)=\\dim\\AAut(\\mathcal{C})$ and, since this latter group scheme is smooth (this follows from \\cite[Proposition 2.2.3]{SV00}, see also \\cite[Proof of Theorem 4.35]{EKmon}), \nwe obtain $\\dim\\AAut(\\frg)=\\dim\\Der(\\mathcal{C})=\\dim\\Der(\\frg)$ by Lemma \\ref{le:inner}. Therefore, $\\AAut(\\frg)$ is smooth.\n\nSince the differential map $\\textup{d}(\\mathrm{Ad}) : \\Der(\\mathcal{C})\\rightarrow\\Der(\\frg)$, $\\delta\\mapsto \\mathrm{ad}_\\delta$, is injective, \\cite[(22.5)]{KMRT} shows that $\\mathrm{Ad}$ is an isomorphism.\n\\end{proof}\n\nDenote by $\\textup{Isom}(\\textup{Cayley})$, $\\textup{Isom}(G_2)$, and $\\textup{Isom}(\\bar A_2)$, the sets of isomorphism classes of Cayley algebras, twisted forms of the Chevalley algebra of type $G_2$, and twisted forms of ${\\mathfrak{psl}}_3(\\mathbb{F})$, respectively. Theorem \\ref{th:char3} and \\eqref{eq:AutCAutC0} immediately give the following consequence, where $[\\cA]$ denotes the isomorphism class of the algebra $\\cA$.\n\n\\begin{corollary}\\label{co:char3}\nThe maps $[\\mathcal{C}]\\mapsto[\\Der(\\mathcal{C})]$ and $[\\mathcal{C}]\\mapsto [\\mathcal{C}^0]$ give bijections\\\\ $\\textup{Isom}(\\textup{Cayley})\\rightarrow \\textup{Isom}(G_2)$ and $\\textup{Isom}(\\textup{Cayley})\\rightarrow \\textup{Isom}(\\bar A_2)$, respectively.\n\\end{corollary}\n\n\n\\medskip\n\n\n\\section{Characteristic $2$}\\label{se:char2}\n\nIn this section, assume that the characteristic of the ground field $\\mathbb{F}$ is $2$. In \\cite[Corollary 4.32]{EKmon} it is proved that the Chevalley algebra of type $G_2$ (i.e., the Lie algebra $\\Der(\\mathcal{C}_s)$), is isomorphic to the projective special linear Lie algebra ${\\mathfrak{psl}}_4(\\mathbb{F})$. Here we extend this result for the Lie algebra of derivations of any Cayley algebra over $\\mathbb{F}$.\n\nLet $V$ be a finite-dimensional vector space over $\\mathbb{F}$, and let $b$ be a nondegenerate alternating (i.e., $b(u,u)=0$ for any $u \\in V$) bilinear form of $V$. Denote by ${\\mathfrak{sp}}(V,b)$ the corresponding symplectic Lie algebra:\n\\[\n{\\mathfrak{sp}}(V,b)=\\{ f\\in{\\mathfrak{gl}}(V): b(f(u),v)+b(u,f(v))=0\\ \\forall u,v\\in V\\}.\n\\]\nIn particular, ${\\mathfrak{sp}}_{2n}(\\mathbb{F})$ denotes the symplectic Lie algebra ${\\mathfrak{sp}}(\\mathbb{F}^{2n},b_s)$, where $b_s$ is the alternating bilinear form with coordinate matrix $\\begin{pmatrix} 0&I_n\\\\ I_n&0\\end{pmatrix}$ in the canonical basis. (Notice that, as $\\text{char}(\\mathbb{F}) = 2$, there is no need of minus signs.) We identify the elements of ${\\mathfrak{sp}}_{2n}(\\mathbb{F})$ with their coordinate matrices in the canonical basis.\n\nA matrix in $M_n(\\mathbb{F})$ is called \\emph{alternating} if it has the form $a+a^t$ for some $a\\in M_n(\\mathbb{F})$, where $a^t$ denotes the transpose of $a$. These are the coordinate matrices of the alternating bilinear forms.\n\n\\begin{lemma}\\label{le:sp62psl4}\nThe second derived power of the symplectic Lie algebra on a vector space of dimension $6$ is isomorphic to the projective special linear Lie algebra ${\\mathfrak{psl}}_4(\\mathbb{F})$:\n\\[\n{\\mathfrak{sp}}_6(\\mathbb{F})^{(2)}\\cong{\\mathfrak{psl}}_4(\\mathbb{F}).\n\\]\n\\end{lemma}\n\\begin{proof}\nFor any natural number $n$ we have\n\\[\n{\\mathfrak{sp}}_{2n}(\\mathbb{F})=\\left\\{\\begin{pmatrix} a&b\\\\ c&a^t\\end{pmatrix} :\na,b,c\\in M_n(\\mathbb{F}),\\ b^t=b,\\, c^t=c\\right\\}.\n\\]\nA direct computation gives\n\\[\n\\begin{split}\n{\\mathfrak{sp}}_{2n}(\\mathbb{F})^{(1)}&=\\left\\{\\begin{pmatrix} a&b\\\\ c&a^t\\end{pmatrix} :\na,b,c\\in M_n(\\mathbb{F}),\\ \\text{$b$ and $c$ alternating}\\right\\},\\\\\n{\\mathfrak{sp}}_{2n}(\\mathbb{F})^{(2)}&=\\left\\{\\begin{pmatrix} a&b\\\\ c&a^t\\end{pmatrix} :\na,b,c\\in M_n(\\mathbb{F}),\\ a\\in{\\mathfrak{sl}}_n(\\mathbb{F}),\\ \\text{$b$ and $c$ alternating}\\right\\}.\n\\end{split}\n\\]\nThe dimension of ${\\mathfrak{sp}}_{2n}(\\mathbb{F})^{(2)}$ is then $n^2-1+2\\binom{n}{2}=2n^2-n-1$.\n\nLet $V$ be a four-dimensional vector space, and consider the second exterior power $\\bigwedge^2V$ as a module for ${\\mathfrak{sl}}(V)$. Fix a nonzero linear isomorphism $\\det:\\bigwedge^4 V\\rightarrow \\mathbb{F}$ and define the nondegenerate alternating bilinear form\n\\[\n\\begin{split}\nb:\\textstyle{\\bigwedge^2V\\times\\bigwedge^2V}&\\longrightarrow \\mathbb{F}\\\\\n (u_1\\wedge u_2,v_1\\wedge v_2)&\\mapsto \\det(u_1\\wedge u_2\\wedge v_1\\wedge v_2).\n\\end{split}\n\\]\nThen, the action of ${\\mathfrak{sl}}(V)$ on $\\bigwedge^2V$ gives a Lie algebra homomorphism\n\\[\n\\Phi:{\\mathfrak{sl}}(V)\\rightarrow {\\mathfrak{sp}}(\\textstyle{\\bigwedge^2V},b)\\cong{\\mathfrak{sp}}_6(\\mathbb{F}),\n\\]\nwith kernel $\\mathbb{F} I_V$ (where $I_V$ denotes the identity map on $V$), so $\\Phi$ induces an injection ${\\mathfrak{psl}}(V)\\hookrightarrow {\\mathfrak{sp}}_(\\bigwedge^2V,b)$. But ${\\mathfrak{psl}}(V)$ is simple of dimension $14$, so, in particular, ${\\mathfrak{psl}}(V)^{(2)}={\\mathfrak{psl}}(V)$, and the dimension of ${\\mathfrak{sp}}(\\bigwedge^2V,b)^{(2)}\\cong{\\mathfrak{sp}}_6(\\mathbb{F})^{(2)}$ is $2\\times 3^2-3-1=14$. Therefore, $\\Phi$ induces an isomorphism ${\\mathfrak{psl}}(V)\\cong {\\mathfrak{sp}}(\\bigwedge^2V,b)^{(2)}$, as required.\n\\end{proof}\n\nWe will need some extra notation. As above, let $(V,b)$ be a finite-dimensional vector space endowed with a nondegenerate alternating bilinear form. Denote by $\\mathfrak{gsp}(V,b)$ the Lie algebra of the group of similarities (i.e., the general symplectic Lie algebra):\n\\begin{multline*}\n\\mathfrak{gsp}(V,b)=\\left\\{ f\\in{\\mathfrak{gl}}(V): \\exists\\lambda\\in\\mathbb{F}\\text{\\ such that\\ }\\right. \\\\\n\\left. b(f(u),v)+b(u,f(v))=\\lambda b(u,v)\\ \\forall u,v\\in V\\right\\},\n\\end{multline*}\nand by $\\mathfrak{pgsp}(V,b)$ the projective general symplectic Lie algebra (i.e., the quotient of $\\mathfrak{gsp}(V,b)$ by the one-dimensional ideal generated by $I_V$). In particular, after choosing a basis, we get the Lie algebras $\\mathfrak{gsp}_{2n}(\\mathbb{F})$ and $\\mathfrak{pgsp}_{2n}(\\mathbb{F})$.\n\n\\begin{corollary}\\label{co:sp6psl4}\nThe Lie algebra of derivations of ${\\mathfrak{psl}}_4(\\mathbb{F})$ is isomorphic to the projective general symplectic Lie algebra $\\mathfrak{pgsp}_6(\\mathbb{F})$:\n\\[\n\\Der\\bigl({\\mathfrak{psl}}_4(\\mathbb{F})\\bigr)\\cong \\mathfrak{pgsp}_6(\\mathbb{F}).\n\\]\n\\end{corollary}\\label{co:pgsp6_Derpsl4}\n\\begin{proof}\nNote that we have the decomposition\n\\[\n\\mathfrak{gsp}_6(\\mathbb{F})={\\mathfrak{sp}}_6(\\mathbb{F})\\oplus \\mathbb{F}\\begin{pmatrix} I_3&0\\\\ 0&0\\end{pmatrix},\n\\]\nand one may easily check that $\\mathfrak{gsp}_6(\\mathbb{F})^{(1)}={\\mathfrak{sp}}_6(\\mathbb{F})$.\n\nFor any $A\\in\\mathfrak{gsp}_6(\\mathbb{F})$, $\\mathrm{ad}_A:B\\mapsto [A,B]$ leaves invariant $\\mathfrak{gsp}_6(\\mathbb{F})^{(3)}={\\mathfrak{sp}}_6(\\mathbb{F})^{(2)}$, which is isomorphic to ${\\mathfrak{psl}}_4(\\mathbb{F})$ by Lemma \\ref{le:sp62psl4}, so we obtain a Lie algebra homomorphism\n\\[\n\\begin{split}\n\\Phi:\\mathfrak{gsp}_6(\\mathbb{F})&\\longrightarrow \\Der\\bigl({\\mathfrak{sp}}_6(\\mathbb{F})^{(2)}\\bigr)\\, \\Bigl(\\cong\\Der\\bigl({\\mathfrak{psl}}_4(\\mathbb{F})\\bigr)\\,\\Bigr),\\\\\nA\\ &\\mapsto \\ \\mathrm{ad}_A\\vert_{{\\mathfrak{sp}}_6(\\mathbb{F})^{(2)}}.\n\\end{split}\n\\]\nThe kernel of $\\Phi$ is the centralizer in $\\mathfrak{gsp}_6(\\mathbb{F})$ of ${\\mathfrak{sp}}_6(\\mathbb{F})^{(2)}$, which is $\\mathbb{F} I_6$, so $\\Phi$ induces an injection $\\mathfrak{pgsp}_6(\\mathbb{F})\\rightarrow \\Der\\bigl({\\mathfrak{psl}}_4(\\mathbb{F})\\bigr)$.  The dimension of $\\mathfrak{pgsp}_6(\\mathbb{F})$ is $21$, and (as it may be calculated in GAP as in \\cite{Candido_et_al}) this is also the dimension of $\\Der\\bigl({\\mathfrak{psl}}_4(\\mathbb{F})\\bigr)$. The result follows.\n\\end{proof}\n\nIn fact, in order to prove the previous result, the exact computation of the dimension of $\\Der\\bigl({\\mathfrak{psl}}_4(\\mathbb{F})\\bigr)$ is not required; we just need the bound \n\\[\n\\dim\\Der\\bigl({\\mathfrak{psl}}_4(\\mathbb{F})\\bigr)\\leq 21.\n\\] \nFor completeness, let us provide an elementary proof of this fact.\n\n\\begin{lemma}\n$\\dim\\Der\\bigl({\\mathfrak{psl}}_4(\\mathbb{F})\\bigr)\\leq 21$.\n\\end{lemma}\n\\begin{proof}\nFor $1\\leq i,j\\leq 4$, let $E_{ij}$ be the matrix in ${\\mathfrak{gl}}_4(\\mathbb{F})$ with $1$ in the $(i,j)$ entry and $0$'s elsewhere. Denote by $\\bar A$ the class of a matrix $A\\in{\\mathfrak{sl}}_4(\\mathbb{F})$ in ${\\mathfrak{psl}}_4(\\mathbb{F})$.\n\nThen, ${\\mathfrak{sl}}_4(\\mathbb{F})$ is graded by $\\mathbb{Z}^3$, with\n\\begin{gather*}\n\\degree(E_{12})=(1,0,0)=-\\degree(E_{21}),\\\\\n\\degree(E_{23})=(0,1,0)=-\\degree(E_{32}),\\\\\n\\degree(E_{34})=(0,0,1)=-\\degree(E_{43}).\n\\end{gather*}\nAs the identity matrix $I_4$ is homogeneous of degree $(0,0,0)$, this induces a grading by $\\mathbb{Z}^3$ on $\\frg={\\mathfrak{psl}}_4(\\mathbb{F})$:\n\\[\n\\Gamma:\\frg=\\bigoplus_{\\alpha\\in\\mathbb{Z}^3}\\frg_\\alpha,\n\\]\nwith support \n\\[\n\\begin{split}\n\\supp\\Gamma&:= \\{\\alpha\\in\\mathbb{Z}^3: \\frg_\\alpha\\ne 0\\}\\\\\n &=\\{(0,0,0),\\pm(1,0,0),\\pm(0,1,0),\\pm(0,0,1),\\pm(1,1,0),\\pm(0,1,1),\\pm(1,1,1)\\}.\n\\end{split}\n\\]\nLet ${\\mathfrak h}=\\espan{H_1:=\\overline{E_{11}+E_{22}},H_2:=\\overline{E_{22}+E_{33}}}=\\frg_{(0,0,0)}$ be the `diagonal' subalgebra of $\\frg$. Then, the decomposition in eigenspaces for the adjoint action of ${\\mathfrak h}$ is $\\frg={\\mathfrak h}\\oplus\\frg_1\\oplus\\frg_2\\oplus\\frg_3$, with\n\\[\n\\begin{split}\n\\frg_1&=\\espan{\\bar E_{12},\\bar E_{21},\\bar E_{34},\\bar E_{43}}\n =\\frg_{(1,0,0)}\\oplus\\frg_{(-1,0,0)}\\oplus\\frg_{(0,0,1)}\\oplus\\frg_{(0,0,-1)},\\\\\n\\frg_2&=\\espan{\\bar E_{23},\\bar E_{32},\\bar E_{14},\\bar E_{41}}\n =\\frg_{(0,1,0)}\\oplus\\frg_{(0,-1,0)}\\oplus\\frg_{(1,1,1)}\\oplus\\frg_{(-1,-1,-1)},\\\\\n\\frg_3&=\\espan{\\bar E_{13},\\bar E_{31},\\bar E_{24},\\bar E_{42}}\n =\\frg_{(1,1,0)}\\oplus\\frg_{(-1,-1,0)}\\oplus\\frg_{(0,1,1)}\\oplus\\frg_{(0,-1,-1)}.\n\\end{split}\n\\]\nAs $\\frg$ is $\\mathbb{Z}^3$-graded, so is $\\Der(\\frg)$. Several steps are required now:\n\n\\begin{romanenumerate}\n\\item $\\dim\\Der(\\frg)_{(0,0,0)}\\leq 3$, and $d({\\mathfrak h})=0$ for any $d\\in \\Der(\\frg)_{(0,0,0)}$.\n\n\\begin{proof}\nAny $d\\in \\Der(\\frg)_{(0,0,0)}$ preserves the one-dimensional spaces $\\frg_\\alpha$, for $\\alpha\\in\\supp\\Gamma\\setminus\\{(0,0,0)\\}$. Then $d$ and $\\mathrm{ad}_{\\mathfrak h}$ commute, so $d({\\mathfrak h})=0$. Also $d(\\bar E_{12})=\\lambda \\bar E_{12}$ and $d(\\bar E_{23})=\\mu \\bar E_{23}$ for some $\\lambda,\\mu\\in\\mathbb{F}$. From $d({\\mathfrak h})=0$, we obtain that $d(\\bar E_{21})=-\\lambda \\bar E_{21}$ and $d(\\bar E_{32})=-\\mu\\bar E_{32}$. Hence $\\tilde d:= d-\\mathrm{ad}_{\\mu H_2+\\lambda H_1}$ annihilates $\\bar E_{12}$ and $\\bar E_{23}$ (and $\\bar E_{21}$ and $\\bar E_{32}$). Since the elements $\\bar E_{12}$, $\\bar E_{21}$, $\\bar E_{23}$, $\\bar E_{32}$, $\\bar E_{34}$ and $\\bar E_{43}$ generate $\\frg$, it follows that $\\tilde d$ is determined by the value $\\tilde d(E_{34})$. We conclude that $\\dim\\Der(\\frg)_{(0,0,0)}-\\dim\\mathrm{ad}_{\\mathfrak h}\\leq 1$.\n\\end{proof}\n\n\\item $\\Der(\\frg)=\\mathrm{ad}_\\frg +\\{d\\in\\Der(\\frg): d({\\mathfrak h})=0\\}$.\n\n\\begin{proof}\nWe already have $\\Der(\\frg)_{(0,0,0)}\\subseteq \\{d\\in\\Der(\\frg):d({\\mathfrak h})=0\\}$, and it is clear that $d\\in\\Der(\\frg)_\\alpha$, with $\\alpha \\in \\mathbb{Z}^3 \\setminus \\supp\\Gamma$, implies $d({\\mathfrak h})\\subseteq \\frg_\\alpha=0$. On the other hand, if $d\\in \\Der(\\frg)_\\alpha$, with $\\alpha \\in\\supp\\Gamma\\setminus \\{(0,0,0)\\}$, then the restriction of $d$ to ${\\mathfrak h}$ defines a linear map $\\beta :{\\mathfrak h}\\rightarrow \\mathbb{F}$ by $d(H)=\\beta(H)\\bar E_{rs}$, where $\\bar E_{rs}$ is the basic element in the one-dimensional space $\\frg_\\alpha$. But for any $H,H'\\in{\\mathfrak h}$ we get:\n\\[\n\\begin{split}\n0=d([H,H'])&=[d(H),H']+[H,d(H')]\\\\\n &= \\beta(H)[\\bar E_{rs},H']+ \\beta(H')[H,\\bar E_{rs}]\\\\\n &=\\bigl(-\\beta(H)\\gamma(H')+\\beta(H')\\gamma(H)\\bigr)\\bar E_{rs},\n\\end{split}\n\\]\nwhere $\\gamma:{\\mathfrak h}\\rightarrow \\mathbb{F}$ is the nonzero linear form such that $[H,\\bar E_{rs}]=\\gamma(H)\\bar E_{rs}$ for any $H\\in{\\mathfrak h}$. Hence $\\beta$ is a scalar multiple of $\\gamma$, and if $\\beta=\\nu\\gamma$ with $\\nu\\in \\mathbb{F}$, then $(d-\\nu\\mathrm{ad}_{\\bar E_{rs}})({\\mathfrak h})=0$. (This argument is similar to the one in \\cite[Proposition 8.1]{EK12}.)\n\\end{proof}\n\n\\item Now note that $\\bar E_{12}$, $\\bar E_{23}$, $\\bar E_{34}$ and $\\bar E_{41}$ generate $\\frg$. Let $0\\ne d\\in \\Der(\\frg)_\\alpha$, with $\\alpha \\in \\mathbb{Z}_3 \\setminus \\{(0,0,0)\\}$, and $d({\\mathfrak h})=0$. Then $d(\\frg_i)\\subseteq \\frg_i$, for $i=1,2,3$, and\n\\[\n\\begin{split}\n&d(\\bar E_{12})\\in\\frg_1\\ \n\\text{so either $d(\\bar E_{12})=0$ or $\\alpha\\in\\{(-2,0,0),(-1,0,1),(-1,0,-1)\\}$,}\\\\\n&d(\\bar E_{23})\\in\\frg_2\\ \n\\text{so either $d(\\bar E_{23})=0$ or $\\alpha \\in\\{(0,-2,0),(1,0,1),(-1,-2,-1)\\}$,}\\\\\n&d(\\bar E_{34})\\in\\frg_1\\ \n\\text{so either $d(\\bar E_{34})=0$ or $\\alpha\\in\\{(0,0,-2),(1,0,-1),(-1,0,1)\\}$,}\\\\\n&d(\\bar E_{41})\\in\\frg_2\\ \n\\text{so either $d(\\bar E_{41})=0$ or $\\alpha\\in\\{(2,2,2),(1,2,1),(1,0,-1)\\}$.}\n\\end{split}\n\\]\nBut $\\Der(\\frg)_\\alpha=0$ for $\\alpha\\in\\{(-2,0,0),(0,-2,0),(0,0,-2),(2,2,2)\\}$, because any $d$ in one of these homogeneous spaces annihilates the generators $\\bar E_{21}$, $\\bar E_{32}$, $\\bar E_{43}$ and $\\bar E_{14}$. Hence, our homogeneous $d$ belongs to $\\Der(\\frg)_\\alpha$ with $\\alpha \\in X:=\\{\\pm(1,0,1),\\pm(1,0,-1),\\pm(1,2,1)\\}$. Now if, for instance, $d\\in\\Der(\\frg)_{(-1,0,-1)}$, then $d$ annihilates $\\bar E_{23}\\in \\frg_{(0,1,0)}$ (because $(0,1,0)+(-1,0,-1)\\not\\in\\supp\\Gamma$),  $\\bar E_{41}\\in\\frg_{(-1,-1,-1)}$, and $\\bar E_{24}\\in\\frg_{(1,1,0)}$. Moreover, since $\\bar E_{34}=[[\\bar E_{31},\\bar E_{12}],\\bar E_{24}]$, it follows that $d$ is determined by the value $d(\\bar E_{12})$. Thus, we get $\\dim\\Der(\\frg)_{(-1,0,-1)}\\leq 1$. A similar argument applies to the other possibilities, so $\\dim\\Der(\\frg)_\\alpha\\leq 1$ for any $\\alpha\\in X$.\n\n\\item Therefore, we conclude that\n\\[\n\\dim\\Der(\\frg)-\\dim\\frg=\\Bigl(\\dim\\Der(\\frg)_{(0,0,0)}-\\dim{\\mathfrak h}\\Bigr)+\\hspace{-5pt}\\sum_{\\alpha \\in X}\\hspace{-5pt}\\dim\\Der(\\frg)_\\alpha \\leq 7,\n\\]\nso that $\\dim\\Der(\\frg)\\leq \\dim\\frg+7=21$. \\qedhere\n\\end{romanenumerate}\n\\end{proof}\n\n\\smallskip\n\nOur next result extends \\cite[Corollary 4.32]{EKmon}.\n\n\\begin{theorem}\\label{th:DerC_psl4}\nLet $\\mathcal{C}$ be a Cayley algebra over a field $\\mathbb{F}$ of characteristic $2$. The Lie algebra of derivations $\\Der(\\mathcal{C})$ is isomorphic to the projective special linear Lie algebra ${\\mathfrak{psl}}_4(\\mathbb{F})$. (Independently of the isomorphism class of $\\mathcal{C}$!)\n\\end{theorem}\n\\begin{proof}\nRecall that $\\Der(\\mathcal{C})$ is a $14$-dimensional simple Lie algebra, a twisted form of $\\Der(\\mathcal{C}_s)$, which is the Chevalley algebra of type $G_2$.\n\nAny $d\\in\\Der(\\mathcal{C})$ leaves the norm $q$ invariant, annihilates the unity $1$ and preserves $\\mathcal{C}^0$, the subspace of trace zero elements. Since the characteristic is $2$, the unity $1$ is in $\\mathcal{C}^0$, so $d$ induces an element $\\tilde d$ in the symplectic Lie algebra ${\\mathfrak{sp}}\\bigl(\\mathcal{C}^0\/\\mathbb{F} 1,\\tilde b_q\\bigr)\\cong{\\mathfrak{sp}}_6(\\mathbb{F})$, where $\\tilde b_q$ is the nondegenerate alternating bilinear form on $\\mathcal{C}^0$ induced by $b_q$. (Note that $\\mathcal{C}^0=\\{x\\in\\mathcal{C}: b_q(x,1)=0\\}$, so $\\tilde b_q$ is nondegenerate.)\n\nTherefore, we have a homomorphism of Lie algebras\n\\[\n\\begin{split}\n\\Phi:\\Der(\\mathcal{C})&\\longrightarrow {\\mathfrak{sp}}\\bigl(\\mathcal{C}^0\/\\mathbb{F} 1,\\tilde b_q\\bigr),\\\\\n d\\quad &\\mapsto\\quad \\tilde d.\n\\end{split}\n\\]\nThe simplicity of $\\Der(\\mathcal{C})$ implies that $\\Phi$ is injective, and hence \n\\[\n\\Phi\\bigl(\\Der(\\mathcal{C})\\bigr)=\\Phi\\bigl(\\Der(\\mathcal{C})^{(2)}\\bigr)\\subseteq \n {\\mathfrak{sp}}\\bigl(\\mathcal{C}^0\/\\mathbb{F} 1,\\tilde b_q\\bigr)^{(2)}\\cong{\\mathfrak{sp}}_6(\\mathbb{F})^{(2)}.\n\\]\nBy dimension count, the image of $\\Phi$ is ${\\mathfrak{sp}}\\bigl(\\mathcal{C}^0\/\\mathbb{F} 1,\\tilde b_q\\bigr)^{(2)}$, which is isomorphic to ${\\mathfrak{sp}}_6(\\mathbb{F})^{(2)}$, and hence to ${\\mathfrak{psl}}_4(\\mathbb{F})$ by Lemma \\ref{le:sp62psl4}.\n\\end{proof}\n\n\\smallskip\n\nThe previous theorem shows that, in characteristic $2$, it is no longer true that two Cayley algebras are isomorphic if and only if their Lie algebras of derivations are isomorphic.\n\n\\begin{remark}\nGiven an irreducible root system of type $X_r$ and its corresponding Chevalley algebra $\\frg$ over a field $\\mathbb{F}$, the quotient $\\frg\/Z(\\frg)$ (where $Z(\\frg)$ is the center of $\\frg$) is usually called the \\emph{classical Lie algebra of type $X_r$}. In particular, in characteristic $2$, Theorem \\ref{th:DerC_psl4} implies that the classical Lie algebras of type $A_3$ and $G_2$ coincide.\n\\end{remark}\n\nWrite $\\cA :=M_6(\\mathbb{F})$, and let $\\sigma$ be the symplectic involution (attached to the standard alternating form $b_s$), such that, for any $X\\in \\cA$, $\\sigma(X)$ is the adjoint relative to $b_s$.\n\n\\begin{theorem}\\label{th:AAutDerC}\nThe affine group scheme of automorphisms of $\\Der(\\mathcal{C}_s)$ is isomorphic to the affine group scheme of automorphisms of the algebra with involution $(\\cA,\\sigma)$.\n\\end{theorem}\n\nOver an algebraic closure $\\mathbb{F}_{\\text{alg}}$, the group $\\Aut({\\mathfrak{psl}}_4(\\mathbb{F}_{\\text{alg}})$ is the adjoint Chevalley group of type $C_3$ (see \\cite{HogewejII}), and hence it is isomorphic to projective general symplectic group $\\textrm{PGSp}_6(\\mathbb{F}_{\\text{alg}})\\cong\\Aut\\bigl(\\cA,\\sigma\\bigr)$. However, Theorem \\ref{th:AAutDerC} considers arbitrary fields, and we shall give an explicit isomorphism of schemes in its proof. Note that this result over $\\mathbb{F}_{\\text{alg}}$ shows that $\\AAut({\\mathfrak{psl}}_4(\\mathbb{F}))$ is connected, and Corollary \\ref{co:pgsp6_Derpsl4} shows that it is smooth.\n\n\\begin{proof}[Proof of Theorem \\ref{th:AAutDerC}]\nAs in the proof of Theorem \\ref{th:DerC_psl4}, we may identify $\\Der(\\mathcal{C}_s)\\cong{\\mathfrak{psl}}_4(\\mathbb{F})$ with the Lie algebra $\\Skew(\\cA,\\sigma)^{(2)}$, where $\\Skew(\\cA,\\sigma):= \\{x\\in\\cA: \\sigma(x)=x\\}$ (as the characteristic is two!) Consider the morphism of affine group schemes\n\\[\n\\begin{split}\n\\varphi:\\AAut(\\cA,\\sigma)&\\longrightarrow \\AAut\\Bigl(\\Skew(\\cA,\\sigma)^{(2)}\\Bigr),\\\\\nf:\\cA_R\\rightarrow\\cA_R&\\mapsto f\\vert_{\\Skew(\\cA_R,\\sigma_R)^{(2)}},\n\\end{split}\n\\]\nwhere $R$ is a unital, commutative and associative $\\mathbb{F}$-algebra, $\\cA_R=\\cA\\otimes_\\mathbb{F} R$, and $\\sigma_R=\\sigma\\otimes\\id$ is the induced involution in $\\cA_R$, so $\\Skew(\\cA_R,\\sigma_R)^{(2)}=\\Skew(\\cA,\\sigma)^{(2)}\\otimes_\\mathbb{F} R$.\n\nThe group homomorphism on points in an algebraic closure $\\mathbb{F}_{\\text{alg}}$\n\\[\n\\varphi_{\\mathbb{F}_{\\text{alg}}}:\\Aut(\\cA_{\\mathbb{F}_{\\text{alg}}})\\rightarrow \\Aut\\bigl(\\Skew(\\cA_{\\mathbb{F}_{\\text{alg}}},\\sigma_{\\mathbb{F}_{\\text{alg}}})^{(2)}\\bigr)\n\\]\nis injective, because $\\Skew(\\cA_{\\mathbb{F}_{\\text{alg}}},\\sigma_{\\mathbb{F}_{\\text{alg}}})^{(2)}= {\\mathfrak{sp}}_6(\\mathbb{F}_{\\text{alg}})^{(2)}$ generates $\\cA_{\\mathbb{F}_{\\text{alg}}}=M_6(\\mathbb{F}_{\\text{alg}})$ as an algebra. Also, the differential $\\text{d}\\varphi$ is an isomorphism: it is the isomorphism $\\Phi$ in the proof of Corollary \\ref{co:pgsp6_Derpsl4}. Therefore, $\\varphi$ is a closed imbedding (\\cite[(22.2)]{KMRT}). But both schemes are smooth, connected, and of the same dimension, so $\\varphi$ is an isomorphism.\n\\end{proof}\n\nSince $\\AAut({\\mathfrak{psl}}_4(\\mathbb{F})$ is smooth, the set of isomorphism classes of twisted forms of ${\\mathfrak{psl}}_4(\\mathbb{F})$ is in bijection with $H^1(\\mathbb{F},\\AAut({\\mathfrak{psl}}_4(\\mathbb{F}))$ (see \\cite[Chapters 17 and 18]{Waterhouse}), and also the set of isomorphism clases of central simple associative algebras of degree $6$ endowed with a symplectic involution is in bijection with $H^1(\\mathbb{F},\\AAut(M_6(\\mathbb{F}),\\sigma)=H^1(\\mathbb{F},\\textrm{PGSp}_6(\\mathbb{F}))$. Our last result is then a direct consequence of Theorem \\ref{th:AAutDerC}.\n\n\\begin{corollary}\nLet $\\mathbb{F}$ be a field of characteristic $2$. The map that sends any central simple associative algebra of degree $6$ over $\\mathbb{F}$ endowed with a symplectic involution $(\\mathcal{B},\\tau)$ to the Lie algebra $\\Skew(\\mathcal{B},\\tau)^{(2)}$ gives a bijection between the set of isomorphism classes of such pairs $(\\mathcal{B},\\tau)$ to the set of twisted forms over $\\mathbb{F}$ of the Lie algebra ${\\mathfrak{psl}}_4(\\mathbb{F})$.\n\\end{corollary}\n\nRecall that ${\\mathfrak{psl}}_4(\\mathbb{F})$ is (isomorphic to) the Chevalley algebra of type $G_2$, so this corollary gives the twisted forms of the classical simple Lie algebras of type $G_2$ in characteristic $2$.\n\n\\bigskip\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section*{}\n\n\\begin{center}\n{\\tiny Padova, Aula Magna DEI, October 28, 2004}\\\\\n\\vspace{2cm}\n{\\bf \\Large Computational Aspects of a Numerical Model for Combustion Flow}\\\\\n\\vspace{1cm}\n{\\bf {\\large Gianluca Argentini}}\\\\\n\\vspace{0.2cm}\n\\normalsize [email protected] \\\\\n\\vspace{0.2cm}\n{\\large Advanced Computing Laboratory}\\\\\n\\vspace{0.2cm}\n{\\it Riello Group}, Legnago (Verona), Italy\n\\end{center}\n\n\\newpage\n\n\\section*{Position of the problem}\n\nDesign, development and engineering of industrial power burners have \nstrong mathematical requests:\n\\begin{itemize}\n\t\\item numerical resolution of a PDEs system involving {\\it Navier-Stokes} \n\tequations for velocity and pressure fields, {\\it energy conservation} law\n\tfor temperature field, {\\it Fick}'s law for diffusion of all the chemical\n\tspecies in the combustion chamber;\n\t\\item geometrical design of the combustion head for a correct shape\n\tand optimal efficiency of {\\it flame};\n\t\\item geometrical design of {\\it ventilation fans} and computation of \n\ta correct air inflow for optimal combustion.\n\\end{itemize}\n\n\\begin{figure}[h]\n\t\\begin{center}\n\t\\includegraphics[width=6cm]{combustionhead.eps}\n\t\\caption{Combustion head and chamber for burner.}\n\t\\end{center}\n\\end{figure}\n\n\\section*{Computational complexity analysis\\\\for~a~flow {\\small (1)}}\n\n{\\bf {\\it Simple example}} for a detailed knowledge of the velocity field\nof fluid particles in the combustion chamber:\n\n\\begin{itemize}\n\t\\item {\\bf M} is the number of flow streamlines to compute;\n\t\\item {\\bf S} is the number of geometrical points for every streamline.\n\\end{itemize}\nHigh values for {\\bf M} are important for a {\\it realistic simulation}\nof the flow, high values for {\\bf S} are important for a fine\n{\\it graphic resolution}: minimal values are of order {\\it O}($10^3-10^4$).\\\\\n\n\\noindent Suppose to use a 3D grid 10 x 10 x 1000 cm (hence {\\bf M} = 100, {\\bf S} = 1000),\na medium value {\\it $v_i$} = 50 cm\/sec for every cartesian component\nof velocity vector field, and a space resolution {\\it h} = 0.5 cm.\n\n\\newpage\n\n\\section*{Computational complexity analysis\\\\for~a~flow {\\small (2)}}\n\nFor numeric resolution of time-dependent advective PDEs, the {\\it Courant-Friedrichs-Lewy}\n({\\it CFL}) {\\it condition} gives an upper limit for the time step:\\\\\n\n$\\Delta t \\leq \\frac{ch}{v}$ \\\\\n\n\\noindent where {\\it c} is a costant, usually $\\leq 1$, depending on the\nused numeric method, and ${\\it v} = \\sup |{\\it v_j}|$.\nThe quantity $\\frac{v \\Delta t}{h}$ is called {\\it CFL number}. \nLet {\\it c} = 1; then\\\\\n\n$\\Delta t \\leq \\frac{0.5 cm}{50 \\frac{cm}{sec}} = 0.01 \\; sec$. \\\\\n\n\\noindent As consequence, for {\\bf 1} real minute of simulation the flops are\nof order {\\bf {\\it O}($10^{10}$)} and the occupation of RAM is {\\bf {\\it O}($10^0$) GB}:\\\\\n{\\it the computation is CPU expensive, RAM consuming and produces a lot of unuseful data}\n(100 snapshots of the flow every second).\n\n\\section*{A Finite Differences method and Interpolations}\nIn the effort of minimize the relevance of these problems, we have studied \na numeric model based on\n\n\\begin{itemize}\n\t\\item a Finite Differences schema with a not too restrictive CFL condition;\n\t\\item an appropriate interpolation of the numeric FD velocity-field for a finer\n\tresolution without modifying the grid step.\n\\end{itemize}\n\n\\noindent This model gives a numeric solution comparable with the solutions based\non finer grids: we present an {\\it estimate} of its goodness and a mathematical \njustification.\\\\\nThe FD method is based on {\\it Lax-Friedrichs} schema:\n\n\\begin{itemize}\n\t\\item discretization in time: ${\\partial_t u_j^n} = \\frac{1}{\\Delta t}(u_j^{n+1} - u_j^n)$,\\\\\n\twhere $u_j^n \\leftarrow \\frac{1}{3}(u_{j+1}^n + u_j^n + u_{j-1}^n)$\\\\\n\t(for a better approximation we compute the mean on three values, \n\ttwo in LF original form);\n\t\\item discretization in space: ${\\partial_x u_j^n}  = \\frac{1}{2h}(u_{j+1}^n - u_{j-1}^n)$;\n\\end{itemize} \n\n\\noindent where {\\it u} is a velocity component, {\\it n} the time step, {\\it j} a value\non the cartesian coordinate {\\it x}.\n\n\\newpage\n\n\\section*{Computational aspects of Lax-Friedrichs schema}\nFor this schema the CFL condition has costant {\\it c} = 1; \n{\\it the Finite Elements method with the same schema for discretization in time\nhas a more restrictive costant c} $< 1$.\\\\\n\n\\noindent If {\\it K} $\\in \\mathbb{R^+}$, {\\it K} $\\leq \\frac{1}{2}$, we can\ndefine the norm ~$\\parallel${\\it u}$\\parallel$ = {\\it K} $sup_j |{\\it u_j}|$;\nthen the modified LF schema is {\\it strongly stable}:\n$\\parallel$${\\it u^n}$$\\parallel$ $\\leq$ $\\parallel$${\\it u^{n-1}}$$\\parallel$ $\\forall {\\it n} \\in \\mathbb{N}$;\nhence there is not the {\\it blowing up} of the numeric solution.\n\n\\noindent Suppose we want to compute at most 10 snapshots for every second;\nthen, in the hypothesis {\\it v} = 50 cm\/sec as the previous example, from\\\\\n\n$v \\Delta t \\leq h$\\\\\n\n\\noindent we must use as minimum a grid step {\\it h} = 5 cm.\\\\\n\\noindent This case gives {\\bf S} = 200, the total flops for 1 minute\nof simulation is now of order {\\it O}($10^8$) and the occupation of RAM\nis of order {\\it O}($10^{-2}$) GB.\\\\\nThe gain is of order {\\it O}($10^2$).\\\\\nThe grid step {\\it h} = 5 cm is too big for a good resolution of\nstreamlines for flows into the combustion head: for better final results,\nit can be useful a method based on {\\it interpolations} of the computed LF values.\n\n\\section*{Interpolation of trajectories {\\small (1)}}\nEvery streamline of LF solution is divided into {\\bf N} couples of points,\\\\\n\\{$(P_1,P_2),(P_2,P_3),...,(P_{{\\bf N}-1},P_{\\bf N})$\\}, so that {\\bf S} = {\\bf N}+1.\\\\\nWe use for every couple a cubic polynomial ({\\it spline}) imposing\nthe following four analytical conditions ({\\bf v} is the LF solution):\n\n\\begin{itemize}\n\t\\item passage at ${\\it P_k}$ point, $1 \\leq {\\it k} \\leq {\\bf N}-1$;\n\t\\item passage at ${\\it P_{k+1}}$ point;\n\t\\item the first derivative at ${\\it P_k}$ is equal to ${\\bf v_{\\it k}}$;\n\t\\item the first derivative at ${\\it P_{k+1}}$ is equal to ${\\bf v_{\\it {k+1}}}$.\n\\end{itemize}\n\n\\noindent In this way we can construct a set of class ${\\it C}^1$ \nnew trajectories; we want to estimate\n\n\\begin{enumerate}\n\t\\item the overload for finding and valuating all the cubics;\n\t\\item the difference compared to the real LF solution of the smaller grid step.\n\\end{enumerate}\n\n\\newpage\n\n\\section*{Interpolation of trajectories {\\small (2)}}\nFor simplicity, consider a single component of a cubic:\\\\\n$s(t) = at^3 + bt^2 + ct + d$, where $0 \\leq t \\leq 1$;\\\\\nif {\\bf T} is the $4\\times4$ matrix\\\\\n\n${\\bf T} = \\left( \\begin{array}{cccc}\n2  & -2 & 1  & 1 \\\\\n-3 & 3  & -2 & -1 \\\\\n0  & 0  & 1  & 0 \\\\\n1  & 0  & 0  & 0 \n\\end{array} \\right)$\\\\\n\n\\noindent and $(p_1, p_2, v_1, v_2)$ is the vector of cartesian coordinates and\nvelocities components of points $P_1$ and $P_2$, we have\\\\\n\n$(a, b, c, d) = {\\bf T} (p_1, p_2, v_1, v_2)$.\n\n\\section*{Interpolation of trajectories {\\small (3)}}\nWe define the $4{\\bf M}\\times4{\\bf M}$ {\\it global matrix}\\\\\n\n${\\bf G} = \\left( \\begin{array}{ccccc}\n{\\bf T}  & {\\bf 0} & . & . &  {\\bf 0} \\\\\n{\\bf 0}  & {\\bf T} & . & . &  {\\bf 0} \\\\\n. & . & . & . & . \\\\\n. & . & . & . & . \\\\\n{\\bf 0}  & {\\bf 0} & . & . &  {\\bf T}\n\\end{array} \\right)$\\\\\n\n\\noindent where {\\bf 0} is the $4\\times4$ zero-matrix. Then\n\n\\begin{itemize}\n\t\\item {\\bf G} is a {\\it sparse} matrix with density number $\\leq \\frac{1}{\\bf M}$;\n\t\\item if ${\\bf p} = (p_{(1,1)}, p_{(1,2)}, . . ., v_{({\\bf M},1)}, v_{({\\bf M},2)})$,\n\twe can compute the cubics, between two points, for all the {\\bf M} trajectories \n\tby the product {\\bf G}{\\bf p}.\n\\end{itemize}\n\n\\newpage\n\n\\section*{Interpolation of trajectories {\\small (4)}}\nThe theoric number of flops for computing the coefficients of all the splines is\nof order ${\\it O}(10 {\\bf M}^2{\\bf N})$. If {\\bf M} = $10^4$ and {\\bf N} = $10^3$,\nthe total number of flops is ${\\it O}(10^{12})$.\\\\\n\n\\noindent With a single processor having a clock frequency of {\\it O}(1) {\\bf GHz},\nthe total time can require some hundreds of seconds, a performance not very good\nfor practical purposes; using\n\n\\begin{itemize}\n\t\\item some mathematical libraries as {\\it LAPACK} routines with {\\bf Fortran}\n\tcalls or {\\bf Matlab} environment,\n\t\\item distributed computation on a multinode cluster,\n\\end{itemize}\n\n\\noindent we have reached a computation time of some tens of seconds.\\\\\n\n\\noindent {\\it Example}: {\\bf Matlab} has internal Lapack level 3 {\\bf BLAS} routines\nfor fast matrix-matrix multiplication and treatment of sparse matrices.\n\n\\section*{Interpolation of trajectories {\\small (5)}}\n\n\\begin{figure}[h]\n\t\\begin{center}\n\t\\includegraphics[width=6.5cm]{matrix-vector-multiplyBW.eps}\n\t\\end{center}\n\\end{figure}\n\n\\noindent Performances for a single {\\bf Gp} multiplication using an Intel\nXeon 3.2 GHz with 1 MB internal {\\it cache}: for {\\bf M}=$10^4$ the memory occupied\nby the sparse version of {\\bf G} is only {\\it O}($10^2$) KB instead of theoric\n{\\it O}($10^6$): {\\bf G} can be stored in processor cache.\n\n\\newpage\n\n\\section*{Computation of splines values {\\small (1)}}\nNow we need a fast method for computing the splines values in a set of\n{\\it parameter~ticks} with fine sampling.\\\\\nLet ${\\it r} \\in \\mathbb{N}^+$ the number of ticks for each cubic: then the values of the\nparameter {\\it t} in these ticks are $(0, \\frac{1}{r}, \\frac{2}{r}, . . ., \\frac{r-1}{r}, 1)$;\nthe value of a cubic at ${\\it t_0}$ is a {\\it scalar product}:\\\\\n\n${at_0^3 + bt_0^2 + ct + d = (a, b, c, d)\\cdot(t_0^3, t_0^2, t_0, 1)}$.\\\\\n\n\\noindent Consider the constant $4\\times{\\it (r+1)}$ {\\bf R} matrix\nand the (${\\bf M}\\times4$) {\\bf C} matrix:\\\\\n\n${\\bf R} = \\left( \\begin{array}{cccccc}\n0 & (\\frac{1}{r})^3 & . & . & (\\frac{r-1}{r})^3 & 1 \\\\\n0 & (\\frac{1}{r})^2 & . & . & (\\frac{r-1}{r})^2 & 1 \\\\\n0 & (\\frac{1}{r})^1 & . & . & (\\frac{r-1}{r})^1 & 1\\\\\n1 & 1 & . & . & 1 & 1\n\\end{array} \\right)$\\\\\n\n\n${\\bf C} = \\left( \\begin{array}{cccc}\na_1 & b_1 & c_1 & d_1 \\\\\na_2 & b_2 & c_2 & d_2 \\\\\n. & . & . & . \\\\\n. & . & . & . \\\\\na_{\\bf M} & b_{\\bf M} & c_{\\bf M} & d_{\\bf M}\n\\end{array} \\right)$\\\\\n\n\\section*{Computation of splines values {\\small (2)}}\nThen the ${\\bf M}\\times{\\it (r+1)}$ matrix {\\bf C}{\\bf R} contains for each row the\nvalues of a cubic between two points, for all the trajectories ({\\it eulerian}\nmethod: computation of all the position and velocity at a fixed instant). The\nflops for one multiplication are of order {\\it O}(10{\\bf M}{\\it r}).\\\\\n\n\\noindent Tests with Xeon 3.2 GHz processor, {\\bf M} = $10^4$, {\\it r} = 10 and\n{\\it GNU} {\\bf Fortran77} show a time of {\\it 0.01 seconds} for a multiplication.\\\\\nWith {\\bf N} = $10^2$, the time for computing the values of all the splines\nof a single time step is {\\it 4.5} seconds (theoric for 3D: $0.01\\times10^2\\times3 = {\\it 3}$ secs).\\\\\n\n\\noindent If {\\bf p} is the number of available processors and {\\it mod}({\\bf M}, {\\bf p}) = 0,\nthe computation can be parallelized distributing $\\frac{\\bf M}{\\bf p}$ rows\nof matrix {\\bf C} to each processor: there is no need of communication among\nprocesses.\\\\\nA version of {\\bf High Performance Fortran} on a SMP system with 4 ItaniumII\nprocessors shows a quasilinear speedup for {\\bf M}, {\\bf N} of order {\\it O}($10^3$).\n\n\\newpage\n\n\\section*{Time for computation}\n\n\\begin{figure}[h]\n\t\\begin{center}\n\t\\includegraphics[width=7cm]{timeExecBW.eps}\n\t\\end{center}\n\\end{figure}\n\n\\noindent These are the total time of computation for the two methods in the case of a cilinder of length\n{\\bf L} = 1m, a flow with a max. speed {\\it v} = 10cm\/sec, {\\bf M} = $10^4$, {\\it r} = 10 and 1 minute\nof real simulation. The space grid is {\\it h} = $\\frac{{\\bf L}}{10{\\bf N}}$.\n\n\\section*{Estimate of LF+interpolations vs normal LF {\\small (1)}}\n{\\it But what is the difference between the modified LF solution and\nnormal LF solution?}\\\\\nConsider the one-dimensional case.\nLet {\\bf u}=$(u_k)$ the solution of normal LF schema with grid step {\\it h} and\ninitial value {$\\bf u_0$}; {\\bf w}=$(w_m)$ the solution of normal LF schema \nwith grid step {\\it $s\\times h$}, $s \\in \\mathbb{N}^+$, and initial value {$\\bf w_0$} $\\subset$ {$\\bf u_0$};\n{\\bf v}=$(v_n)$ the solution of modified LF schema obtained by interpolation\nof {\\bf w} and valuation on {\\it s} points per cubic; for a cubic, let ${v_k}$,\n$k \\leq s$, the value of {\\bf v} at {\\it t}=$\\frac{k}{s}$ and ${u_k}$ the\nvalue of {\\it u} at the corresponding node of the finer grid; $\\frac{v \\Delta t}{h}$ the\nCFL number and {\\it N} the {\\it N}-th time step. Let\n$${\\it M_0} = \\max_{|m-n|=1}|u_{0,m}-u_{0,n}|$$\nThen it is possible to prove this result:\n\\newtheorem{teorema}{Theorem}\n\\begin{teorema}\nIf ${\\it M_0} >$ {\\normalsize 0}, there are two positive constants {\\it A} and {\\it B} such that\\\\\n$$|v_n - u_n| \\leq (A + Bs)M_0\\sum_{i=0}^{\\it N}(\\frac{v \\Delta t}{2h})^i\\qquad \\forall n\\in\\{grid\\;indexes\\}, \\forall N\\in\\mathbb{N}.$$\n\\end{teorema}\n\n\\newpage\n\n\\section*{Estimate of LF+interpolations vs normal LF {\\small (2)}}\nThe CFL number $\\frac{v \\Delta t}{h}$ is usually indicated by $\\lambda v$.\nFrom the previous theorem it follows:\n\\newtheorem{corollario}{Corollary}\n\\begin{corollario}\nIf $\\lambda v < 2$, then\\\\\n$$|v_n - u_n| < \\frac{2(A + Bs)M_0}{2 - \\lambda v}.$$\n\\end{corollario}\n\\noindent The CFL condition satisfies the hypothesis of the corollary.\\\\\nHence, for a realistic solution from the LF+interpolations model,\nthe conditions are:\n\n\\begin{itemize}\n\t\\item a small $(\\ll 2)$ CFL number,\n\t\\item a not too big number {\\it s} of valuations for the cubics; note that \n\t{\\it s} has the inverse logical meaning of the previous {\\it r} parameter.\n\\end{itemize}\n\n\\noindent Note that if ${\\it M_0}$ is very big, as in the case of very caotic\nflows, the LF+interpolations solution can be not very realistic.\n\n\\section*{Estimate of LF+interpolations vs normal LF {\\small (3)}}\nTesting the estimate: {\\it example} for one-dimensional non linear Navier-Stokes equation,\n$\\lambda v$ = 1, {\\it s} = 10, after {\\it N} = $10^5$ time steps; graphic of the error between\nLF+interpolations and normal LF solutions.\n\n\\begin{figure}[h]\n\t\\begin{center}\n\t\\includegraphics[width=6.0cm]{errorBW.eps}\n\t\\end{center}\n\\end{figure}\n\n\\noindent In this case it can be shown that {\\it A}~= 8, {\\it B} = 2 is a \nfirst, not optimized, approximation for the two constants.\nThe picture shows that the estimate is correct but large.\n\n\\newpage\n\n\\section*{Conclusions}\nThe numeric LF schema can be modified using the interpolations\nmethod so that:\n\n\\begin{itemize}\n\t\\item the time spent on computation is much lower than the time of the LF based\non the corresponding finer grid;\n\t\\item the computation can be parallelized on multiprocessors environment with\n\tvery reduced need of communication;\n\t\\item the error on normal LF solution can be estimated and depends on the \n\tinitial value ${\\bf u_0}$ of the problem;\n\t\\item the estimate is compatible with CFL condition.\n\\end{itemize}\n\n$\\;$ \\\\\n$\\;$ \\\\\n\\noindent {\\textbf{\\textit{Thanks}}\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section*{Introduction}\\label{s:intro}\n\nCombinatorial structures are often equipped with operations which allow to\ncombine two given structures of a given type into a third and vice versa. \nThis leads to the construction of algebraic structures, particularly that of graded\nHopf algebras. When the former are formalized through\nthe notion of species, which keeps track of the underlying ground set\nof the combinatorial structure,\nit is possible to construct finer algebraic structures than the latter.\nThis leads to Hopf monoids in the category of species.\nThe basic theory of these objects is laid out in~\\cite[Part~II]{AguMah:2010},\nalong with the discussion of several examples.\n Section~\\ref{s:hopf} reviews basic material concerning species and Hopf monoids.\n\nFree monoids are the subject of Section~\\ref{s:freemonoid}. Just as the tensor algebra\nof a vector space carries a canonical structure of Hopf algebra, the free monoid on\na positive species carries one of Hopf monoid. In fact, this structure admits a one\nparameter deformation, meaningful even when the parameter $q$ is set to zero. \nThe deformation only concerns the comonoid structure;\nthe monoid structure stays fixed throughout. \nA rigidity result (Theorem~\\ref{t:0free}) applies when $q=0$ and makes\nthis case of particular importance. It states that a connected $0$-Hopf monoid is\nnecessarily free as a monoid. This is a version of a result for Hopf algebras\nof Loday and Ronco~\\cite[Theorem~2.6]{LodRon:2006}.\n\nSection~\\ref{s:freeness} contains our two main results;\nthey concern freeness under Hadamard products.\nThe Hadamard product is a basic operation on species which reflects into the\nfamiliar Hadamard product of the dimension sequences. While there is also a\nversion of this operation for graded (co)algebras, the case of species is distinguished\nby the fact that the Hadamard product of two Hopf monoids is another Hopf monoid\n(Proposition~\\ref{p:hadamard}). In fact, the Hadamard product \nof a $p$-Hopf monoid $\\thh$ and a $q$-Hopf monoid $\\mathbf{k}$ is a $pq$-Hopf monoid $\\thh\\times\\mathbf{k}$. \nCombining this result with rigidity for  connected $0$-Hopf monoids\nwe obtain our first main result (Theorem~\\ref{t:freeness}).\nIt states that if $\\thh$ is connected and $\\mathbf{k}$ is free as a monoid,\nthen $\\thh\\times\\mathbf{k}$ is free as a monoid.\nA number of freeness results in the literature (for certain Hopf monoids as well as Hopf algebras)\nare consequences of this fact.\nThe second main result (Theorem~\\ref{t:had-free-monoid})\nprovides an explicit basis for the Hadamard\nproduct when both factors are free monoids.\nTo this end, we introduce an operation on species which intertwines\nwith the Hadamard product via the free monoid functor.\n\nThe previous results entail enumerative implications on the dimension sequence of a Hopf monoid.\nThese are explored in Section~\\ref{s:genfun}. They can be conveniently\nformulated in terms of the Boolean transform of a sequence (or power series),\nsince the type generating function of a positive species $\\tp$ is the Boolean transform of that of the free monoid on $\\tp$.\nWe deduce that the Boolean transform of the\ndimension sequence of a connected Hopf monoid\nis nonnegative (Theorem~\\ref{t:ordi-hopf}).\nThis turns out to be stronger than several previously known conditions on the\ndimension sequence of a connected Hopf monoid. \nWe provide examples of sequences with nonnegative Boolean transform which\ndo not arise as the dimension sequence of any connected Hopf monoid,\nshowing that the converse of Theorem~\\ref{t:ordi-hopf} does not hold (Proposition~\\ref{p:ordi-hopf}).\n\nAppendix~\\ref{s:boolean} contains additional information on Boolean transforms;\nin particular, Proposition~\\ref{p:bool-had} provides\nan explicit formula for the Boolean transform of the Hadamard product\nof two sequences (in terms of the transforms of the factors). This implies that\nthe set of real sequences with nonnegative Boolean transform is closed under Hadamard products.\n\n\\section{Species and Hopf monoids}\\label{s:hopf}\n\nWe briefly review Joyal's notion of species~\\cite{BerLabLer:1998,Joy:1981}\nand of Hopf monoid in the category of species. For more details on the latter,\nsee~\\cite{AguMah:2010}, particularly Chapters 1, 8 and 9.\n\n\\subsection{Species and the Cauchy product}\\label{ss:cauchy}\n\nLet $\\mathsf{set^{\\times}}$ denote the category whose objects are finite sets and \nwhose morphisms are bijections.\nLet $\\Bbbk$ be a field and let $\\Vect$ denote the category whose objects are vector spaces over $\\Bbbk$\nand whose morphisms are linear maps.\n\nA \\emph{(vector) species} is a functor\n\\[\n\\mathsf{set^{\\times}} \\longrightarrow \\Vect.\n\\]\nGiven a species $\\tp$, its value on a finite set $I$ is denoted by $\\tp[I]$.\nA morphism between species $\\tp$ and $\\mathbf{q}$\nis a natural transformation between the functors $\\tp$ and $\\mathbf{q}$.\nLet $\\mathsf{Sp}$ denote the category of species.\n\nGiven a set $I$ and subsets $S$ and $T$ of $I$, the notation $I=S\\sqcup T$\nindicates that\n\\[\nI=S\\cup T \\quad\\text{and}\\quad S\\cap T = \\emptyset.\n\\]\nWe say in this case that the ordered pair $(S,T)$ is a \\emph{decomposition} of $I$.\n\nGiven species $\\tp$ and $\\mathbf{q}$, their \\emph{Cauchy product} is the species \n$\\tp \\bm\\cdot \\mathbf{q}$ defined on a finite set $I$ by\n\\begin{equation}\\label{e:cau} \n(\\tp \\bm\\cdot \\mathbf{q})[I] \n := \\bigoplus_{I = S \\sqcup T}  \\tp[S] \\otimes \\mathbf{q}[T].\n \\end{equation}\n The direct sum is over all decompositions $(S,T)$ of $I$,\n or equivalently over all subsets $S$ of $I$.\n On a bijection $\\sigma: I\\to J$, \n $(\\tp \\bm\\cdot \\mathbf{q})[\\sigma]$ is defined to be the direct sum of the maps\n \\[\n\\tp[S] \\otimes \\mathbf{q}[T]\\map{\\tp[\\sigma|_S]\\otimes\\tp[\\sigma|_T]}\n\\tp[\\sigma(S)] \\otimes \\mathbf{q}[\\sigma(T)]\n\\]\nover all decompositions $(S,T)$ of $I$, where $\\sigma|_S$ denotes the restriction of $\\sigma$ to $S$.\n\nThe operation~\\eqref{e:cau} turns $\\mathsf{Sp}$ into a monoidal category.\nThe unit object is the species $\\mathbf{1}$ defined by\n\\[\n\\mathbf{1}[I]  := \\begin{cases}\n\\Bbbk & \\text{if $I$ is empty,} \\\\\n0 & \\text{otherwise.}\n\\end{cases}\n\\]\n\nLet $q \\in \\Bbbk$ be a fixed scalar, possibly zero.\nConsider the natural transformation\n\\[\n\\beta_q \\colon  \\tp \\bm\\cdot \\mathbf{q} \\to \\mathbf{q} \\bm\\cdot \\tp\n\\]\nwhich on a finite set $I$ is the direct sum of the maps\n\\begin{equation*\n  \\tp[S] \\otimes \\mathbf{q}[T] \\to \\mathbf{q}[T] \\otimes \\tp[S],\n\\qquad\nx \\otimes y \\mapsto q^{\\abs{S}\\abs{T}}   y \\otimes x\n\\end{equation*}\nover all decompositions $(S,T)$ of $I$. The notation $\\abs{S}$ stands\nfor the cardinality of the set $S$.\n\nIf $q$ is nonzero, then $\\beta_q$ is a (strong) braiding\nfor the monoidal category $(\\mathsf{Sp},\\bm\\cdot)$.\nIn this case, the inverse braiding is $\\beta_{q^{-1}}$,\nand $\\beta_q$ is a symmetry if and only if $q=\\pm 1$.\nThe natural transformation $\\beta_0$ is a lax braiding\nfor $(\\mathsf{Sp},\\bm\\cdot)$.\n\n\\subsection{Hopf monoids in species}\\label{ss:hopf}\n\nWe consider monoids and comonoids in the monoidal category $(\\mathsf{Sp},\\bm\\cdot)$\nand bimonoids and Hopf monoids in the braided monoidal category $(\\mathsf{Sp},\\bm\\cdot,\\beta_q)$.\nWe refer to the latter as $q$-\\emph{bimonoids} and $q$-\\emph{Hopf monoids}.\nWhen $q=1$, we speak simply of \\emph{bimonoids} and \\emph{Hopf monoids}.\n\nThe structure of a monoid $\\tp$ consists of morphisms of species $\\mu:\\tp\\bm\\cdot\\tp\\to\\tp$\nand $\\iota:\\mathbf{1}\\to\\tp$ subject to the familiar associative and unital axioms.\nIn view of~\\eqref{e:cau}, the product $\\mu$ consists of a collection of linear maps\n\\[\n\\mu_{S,T} : \\tp[S]\\otimes\\tp[T] \\to \\tp[I],\n\\]\none for each finite set $I$ and each decomposition $(S,T)$ of $I$.\nThe unit $\\iota$ reduces to a linear map\n\\[\n\\iota_\\emptyset: \\Bbbk \\to \\tp[\\emptyset].\n\\]\nSimilarly, the structure of a comonoid $\\mathbf{q}$ consists of linear maps\n\\[\n\\Delta_{S,T}: \\mathbf{q}[I]\\to\\mathbf{q}[S]\\otimes\\mathbf{q}[T]\n\\quad\\text{and}\\quad\n\\epsilon_\\emptyset: \\mathbf{q}[\\emptyset] \\to \\Bbbk.\n\\]\n\nLet $I=S\\sqcup T=S'\\sqcup T'$ be two decompositions of a finite set.\nThe compatibility axiom for $q$-Hopf monoids\nstates that the diagram \n\\begin{equation}\\label{e:compr}\n\\begin{gathered}\n\\xymatrix@R+2pc@C-5pt{\n\\thh[A] \\otimes \\thh[B] \\otimes \\thh[C] \\otimes \\thh[D] \\ar[rr]^{\\mathrm{id}\n\\otimes \\beta_q \\otimes \\mathrm{id}} & &\n\\thh[A] \\otimes \\thh[C] \\otimes \\thh[B] \\otimes \\thh[D] \\ar[d]^{\\mu_{A,C}\n\\otimes \\mu_{B,D}}\\\\\n\\thh[S] \\otimes \\thh[T] \\ar[r]_-{\\mu_{S,T}}\\ar[u]^{\\Delta_{A,B} \\otimes\n\\Delta_{C,D}} & \\thh[I] \\ar[r]_-{\\Delta_{S',T'}} & \\thh[S'] \\otimes\n\\thh[T']\n}\n\\end{gathered}\n\\end{equation}\ncommutes, where $A=S\\cap S'$, $B=S\\cap T'$, $C=T\\cap S'$, $D=T\\cap T'$.\nFor more details, see~\\cite[Sections~8.2 and~8.3]{AguMah:2010}.\n\n\\subsection{Connected species and Hopf monoids}\\label{ss:con-hopf}\n\nA species $\\tp$ is \\emph{connected} if $\\dim_{\\Bbbk} \\tp[\\emptyset]=1$.\nIn a connected monoid, the map $\\iota_\\emptyset$ is an isomorphism $\\Bbbk\\cong\\tp[\\emptyset]$, and the resulting maps\n\\[\n\\tp[I] \\cong \\tp[I]\\otimes\\tp[\\emptyset] \\map{\\mu_{I,\\emptyset}} \\tp[I]\n\\quad\\text{and}\\quad\n\\tp[I] \\cong \\tp[\\emptyset]\\otimes\\tp[I] \\map{\\mu_{\\emptyset,I}} \\tp[I]\n\\]\nare identities. Thus, to provide a monoid structure on a connected species\nit suffices to specify the maps $\\mu_{S,T}$ when $S$ and $T$ are nonempty.\nA similar remark applies to connected comonoids.\n\nChoosing $S=S'$ and $T=T'$ in~\\eqref{e:compr} one obtains that\nfor a connected $q$-bimonoid $\\thh$\nthe composite\n\\[\n\\thh[S]\\otimes\\thh[T] \\map{\\mu_{S,T}} \\thh[I] \\map{\\Delta_{S,T}} \\thh[S]\\otimes\\thh[T]\n\\]\nis the identity.\n\nA connected $q$-bimonoid is automatically a $q$-Hopf monoid; see~\\cite[Sections~8.4 and~9.1]{AguMah:2010}. The \\emph{antipode} of a Hopf monoid will not concern us in this paper.\n\n\n\n\\subsection{The Hopf monoid of linear orders}\\label{ss:linear}\n\nThe $q$-Hopf monoid $\\wL_q$ is defined as follows. The vector space  $\\wL_q[I]$\nhas for basis the set of linear orders on the finite set $I$. The product and coproduct\nare defined by \\emph{concatenation} and \\emph{restriction}, respectively:\n\\begin{align*}\n\\mu_{S,T} : \\wL_q[S] \\otimes \\wL_q[T] & \\to \\wL_q[I] & \\Delta_{S,T}: \\wL_q[I] & \\to \\wL_q[S] \\otimes \\wL_q[T] \\\\\nl_1 \\otimes l_2 & \\mapsto l_1 \\cdot l_2 & l & \\mapsto q^{\\area_{S,T}(l)}\\, l|_S \\otimes l|_T.\n\\end{align*}Here $l_1 \\cdot l_2$ is the linear order on $I$\nwhose restrictions to $S$ and $T$ are $l_1$ and $l_2$\nand in which the elements of $S$ precede the elements of $T$,\nand $l|_S$ is the restriction \nof the linear order $l$ on $I$\nto the subset $S$. \nThe \\emph{Schubert cocycle} is\n\\begin{equation}\\label{e:schubert-linear}\n\\area_{S,T}(l) := \\abs{\\{(i,j)\\in S\\times T \\mid \\text{$i>j$\naccording to $l$}\\}}.\n\\end{equation}\n\nWe write $\\wL$ instead of $\\wL_1$.\nNote that the monoid structure of $\\wL_q$ is independent of $q$. \nThus, $\\wL=\\wL_q$ as monoids. The comonoid $\\wL$ is cocommutative,\nbut, for $q\\neq 1$, $\\wL_q$ is not.\n\n\n\\section{The free monoid on a positive species}\\label{s:freemonoid}\n\nWe review the explicit construction of the free monoid on a positive species,\nfollowing~\\cite[Section~11.2]{AguMah:2010}. The free monoid carries\na canonical structure of $q$-Hopf monoid. The case $q=0$ is of particular\ninterest for our purposes, in view of the fact that any connected $0$-Hopf monoid is free\n(Theorem~\\ref{t:0free} below).\n\n\\subsection{Set compositions}\\label{ss:compositions}\n\nA \\emph{composition} of a finite set $I$ is an ordered sequence $F=(I_1,\\dots,I_k)$\nof disjoint nonempty subsets of $I$ such that\n\\[\nI  = \\bigcup_{i=1}^k I_i.\n\\]\nThe subsets $I_i$ are the \\emph{blocks}\nof $F$. \nWe write $F\\vDash I$ to indicate that $F$ is a composition of~$I$.\n\nThere is only one composition of the empty set\n(with no blocks).\n\nGiven $I=S\\sqcup T$ and compositions $F=(S_1,\\dots,S_j)$ of $S$\nand $G=(T_1,\\dots,T_k)$ of $T$, their \\emph{concatenation}\n\\[\nF\\cdot G := (S_1,\\dots,S_j,T_1,\\dots,T_k)\n\\]\nis a composition of $I$.\n\nGiven $S\\subseteq I$ and a composition $F=(I_1,\\dots,I_k)$ of $I$,\nwe say that $S$ is \\emph{$F$-admissible} if for each $i=1,\\ldots,k$, either\n\\[\nI_i\\subseteq S \\quad\\text{or}\\quad I_i\\cap S=\\emptyset.\n\\]\nIn this case, we let $i_1<\\cdots<i_j$ be the subsequence of $1<\\cdots<k$\nconsisting of those indices $i$ for which $I_i\\subseteq S$, and\ndefine the \\emph{restriction} of $F$ to $S$ by\n\\[\nF|_S = (I_{i_1},\\ldots,I_{i_j}).\n\\]\nIt is a composition of $S$.\n\n\nGiven $I=S\\sqcup T$ and a composition $F=(I_1,\\dots,I_k)$ of $I$,\nlet\n\\begin{equation}\\label{e:schubert-comp}\n\\area_{S,T}(F)  := \\abs{\\{(i,j)\\in S\\times T \\mid \\text{$i$ appears in a strictly later block of $F$ than $j$}\\}}.\n\\end{equation}\nAlternatively,\n\\[\n\\area_{S,T}(F)  = \\sum_{1 \\leq i < j \\leq k} \\abs{I_i \\cap T}\\,\\abs{I_j \\cap S}.\n\\]\n\nStill in the preceding situation, note that $S$ is $F$-admissible if and only if $T$ is.\nThus $F|_S$ and $F|_T$ are defined simultaneously.\n\nIf the blocks of $F\\vDash I$ are singletons, then $F$ amounts to a linear order on $I$.\nConcatenation and restriction of set compositions reduce in this case to the\ncorresponding operations for linear orders (Section~\\ref{ss:linear}). In addition,\n\\eqref{e:schubert-comp} reduces to~\\eqref{e:schubert-linear}.\n\nThe set of compositions of $I$ is a partial order under \\emph{refinement}: we set\n$F\\leq G$ if each block of $F$ is obtained by merging a number of adjacent blocks of $G$. The composition $(I)$ is the unique minimum element, and linear orders are the maximal elements.\n\nSet compositions of $I$ are in bijection with flags of subsets of $I$ via\n\\[\n(I_1,\\ldots,I_k) \\mapsto (\\emptyset\\subset I_1\\subset I_1\\cup I_2\\subset \\cdots \\subset\nI_1\\cup \\cdots\\cup I_k = I).\n\\]\nRefinement of compositions corresponds to inclusion of flags. In this manner the poset of set compositions is\na lower set of the Boolean poset $2^{2^I}$, and hence a meet-semilattice.\nThe meet operation and concatenation interact as follows:\n\\begin{equation}\\label{e:meet-conc-set}\n(F\\cdot F')\\wedge(G\\cdot G') = (F\\wedge G)\\cdot(F'\\wedge G'),\n\\end{equation}\nwhere $F,G\\vDash S$ and $F',G'\\vDash T$, $I=S\\sqcup T$.\n\n\\begin{remark}\nSet compositions of $I$ are in bijection with faces of the\n\\emph{braid arrangement} in $\\mathbb{R}^I$. Refinement of compositions corresponds to inclusion of faces, meet to\nintersection, linear orders to chambers, and $(I)$ to the central face. \nWhen $S$ and $T$ are nonempty, the statistic $\\area_{S,T}(F)$ counts the number of hyperplanes that separate the face $(S,T)$ from $F$. For more details, see~\\cite[Chapter~10]{AguMah:2010}.\n\\end{remark}\n\n\n\\subsection{The free monoid}\\label{ss:free}\n\nA species $\\mathbf{q}$ is \\emph{positive} if $\\mathbf{q}[\\emptyset]=0$.\n\nGiven a positive species $\\mathbf{q}$ and a composition $F=(I_1,\\ldots,I_k)$ of $I$, write\n\\begin{equation}\\label{e:sp-comp}\n\\mathbf{q}(F)  := \\mathbf{q}[I_1] \\otimes \\dots \\otimes \\mathbf{q}[I_k].\n\\end{equation}\nWe define a new species $\\mathcal{T}(\\mathbf{q})$ by\n\\[\n\\mathcal{T}(\\mathbf{q})[I]  := \\bigoplus_{F\\vDash I} \\mathbf{q}(F).\n\\]\nA bijection $\\sigma:I\\to J$ transports a composition $F=(I_1,\\dots,I_k)$ of $I$\ninto a composition $\\sigma(F):=\\bigl(\\sigma(I_1),\\dots,\\sigma(I_k)\\bigr)$ of $J$.\nThe map $\\mathcal{T}(\\mathbf{q})[\\sigma]:\\mathcal{T}(\\mathbf{q})[I]\\to\\mathcal{T}(\\mathbf{q})[J]$ is the direct sum of the maps\n\\[\n\\mathbf{q}(F) = \\mathbf{q}[I_1] \\otimes \\dots \\otimes \\mathbf{q}[I_k] \\map{\\mathbf{q}[\\sigma|_{I_1}] \\otimes \\dots \\otimes \\mathbf{q}[\\sigma|_{I_k}]} \n\\mathbf{q}[\\sigma(I_1)] \\otimes \\dots \\otimes \\mathbf{q}[\\sigma(I_k)] = \\mathbf{q}\\bigl(\\sigma(F)\\bigr).\n\\]\n\nWhen $F$ is the unique composition of $\\emptyset$, we have $\\mathbf{q}(F)=\\Bbbk$.\nThus, the species $\\mathcal{T}(\\mathbf{q})$ is connected.\n\nEvery nonempty $I$ admits a unique composition with one block; namely, $F=(I)$.\nIn this case, $\\mathbf{q}(F)=\\mathbf{q}[I]$. This yields an embedding\n$\n\\mathbf{q}[I] \\hookrightarrow \\mathcal{T}(\\mathbf{q})[I]\n$\nand thus an embedding of species \n\\[\n\\eta_\\mathbf{q}: \\mathbf{q} \\hookrightarrow \\mathcal{T}(\\mathbf{q}).\n\\]\nOn the empty set, $\\eta_\\mathbf{q}$ is (necessarily) zero.\n\nGiven $I=S\\sqcup T$ and compositions $F\\vDash S$ and\n$G\\vDash T$, we have a canonical isomorphism\n\\[\n \\mathbf{q}(F)\\otimes \\mathbf{q}(G)\\cong \\mathbf{q}(F\\cdot G)\n\\]\nobtained by concatenating the factors in~\\eqref{e:sp-comp}.\nThe sum of these over all $F\\vDash S$ and\n$G\\vDash T$ yields a map\n\\[\n\\mu_{S,T}: \\mathcal{T}(\\mathbf{q})[S]\\otimes\\mathcal{T}(\\mathbf{q})[T] \\to \\mathcal{T}(\\mathbf{q})[I].\n\\]\nThis turns $\\mathcal{T}(\\mathbf{q})$ into a monoid. In fact, $\\mathcal{T}(\\mathbf{q})$ is the \\emph{free}\nmonoid on the positive species $\\mathbf{q}$, in view of the following result (a slight reformulation of~\\cite[Theorem 11.4]{AguMah:2010}).\n\n\\begin{theorem}\\label{t:dmunivp}\nLet $\\tp$ be a monoid, $\\mathbf{q}$ a positive species, and\n$\\zeta\\colon \\mathbf{q} \\to \\tp$ a morphism of species.\nThen there exists a unique morphism of monoids\n$\\hat{\\zeta}\\colon \\mathcal{T}(\\mathbf{q}) \\to \\tp$ such that\n\\begin{equation*\n\\begin{gathered}\n\\xymatrix@C+20pt{\n\\mathcal{T}(\\mathbf{q}) \\ar@{.>}[r]^-{\\hat{\\zeta}} &  \\tp\\\\\n \\mathbf{q} \\ar[ru]_{\\zeta}\\ar[u]^{\\eta_\\mathbf{q}}\n}\n\\end{gathered}\n\\end{equation*}\ncommutes.\n\\end{theorem}\n\nThe map $\\hat{\\zeta}$ is as follows. On the empty set, it is the unit map of $\\tp$:\n\\[\n\\mathcal{T}(\\mathbf{q})[\\emptyset] = \\Bbbk \\map{\\iota_\\emptyset} \\tp[\\emptyset].\n\\]\nOn a nonempty set $I$, it is the sum of the maps\n\\[\n\\mathbf{q}(F) = \\mathbf{q}[I_1]\\otimes\\cdots\\otimes\\mathbf{q}[I_k] \\map{\\zeta_{I_1}\\otimes\\cdots\\otimes\\zeta_{I_k}}\n\\tp[I_1]\\otimes\\cdots\\otimes\\tp[I_k]\n\\map{\\mu_{I_1,\\ldots,I_k}} \\tp[I],\n\\]\nwhere $\\mu_{I_1,\\ldots,I_k}$ denotes an iteration of the product of $\\tp$\n(well-defined by associativity).\n\n\\smallskip \n\nWhen there is given an isomorphism of monoids $\\tp\\cong\\mathcal{T}(\\mathbf{q})$, we say that\nthe positive species $\\mathbf{q}$ is a \\emph{basis} of the (free) monoid $\\tp$.\n\n\\begin{remark}\nThe free monoid $\\mathcal{T}(\\mathbf{q})$ on an arbitrary species $\\mathbf{q}$ exists~\\cite[Example~B.29]{AguMah:2010}. One has that $\\mathcal{T}(\\mathbf{q})[\\emptyset]$ is the free associative unital algebra\non the vector space $\\mathbf{q}[\\emptyset]$. Thus, $\\mathcal{T}(\\mathbf{q})$ is connected if and only if\n$\\mathbf{q}$ is positive. We only consider this case in this paper.\n\\end{remark}\n\n\\subsection{The free monoid as a Hopf monoid}\\label{ss:freeHopf}\n\nLet $q\\in\\Bbbk$ and $\\mathbf{q}$ a positive species. The species $\\mathcal{T}(\\mathbf{q})$ admits a\ncanonical $q$-Hopf monoid structure, which we denote by $\\Tc_q(\\mathbf{q})$, as follows.\n\nAs monoids, $\\Tc_q(\\mathbf{q})=\\mathcal{T}(\\mathbf{q})$. In particular, $\\Tc_q(\\mathbf{q})$ and $\\mathcal{T}(\\mathbf{q})$ are the same\nspecies. The comonoid structure depends on $q$. Given $I=S\\sqcup T$,\nthe coproduct\n\\[\n\\Delta_{S,T}: \\Tc_q(\\mathbf{q})[I]\\to\\Tc_q(\\mathbf{q})[S]\\otimes\\Tc_q(\\mathbf{q})[T]\n\\]\nis the sum of the maps\n\\begin{align*}\n\\mathbf{q}(F) & \\to \\mathbf{q}(F|_S)\\otimes \\mathbf{q}(F|_T) \\\\\nx_1\\otimes\\cdots\\otimes x_k & \\mapsto \n\\begin{cases} \nq^{\\area_{S,T}(F)} (x_{i_1}\\otimes\\cdots\\otimes x_{i_j})\\otimes (x_{i'_1}\\otimes\\cdots\\otimes x_{i'_k}) & \\text{ if $S$ is $F$-admissible,}\\\\\n0 & \\text{ otherwise.}\n\\end{cases}\n\\end{align*}\nHere $F=(I_1,\\ldots,I_k)$ and $x_i\\in \\mathbf{q}[I_i]$ for each $i$.\nIn the admissible case, we have written $F|_S=(I_{i_1},\\ldots,I_{i_j})$ and\n$F|_T=(I_{i'_1},\\ldots,I_{i'_k})$.\n\nThe preceding turns $\\Tc_q(\\mathbf{q})$ into a $q$-bimonoid. Since it is connected,\nit is a $q$-Hopf monoid. \n\n\\subsection{Freeness of the Hopf monoid of linear orders}\\label{ss:free-linear}\n\nLet $\\mathbf{X}$ be the species defined by\n\\[\n\\mathbf{X}[I]  := \\begin{cases}\n\\Bbbk & \\text{if $I$ is a singleton,} \\\\\n0 & \\text{otherwise.}\n\\end{cases}\n\\]\nIt is positive. Note that\n\\begin{equation}\\label{e:XF}\n\\mathbf{X}(F) \\cong \n\\begin{cases}\n\\Bbbk & \\text{ if all blocks of $F$ are singletons,} \\\\\n 0       & \\text{ otherwise.}\n\\end{cases}\n\\end{equation}\nSince a set composition of $I$ into singletons amounts to a linear order on $I$, we have \n$\\mathcal{T}(\\mathbf{X})[I]\\cong \\wL[I]$ for all finite sets $I$.\nThis gives rise to a canonical isomorphism of species \n\\[\n\\mathcal{T}(\\mathbf{X}) \\cong \\wL.\n\\]\nMoreover, the closing remarks in Section~\\ref{ss:compositions}\nimply that this is an isomorphism of $q$-Hopf monoids\n\\[\n\\Tc_q(\\mathbf{X}) \\cong \\wL_q.\n\\]\nIn particular, $\\wL$ is the free monoid on the species $\\mathbf{X}$.\n\n\n\\subsection{Loday-Ronco freeness for $0$-Hopf monoids}\\label{ss:0free}\n\nThe $0$-Hopf monoid \n$\\mathcal{T}_0(\\mathbf{q})$ has the same underlying species and the same product\nas the Hopf monoid $\\mathcal{T}(\\mathbf{q})$ (Section~\\ref{ss:free}).\nWe now discuss the coproduct, by setting $q=0$ in the description\nof Section~\\ref{ss:freeHopf}.\nFix a decomposition $I=S\\sqcup T$.\nThe compositions $F\\vDash I$ that contribute to $\\Delta_{S,T}$\nare those for which\n$S$ is $F$-admissible and in addition $\\area_{S,T}(F)=0$. This happens if and only if\n\\[\nF = F|_S \\cdot F|_T.\n\\]\nWhen $S,T\\neq\\emptyset$, the preceding is in turn equivalent to\n\\begin{equation}\\label{e:0free}\n(S,T)\\leq F.\n\\end{equation}\nTherefore, the coproduct $\\Delta_{S,T}$ of $\\mathcal{T}_0(\\mathbf{q})$\nis the direct sum over all $F\\vDash I$ of the above form of the maps \n\\begin{align*}\n\\mathbf{q}(F) & \\to \\mathbf{q}(F|_S)\\otimes \\mathbf{q}(F|_T) \\\\\nx_1\\otimes\\cdots\\otimes x_k & \\mapsto  \n (x_{1}\\otimes\\cdots\\otimes x_{j})\\otimes (x_{j+1}\\otimes\\cdots\\otimes x_{k}).\n \\end{align*}\nHere $F=(I_1,\\ldots,I_k)$, $S=I_1\\cup\\cdots\\cup I_j$, and $T=I_{j+1}\\cup\\cdots\\cup I_k$.\n\n\n\\begin{theorem}\\label{t:0free}\nLet $\\thh$ be a connected $0$-Hopf monoid. Then there exist a positive species\n$\\mathbf{q}$ and an isomorphism of $0$-Hopf monoids\n\\[\n\\thh \\cong \\mathcal{T}_0(\\mathbf{q}).\n\\]  \n\\end{theorem}\n\nThe species $\\mathbf{q}$ can be obtained as the \\emph{primitive part} of $\\thh$.\n\nThere is a parallel result for connected graded $0$-Hopf algebras which is\ndue to Loday and Ronco~\\cite[Theorem~2.6]{LodRon:2006}.\nAn adaptation of their proof yields the result for connected $0$-Hopf monoids;\nthe complete details are given in~\\cite[Theorem~11.49]{AguMah:2010}.\n\n\\begin{remark}\nTheorem~\\ref{t:0free} states that any connected $0$-Hopf monoid is free as a\nmonoid. It is also true that it is \\emph{cofree} as a comonoid; in addition, if $\\mathbf{q}$ is finite-dimensional,\nthen the $0$-Hopf monoid $\\mathcal{T}_0(\\mathbf{q})$ is \\emph{self-dual}. See~\\cite[Section~11.10.3]{AguMah:2010} for more details.\n\\end{remark}\n\n\n\n\n\\section{Freeness under Hadamard products}\\label{s:freeness}\n\nThe Hadamard product of two Hopf monoids is another Hopf monoid. \nWe review this construction and\nwe prove in Theorem~\\ref{t:freeness} \nthat if one of the Hopf monoids is free as a monoid, then\nthe Hadamard product is also free as a monoid \n(provided the other Hopf monoid is connected).\nWe introduce an operation on positive species which\nallows us to describe a basis for the Hadamard\nproduct of two free monoids in terms of bases of the factors (Theorem~\\ref{t:had-free-monoid}).\n\n\\subsection{The Hadamard product of Hopf monoids}\\label{ss:hadamard}\n\nThe \\emph{Hadamard product} of two species $\\tp$ and $\\mathbf{q}$ is the species\n$\\tp\\times\\mathbf{q}$ defined on a finite set $I$ by\n\\[\n(\\tp\\times\\mathbf{q})[I] := \\tp[I]\\otimes\\mathbf{q}[I],\n\\]\nand on bijections similarly. \n\nIf $\\tp$ and $\\mathbf{q}$ are connected, then so is $\\tp\\times\\mathbf{q}$.\n\n\\begin{proposition}\\label{p:hadamard}\nLet $p,q\\in\\Bbbk$ be arbitrary scalars.\nIf $\\thh$ is a $p$-bimonoid and $\\mathbf{k}$ is a $q$-bimonoid,\nthen $\\thh\\times\\mathbf{k}$ is a $pq$-bimonoid.\n\\end{proposition}\nThe proof is given in~\\cite[Corollary~9.6]{AguMah:2010}.\nThe corresponding statement for Hopf monoids holds as well.\n\nThe product of $\\thh\\times\\mathbf{k}$ is defined by\n\\[\n\\xymatrix@C+11pt{\n(\\thh\\times\\mathbf{k})[S]\\otimes(\\thh\\times\\mathbf{k})[T] \\ar@{.>}[rr]^-{\\mu_{S,T}} \\ar@{=}[d] & & \n(\\thh\\times\\mathbf{k})[I] \\ar@{=}[d]\\\\\n(\\thh[S]\\otimes\\mathbf{k}[S])\\otimes(\\thh[T]\\otimes\\mathbf{k}[T]) \\ar[r]_-{\\cong} & \n(\\thh[S]\\otimes\\thh[T])\\otimes(\\mathbf{k}[S]\\otimes\\mathbf{k}[T]) \\ar[r]_-{\\mu_{S,T}\\otimes\\mu_{S,T}} &\n\\thh[I] \\otimes\\mathbf{k}[I] \n}\n\\]\nwhere the first map on the bottom simply switches the middle tensor factors.\nThe coproduct is defined similarly.\n\nIn particular, if $\\thh$ and $\\mathbf{k}$ are bimonoids ($p=q=1$), then so is $\\thh\\times\\mathbf{k}$.\n\n\\begin{remark}\nThere is a parallel between the notions of species on the one hand,\nand of graded vector spaces on the other. This extends to a parallel between\nHopf monoids in species and graded Hopf algebras. These topics are\nstudied in detail in~\\cite[Part III]{AguMah:2010}.\n\nThe Hadamard product of graded vector spaces can be defined, but does not enjoy\nthe same formal properties of that for species. In particular, the Hadamard product\nof two graded bialgebras carries natural algebra and coalgebra structures, but these\nare not compatible in general; see~\\cite[Remark~8.65]{AguMah:2010}.\nFor this reason, our main result (Theorem~\\ref{t:freeness} below) does not\npossess an analogue for graded bialgebras.\n\\end{remark}\n\n\\subsection{Freeness under Hadamard products}\\label{ss:freeness}\n\nThe following is our main result. Let $p$ and $q\\in\\Bbbk$ be arbitrary scalars.\n\n\\begin{theorem}\\label{t:freeness}\nLet $\\thh$ be a connected $p$-Hopf monoid. \nLet $\\mathbf{k}$ be a $q$-Hopf monoid that is free as a monoid.\nThen $\\thh\\times\\mathbf{k}$ is a connected $pq$-Hopf monoid that is free as a monoid.\n\\end{theorem}\n\\begin{proof}\nSince $\\thh$ and $\\mathbf{k}$ are connected (the latter by freeness), so is $\\thh\\times\\mathbf{k}$.\nWe then know from Proposition~\\ref{p:hadamard} that $\\thh\\times\\mathbf{k}$ is a connected $pq$-Hopf monoid. Now, as monoids, we have\n\\[\n\\mathbf{k}\\cong \\mathcal{T}_q(\\mathbf{q}) = \\mathcal{T}_0(\\mathbf{q})\n\\]\nfor some positive species $\\mathbf{q}$. Hence, as monoids,\n\\[\n\\thh\\times\\mathbf{k} \\cong \\thh \\times \\mathcal{T}_0(\\mathbf{q}).\n\\]\nBut the latter is a $0$-Hopf monoid by Proposition~\\ref{p:hadamard}, and hence\nfree as a monoid by Theorem~\\ref{t:0free}.\n\\end{proof}\n\n\\begin{corollary}\\label{c:freeness}\nLet $\\thh$ be a connected $p$-Hopf monoid. Then $\\thh\\times\\wL_q$ is free as a monoid.\n\\end{corollary}\n\\begin{proof}\nThis is a special case of Theorem~\\ref{t:freeness}, since as discussed in Section~\\ref{ss:free-linear}, $\\wL_q\\cong\\mathcal{T}_q(\\mathbf{X})$.\n\\end{proof}\n\n\n\\begin{example}\\label{eg:LL}\nThe Hopf monoid $\\tLL_q$ of \\emph{pairs of linear orders} is studied in~\\cite[Section~12.3]{AguMah:2010}. There is an isomorphism of $q$-Hopf monoids\n\\[\n \\tLL_q \\cong \\wL^{*} \\times \\wL_q.\n \\]\nCorollary~\\ref{c:freeness} implies that $\\tLL_q$ is free as a monoid.\nThis result was obtained by different means in~\\cite[Section~12.3]{AguMah:2010}.\nIt implies the fact that the Hopf algebra of permutations of Malvenuto\nand Reutenauer~\\cite{MalReu:1995} is free as an algebra, a result known from~\\cite{PoiReu:1995}. See Section~\\ref{ss:livernet} below for more comments regarding \nconnections between Hopf monoids and Hopf algebras.\n\\end{example}\n\n\\begin{example}\\label{eg:super}\nThe Hopf monoid $\\SC(\\mathrm{U})$ of \\emph{superclass functions} \non unitriangular matrices with entries in $\\mathbb{F}_2$ is studied in~\\cite{ABT:2011}.\nThere is an isomorphism of Hopf monoids\n\\[\n\\SC(\\mathrm{U}) \\cong \\tPi\\times\\wL,\n\\]\nwhere $\\tPi$ is the Hopf monoid of \\emph{set partitions} of~\\cite[Section~12.6]{AguMah:2010}. It follows that $\\SC(\\mathrm{U})$ is free as a monoid. This result was obtained\nby different means in~\\cite[Proposition~17]{ABT:2011}. It implies the fact that\nthe Hopf algebra of \\emph{symmetric functions in noncommuting variables}~\\cite{RosSag:2006} is free\nas an algebra, a result known from~\\cite{Wol:1936}.\n\\end{example}\n\n\\subsection{Livernet freeness for certain Hopf algebras}\\label{ss:livernet}\n\n\nIt is possible to associate a number of graded Hopf algebras to\na given Hopf monoid $\\thh$. This is the subject of~\\cite[Chapter~15]{AguMah:2010}.\nIn particular, there are two graded Hopf algebras $\\mathcal{K}(\\thh)$\nand $\\overline{\\Kc}(\\thh)$ related by a canonical surjective morphism\n\\[\n\\mathcal{K}(\\thh) \\twoheadrightarrow \\overline{\\Kc}(\\thh).\n\\]\nMoreover, for any Hopf monoid $\\thh$, there is a canonical isomorphism of graded Hopf algebras~\\cite[Theorem~15.13]{AguMah:2010}\n\\[\n\\overline{\\Kc}(\\wL\\times\\thh) \\cong \\mathcal{K}(\\thh).\n\\]\n\nThe functor $\\overline{\\Kc}$ preserves a number of properties, including freeness:\nif $\\thh$ is free as a monoid, then $\\overline{\\Kc}(\\thh)$ is free as an algebra~\\cite[Proposition~18.7]{AguMah:2010}.\n\nCombining these remarks with Corollary~\\ref{c:freeness} we deduce that for any\nconnected Hopf monoid $\\thh$, the algebra $\\mathcal{K}(\\thh)$ is free. This result is\ndue to Livernet~\\cite[Theorem~4.2.2]{Liv:2010}. A proof similar to the one above\nis given in~\\cite[Section~16.1.7]{AguMah:2010}.\n\nAs an example, for $\\thh=\\tLL$ we obtain that the Hopf algebra of pairs of permutations\nis free as an algebra, a result known from~\\cite[Theorem 7.5.4]{AguMah:2006}.\n\n\n\n\\subsection{The Hadamard product of free monoids}\\label{ss:had-free}\n\nGiven positive species $\\tp$ and $\\mathbf{q}$,\ndefine a new positive species $\\tp \\star \\mathbf{q}$ by\n\\begin{equation}\\label{e:star}\n(\\tp \\star \\mathbf{q})[I] :=\n\\bigoplus_{\\substack{F,G\\vDash I\\\\F\\wedge G=(I)}} \\tp(F)\\otimes\\mathbf{q}(G).\n\\end{equation}\nThe sum is over all pairs $(F,G)$ of compositions of $I$\nsuch that $F\\wedge G=(I)$. We are employing notation~\\eqref{e:sp-comp}.\n\n\\begin{lemma}\\label{l:star-prop}\nFor any composition $H\\vDash I$, there is a canonical isomorphism of vector spaces\n\\begin{equation}\\label{e:star-prop}\n(\\tp \\star \\mathbf{q})(H) \\cong \\bigoplus_{\\substack{F,G\\vDash I\\\\F\\wedge G=H}}\\tp(F)\\otimes\\mathbf{q}(G)\n\\end{equation}\ngiven by rearrangement of the tensor factors.\n\\end{lemma}\n\\begin{proof}\nLet us say that a function $f$ on set compositions with values on vector spaces is\n\\emph{multiplicative} if $f(H_1\\cdot H_2) \\cong f(H_1)\\otimes f(H_2)$ for all\n$H_1\\vDash I_1$, $H_2\\vDash I_2$, $I=I_1\\sqcup I_2$. Such functions are uniquely determined\nby their values on the compositions of the form $(I)$. The isomorphism~\\eqref{e:star-prop} holds when $H=(I)$ by definition~\\eqref{e:star}. It thus suffices to check that both\nsides are multiplicative.\n\nThe left hand side of~\\eqref{e:star-prop} is multiplicative in view of~\\eqref{e:sp-comp}. \n\nIf, for $i=1,2$, $F_i,G_i\\vDash I_i$ are such that $F_i\\wedge G_i=H_i$, then\n\\[\n(F_1\\cdot F_2)\\wedge(G_1\\cdot G_2) = H_1\\cdot H_2\n\\]\nby~\\eqref{e:meet-conc-set}. Moreover, if $F,G\\vDash I_1\\sqcup I_2$ are such that \n$F\\wedge G=H_1\\cdot H_2$, then $F=F_1\\cdot F_2$ and $G=G_1\\cdot G_2$ for\nunique $F_i,G_i$ as above.\nThis implies the multiplicativity of the right hand side.\n\\end{proof}\n\nWe show that the operation~\\eqref{e:star} is associative.\n\n\\begin{proposition}\\label{p:star-asso}\nFor any positive species $\\tp$, $\\mathbf{q}$ and $\\mathbf{r}$,\nthere is a canonical isomorphism\n\\begin{equation}\n(\\tp \\star \\mathbf{q}) \\star \\mathbf{r} \\cong \\tp \\star (\\mathbf{q} \\star \\mathbf{r}).\n\\end{equation}\n\\end{proposition}\n\\begin{proof} Define\n\\[\n(\\tp \\star \\mathbf{q} \\star \\mathbf{r})[I] := \\bigoplus_{\\substack{F,G,H\\vDash I,\\\\F\\wedge G\\wedge H=(I)}} \\tp(F)\\otimes\\mathbf{q}(G)\\otimes\\mathbf{r}(H).\n\\]\nWe make use of the isomorphism~\\eqref{e:star-prop} to build the following.\n\\begin{align*}\n\\bigl(\\tp \\star (\\mathbf{q} \\star \\mathbf{r})\\bigr)[I] & =\n\\bigoplus_{\\substack{F,K\\vDash I\\\\ F\\wedge K=(I)}} \\tp(F)\\otimes (\\mathbf{q}\\star \\mathbf{r})(K)\\\\\n& \\cong \\bigoplus_{\\substack{F,K\\vDash I,\\\\F\\wedge K=(I)}}\\bigoplus_{\\substack{G,H\\vDash I,\\\\G\\wedge H=K}} \\tp(F)\\otimes\\mathbf{q}(G)\\otimes\\mathbf{r}(H)\\\\\n& = \\bigoplus_{\\substack{F,G,H\\vDash I,\\\\F\\wedge G\\wedge H=(I)}} \\tp(F)\\otimes\\mathbf{q}(G)\\otimes\\mathbf{r}(H) = (\\tp \\star \\mathbf{q} \\star \\mathbf{r})[I]\n\\end{align*}\nThe space $\\bigl((\\tp \\star \\mathbf{q}) \\star \\mathbf{r}\\bigr)[I]$ can be identified with\n$(\\tp \\star \\mathbf{q} \\star \\mathbf{r})[I]$ in a similar manner.\n\\end{proof}\n\nThere is also an evident natural isomorphism\n\\begin{equation}\n\\tp \\star \\mathbf{q} \\cong \\mathbf{q} \\star \\tp.\n\\end{equation}\nThus, $\\star$ defines a (nonunital) symmetric monoidal structure on the category of positive species.\n\nOur present interest in the operation $\\star$ stems from the following result, which\nprovides an explicit description for the basis of a Hadamard product of two free monoids\nin terms of bases of the factors.\n\n\\begin{theorem}\\label{t:had-free-monoid}\nFor any positive species $\\tp$ and $\\mathbf{q}$,\nthere is a natural isomorphism of monoids\n\\begin{equation}\\label{e:had-free-monoid}\n\\mathcal{T}(\\tp \\star \\mathbf{q}) \\cong \\mathcal{T}(\\tp)\\times\\mathcal{T}(\\mathbf{q}).\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nWe calculate using~\\eqref{e:star-prop}.\n\\begin{multline*}\n\\mathcal{T}(\\tp \\star \\mathbf{q})[I] =\n\\bigoplus_{H\\vDash I} (\\tp \\star \\mathbf{q})(H)  \\cong\n\\bigoplus_{H\\vDash I} \\bigoplus_{\\substack{F,G\\vDash I\\\\F\\wedge G=H}} \\tp(F)\\otimes\\mathbf{q}(G)\\\\\n= \\bigoplus_{F,G\\vDash I} \\tp(F)\\otimes\\mathbf{q}(G)\n= \\mathcal{T}(\\tp)[I]\\otimes\\mathcal{T}(\\mathbf{q})[I] =\n\\bigl(\\mathcal{T}(\\tp)\\times\\mathcal{T}(\\mathbf{q})\\bigr)[I].\n\\end{multline*}\nThe fact that this isomorphism preserves products follows from~\\eqref{e:meet-conc-set}. \n\\end{proof}\n\n\\begin{example}\\label{eg:basisLL}\nSince $\\mathbf{X}$ is a basis for $\\wL$,\n\\eqref{e:had-free-monoid} implies that $\\mathbf{X}\\star\\mathbf{X}$ is a basis for $\\wL\\times\\wL$.\nFrom~\\eqref{e:XF} we obtain that\n\\[\n\\{(C,D) \\mid C\\wedge D = (I)\\}.\n\\]\nis a linear basis for $(\\mathbf{X}\\star\\mathbf{X})[I]$. (The linear orders $C$ and $D$ are viewed as set compositions into singletons.)\nFor related results, see~\\cite[Section 12.3.6]{AguMah:2010}.\n\\end{example}\n\nRecall that, for each scalar $q\\in\\Bbbk$,\nany free monoid $\\mathcal{T}(\\tp)$ is endowed with a canonical\ncomonoid structure and the resulting $q$-Hopf monoid is denoted $\\Tc_q(\\tp)$ \n(Section~\\ref{ss:freeHopf}).\nIt turns out that, when $q=0$,~\\eqref{e:had-free-monoid} is in fact an isomorphism\nof $0$-Hopf monoids, as we now prove. The proof below also shows that~\\eqref{e:had-free-monoid} is not an isomorphism of comonoids for $q\\neq 0$.\n\n\\begin{proposition}\\label{p:had-free-monoid}\nThe map~\\eqref{e:had-free-monoid}\nis an isomorphism of $0$-Hopf monoids\n\\[\n\\mathcal{T}_0(\\tp \\star \\mathbf{q}) \\cong \\mathcal{T}_0(\\tp)\\times\\mathcal{T}_0(\\mathbf{q}).\n\\]\n\\end{proposition}\n\\begin{proof}\nIn order to prove that coproducts are preserved it  suffices to check that they\nagree on the basis $\\tp\\star\\mathbf{q}$ of $\\mathcal{T}(\\tp \\star \\mathbf{q})$ and on its image in $\\mathcal{T}(\\tp)\\times\\mathcal{T}(\\mathbf{q})$. The image of $(\\tp \\star \\mathbf{q})[I]$ is the direct sum of the spaces $\\tp(F)\\otimes\\mathbf{q}(G)$ over those $F,G\\vDash I$ such that $F\\wedge G =(I)$.\nChoose $S,T\\neq\\emptyset$ such that $I=S\\sqcup T$. \nIn view of the definition of the coproduct on a free monoid (Section~\\ref{ss:freeHopf}),\nthe coproduct $\\Delta_{S,T}$\nof $\\mathcal{T}_q(\\tp \\star \\mathbf{q})$ is zero on $(\\tp \\star \\mathbf{q})[I]$. (This holds for any $q\\in\\Bbbk$.)\nOn the other hand, from~\\eqref{e:0free} we have that the coproduct of\n $\\mathcal{T}_0(\\tp)\\times\\mathcal{T}_0(\\mathbf{q})$ on $\\tp(F)\\otimes\\mathbf{q}(G)$ is also zero, unless both\n \\[\n (S,T)\\leq F \\quad\\text{and}\\quad (S,T)\\leq G.\n \\]\n Since this is forbidden by the assumption $F\\wedge G =(I)$, the coproducts agree.\n \\end{proof}\n\n\n\n\n\\section{The dimension sequence of a connected Hopf monoid}\n\\label{s:genfun}\n\nWe now derive a somewhat surprising application of Theorem~\\ref{t:freeness}. \nIt states that the reciprocal of the ordinary generating function of a connected Hopf monoid\nhas nonpositive (integer) coefficients (Theorem~\\ref{t:ordi-hopf} below).\nWe compare this result with other previously known conditions satisfied by\nthe dimension sequence of a connected Hopf monoid.\n\n\\subsection{Coinvariants}\\label{ss:coinvariants}\n\nLet $G$ be a group and $V$ a $\\Bbbk G$-module. The space of \\emph{coinvariants} $V_{G}$ is\nthe quotient of $V$ by the $\\Bbbk$-subspace spanned by the elements of the form\n\\[\nv-g\\cdot v\n\\]\nfor $v\\in V$, $g\\in G$. If $V$ is a free $\\Bbbk G$-module, then \n\\[\n\\dim_{\\Bbbk} V_G = \\rank_{\\Bbbk G} V.\n\\]\n\nLet $V$ and $W$ be $\\Bbbk G$-modules. Let $U_1$ be the space $V\\otimes W$\nwith \\emph{diagonal} $G$-action:\n\\[\ng\\cdot (v\\otimes w) := (g\\cdot v)\\otimes(g\\cdot w).\n\\]\nLet $U_2$ be the same space but with the following $G$-action:\n\\[\ng\\cdot (v\\otimes w) := v\\otimes (g\\cdot w).\n\\]\nThe following is a well-known basic fact.\n\n\\begin{lemma}\\label{l:diagonal}\nIf $W$ is free as a $\\Bbbk G$-module, then $U_1\\cong U_2$. In particular, \n\\[\n\\dim_{\\Bbbk} (U_1)_G = (\\dim_{\\Bbbk} V)(\\dim_{\\Bbbk} W_G).\n\\]\n\\end{lemma}\n\\begin{proof}\nWe may assume $W=\\Bbbk G$. In this case, the map\n\\[\nU_1\\to U_2, \\quad\nv\\otimes g \\mapsto (g^{-1}\\cdot v) \\otimes g\n\\]\nis an isomorphism of $\\Bbbk G$-modules. The second assertion follows because $U_2$\nis a free module of rank equal to $(\\dim_{\\Bbbk} V)(\\rank_{\\Bbbk G} W)$.\n\\end{proof}\n\n\n\\subsection{The type generating function}\\label{ss:type}\n\nLet $\\tp$ be a species. We write $\\tp[n]$ for the space $\\tp[\\{1,\\ldots,n\\}]$.\nThe symmetric group $\\Sr_n$ acts on $\\tp[n]$ by\n\\[\n\\sigma\\cdot x := \\tp[\\sigma](x)\n\\]\nfor $\\sigma\\in\\Sr_n$, $x\\in\\tp[n]$. For example,\n\\[\n\\wL[n] \\cong \\Bbbk\\Sr_n\n\\]\nas $\\Bbbk\\Sr_n$-modules.\n\nFrom now on, we assume that all species $\\tp$ are \\emph{finite-dimensional}.\nThis means that for each $n\\geq 0$ the space $\\tp[n]$ is finite-dimensional. \nThe \\emph{type generating function} of $\\tp$ is the power series\n\\[\n\\type{\\tp}{x} : = \\sum_{n\\geq 0} \\dim_{\\Bbbk} \\tp[n]_{S_n}\\, x^n.\n\\]\n\nFor example,\n\\[\n\\type{\\wL}{x} = \\sum_{n\\geq 0} x^n=\\frac{1}{1-x}.\n\\]\nMore generally, for any positive species $\\mathbf{q}$,\n\\begin{equation}\\label{e:type-free}\n\\type{\\mathcal{T}(\\mathbf{q})}{x} = \\frac{1}{1-\\type{\\mathbf{q}}{x}}.\n\\end{equation}\nThis follows by a direct calculation or from~\\cite[Theorem~1.4.2.b]{BerLabLer:1998}.\n\nLet $\\tp$ be a free monoid. It follows from~\\eqref{e:type-free} that\n\\begin{equation}\\label{e:type-free2}\n1-\\frac{1}{\\type{\\tp}{x}} \\in \\mathbb{N}{[\\![} x{]\\!]}.\n\\end{equation}\nIn other words, the reciprocal of the type generating function of a free\nmonoid has nonpositive integer coefficients (except for the first, which is $1$).\n\n\n\\subsection{Generating functions for Hadamard products}\\label{ss:gen-had}\n\nThe type generating function of a Hadamard product $\\tp\\times\\mathbf{q}$ is in general not\ndetermined by those of the factors. (It is however determined by their \\emph{cycle indices}; see~\\cite[Proposition~2.1.7.b]{BerLabLer:1998}.) \n\nThe \\emph{ordinary generating function} of a species $\\tp$ is\n\\[\n\\ordi{\\tp}{x} : = \\sum_{n\\geq 0} \\dim_{\\Bbbk} \\tp[n]\\, x^n.\n\\]\n\nThe Hadamard product of power series is defined by\n\\[\n\\bigl(\\sum_{n\\geq 0} a_n x^n\\bigr)\\times\\bigl(\\sum_{n\\geq 0} b_n x^n\\bigr) :=\n\\sum_{n\\geq 0} a_nb_n x^n.\n\\]\n\n\\begin{proposition}\\label{p:free-had}\nLet $\\tp$ be an arbitrary species and $\\mathbf{q}$ a species for which $\\mathbf{q}[n]$ is a free $\\Bbbk\\Sr_n$-module for every $n\\geq 0$. Then\n\\begin{equation}\\label{e:free-had}\n\\type{\\tp\\times \\mathbf{q}}{x} = \\ordi{\\tp}{x}\\times \\type{\\mathbf{q}}{x}. \n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nIn view of Lemma~\\ref{l:diagonal}, we have\n\\[\n\\dim_{\\Bbbk} \\bigl((\\tp\\times\\mathbf{q})[n]\\bigr)_{\\Sr_n} = (\\dim_{\\Bbbk} \\tp[n])(\\dim_{\\Bbbk} \\mathbf{q}[n]_{\\Sr_n})\n\\]\nfrom which the result follows.\n\\end{proof}\n\n\nSince $\\type{\\wL}{x}$ is the unit for the Hadamard product of power series, we have from~\\eqref{e:free-had} that\n\\begin{equation}\\label{e:type-hadamard}\n\\type{\\tp\\times \\wL}{x} = \\ordi{\\tp}{x}.\n\\end{equation}\nMore generally, for any positive species $\\mathbf{q}$,\n\\begin{equation}\\label{e:type-hadamard2}\n\\type{\\tp\\times \\mathcal{T}(\\mathbf{q})}{x} = \\ordi{\\tp}{x}\\times  \\frac{1}{1-\\type{\\mathbf{q}}{x}}.\n\\end{equation}\nThis follows from~\\eqref{e:type-free} and~\\eqref{e:free-had};\nthe $\\Bbbk\\Sr_n$-module $\\mathcal{T}(\\mathbf{q})[n]$ is free by~\\cite[Lemma~B.18]{AguMah:2010}.\n\n\\subsection{The ordinary generating function of a connected Hopf monoid}\\label{ss:ordi-hopf}\n\n Let $\\thh$ be a connected $q$-Hopf monoid. \n By Corollary~\\ref{c:freeness}, $\\thh\\times\\wL$ is a free monoid. \n Let $\\mathbf{q}$ be a basis. Thus, $\\mathbf{q}$ is a positive species such that\n\\[\n\\thh\\times\\wL\\cong \\mathcal{T}(\\mathbf{q})\n\\]\nas monoids. \n\n \\begin{proposition}\\label{p:ordi-type}\n In the above situation,\n \\begin{equation}\\label{e:ordi-type}\n \\ordi{\\thh}{x} = \\frac{1}{1-\\type{\\mathbf{q}}{x}}.\n \\end{equation}\n \\end{proposition}\n \\begin{proof}\n We have, by~\\eqref{e:type-free} and~\\eqref{e:type-hadamard},\n \\[\n \\ordi{\\thh}{x} =\n \\type{\\thh\\times\\wL}{x}=\\type{\\mathcal{T}(\\mathbf{q})}{x}=\\frac{1}{1-\\type{\\mathbf{q}}{x}}.  \\qedhere\n \\]\n \\end{proof}\n\n\\begin{theorem}\\label{t:ordi-hopf}\nLet $\\thh$ be a connected $q$-Hopf monoid. Then\n\\begin{equation}\\label{e:ordi-hopf}\n1-\\frac{1}{\\ordi{\\thh}{x}} \\in \\mathbb{N}{[\\![} x{]\\!]}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFrom~\\eqref{e:ordi-type} we deduce\n\\[\n1-\\frac{1}{\\ordi{\\thh}{x}} = \\type{\\mathbf{q}}{x}\n \\in \\mathbb{N}{[\\![} x{]\\!]}. \\qedhere\n \\]\n\\end{proof}\n\nIn the terminology of Section~\\ref{s:boolean} below, Theorem~\\ref{t:ordi-hopf} states that the Boolean transform of the\ndimension sequence of a connected $q$-Hopf monoid\nis nonnegative; see~\\eqref{e:bool-tran}. Proposition~\\ref{p:ordi-type} states more\nprecisely that the Boolean transform of the ordinary generating function of $\\thh$ is the\ntype generating function of $\\mathbf{q}$.\n\n\\begin{example}\\label{eg:globaldes}\nWe have\n\\[\n1-\\frac{1}{\\sum_{n\\geq 0} n! x^n} =  x + x^2 + 3x^3 + 13x^4 + 71x^5+ 461 x^6 + \\cdots.\n\\]\nThe Boolean transform $b_n$ of the dimension sequence of $\\wL$\nadmits the following description. Say that a linear order on the set $[n]$ is\n\\emph{decomposable} if it is the concatenation of a linear order on $[i]$\nand a linear order on $[n]\\setminus [i]$ for some $i$ such that $1\\leq i<n$.\nEvery linear order is the concatenation of a unique sequence of indecomposable ones.\nIt then follows from~\\eqref{e:bool-tran3} that $b_n$ is the number of indecomposable linear orders on $[n]$.\nThe sequence $b_n$ is~\\cite[A003319]{Slo:oeis}. \n\\end{example}\n\n\\begin{example}\\label{eg:atomic}\nA partition $X$ of the set $[n]$ is \\emph{atomic} if $[i]$\nis not a union of blocks of $X$ for any $i$ such that $1\\leq i<n$.\nThe dimension sequence of the Hopf monoid $\\tPi$ is the sequence of Bell numbers,\nand its Boolean transform counts the number of atomic partitions of $[n]$.\nThe latter is sequence~\\cite[A074664]{Slo:oeis}.\n\\end{example}\n\nLet $a_n:=\\dim_{\\Bbbk} \\thh[n]$. The conditions imposed by~\\eqref{e:ordi-hopf} on the first terms of this sequence are as follows.\n\\begin{gather*}\na_1^2 \\leq a_2,\\\\\n2a_1a_2-a_1^3 \\leq a_3,\\\\\n2a_1a_3 - 3a_1^2a_2+a_2^2 + a_1^4 \\leq a_4.\n\\end{gather*}\n\n\\begin{example}\\label{eg:elements}\nSuppose that the sequence starts with\n\\[\na_1=1,\\quad a_2=2,\\quad\\text{and}\\quad a_3=3.\n\\]\nThe third inequality above then implies $a_4\\geq 5$. It follows that the species $\\te$\n\\emph{of elements} (for which $\\dim_{\\Bbbk} \\te[n]=n$) does not carry a bimonoid structure.\nThis result was obtained by different means in~\\cite[Example~3.5]{AL:2012}. \n\\end{example}\n\n\nThe calculation of Example~\\ref{eg:globaldes} can be generalized to all free\nmonoids in place of $\\wL$. To this end, let us say that a composition $F$ of the set\n$[n]$ is \\emph{decomposable} if $F=F_1\\cdot F_2$ for some $F_1\\vDash [i]$,\n$F_2\\vDash [n]\\setminus[i]$, and some $i$ such that $1\\leq i<n$.\n\n\\begin{proposition}\\label{p:globaldes}\nFor any positive species $\\tp$, the Boolean transform of the dimension sequence of \nthe Hopf monoid $\\mathcal{T}(\\tp)$ is given by\n\\[\nb_n =  \\sum_{\\substack{F\\vDash [n] \\\\ \\text{$F$ indecomposable}}} \\dim_\\Bbbk \\tp(F).\n\\]\n\\end{proposition}\n\\begin{proof}\nWe have from~\\eqref{e:had-free-monoid} that\n\\[\n\\mathcal{T}(\\tp\\star\\mathbf{X}) \\cong \\mathcal{T}(\\tp)\\times \\mathcal{T}(\\mathbf{X}) \\cong \\mathcal{T}(\\tp)\\times\\wL.\n\\]\nHence, by~\\eqref{e:ordi-type},\n\\[\n\\ordi{\\mathcal{T}(\\tp)}{x} = \\frac{1}{1-\\type{\\tp\\star\\mathbf{X}}{x}}.\n\\]\nThus, $\\type{\\tp\\star\\mathbf{X}}{x}$ is the Boolean transform of $\\ordi{\\mathcal{T}(\\tp)}{x}$,\nand hence $b_n = \\dim_{\\Bbbk} \\bigl((\\tp\\star\\mathbf{X})[n]\\bigr)_{\\Sr_n}$.\n\nFrom~\\eqref{e:XF} and~\\eqref{e:star} we have that\n\\[\n(\\tp\\star\\mathbf{X})[I] = \\bigoplus_{(F,C):\\,F\\wedge C=(I)} \\tp(F)\n\\]\nwhere $F$ varies over set compositions and $C$ varies over linear orders on $I$.\nIt follows that $(\\tp\\star\\mathbf{X})[n]$ is a free $\\Bbbk\\Sr_n$-module with\n$\\Sr_n$-coinvariants equal to the space \n\\[\n\\bigoplus_{F\\vDash [n],\\, F\\wedge C_n=([n])} \\tp(F)\n\\]\nwhere $C_n$ denotes the canonical linear order on $[n]$. The result follows since\n$F\\wedge C_n=([n])$ if and only if $F$ is indecomposable.\n\n(Alternatively, we may prove this result by appealing to~\\eqref{e:bool-tran3} as in Example~\\ref{eg:globaldes}.)\n\\end{proof}\n\n\n\nLet $\\thh$ and $\\mathbf{k}$ be connected Hopf monoids.\nThe Boolean transform of the dimension sequence of $\\thh\\times\\mathbf{k}$\ncan be explicitly described \nin terms of the Boolean transforms of the dimension sequences of $\\thh$ and $\\mathbf{k}$;\nsee Proposition~\\ref{p:bool-had} below.\n\nFor example, let $b_n$ be the Boolean transform\nof the dimension sequence of $\\tLL$ (Example~\\ref{eg:LL}).\nThis is sequence~\\cite[A113871]{Slo:oeis} and its\nfirst few terms are $1,3,29,499$.\nRecalling that $\\tLL\\cong \\wL^*\\times\\wL$ and\nemploying~\\eqref{e:bool-had} we readily obtain that $b_n$\ncounts the number of pairs $(l,m)$ of linear orders on $[n]$\nsuch that $\\alpha \\wedge \\beta = (n)$,\nwhere the sequence of indecomposables of $l$ has size $\\alpha$ and\nthat of $m$ has size $\\beta$.\n\n\\begin{remark}\nTheorem~\\ref{t:ordi-hopf} states that if $\\thh$ is a connected $q$-Hopf monoid, then \nthe Boolean transform of $\\ordi{\\thh}{x}$ is nonnegative. This was deduced by considering the Hadamard product of $\\thh$ with $\\wL$.\nOne may also consider the Hadamard product of $\\thh$ with an arbitrary free Hopf monoid.\nThen, using Theorem~\\ref{t:freeness}\ntogether with~\\eqref{e:type-free2} and~\\eqref{e:type-hadamard2}, one obtains that\nfor any series $A(x)\\in \\mathbb{N}{[\\![} x{]\\!]}$ with nonnegative Boolean transform, the Hadamard product $\\ordi{\\thh}{x}\\times A(x)$ also has nonnegative Boolean transform.\nHowever, this does not impose any additional conditions on  $\\ordi{\\thh}{x}$,\nin view of Corollary~\\ref{c:submonoid}.\n\\end{remark}\n\n\n\\subsection{Non-attainable sequences}\\label{ss:non}\n\nThe question arises whether condition~\\eqref{e:ordi-hopf} characterizes\nthe dimension sequence of a connected Hopf monoid. In other words,\ngiven a sequence of nonnegative integers $b_n$, $n\\geq 1$, \nis there a connected $q$-Hopf monoid $\\thh$ such that\n\\begin{equation}\\label{e:ordi-hopf2}\n1-\\frac{1}{\\ordi{\\thh}{x}} = \\sum_{n\\geq 1} b_n x^n\n\\end{equation}\nholds? In other words, the question is\nwhether $b_n$ is the Boolean transform of the\ndimension sequence of a connected $q$-Hopf monoid. The answer is negative,\nas the following result shows.\n\n\\begin{proposition}\\label{p:ordi-hopf}\nConsider the sequence defined by\n\\[\nb_n:= \\begin{cases}\n 1 & \\text{ if $n=2$,} \\\\\n 0        & \\text{ otherwise.}\n\\end{cases}\n\\]\nThen there is no connected $q$-bimonoid $\\thh$ for which~\\eqref{e:ordi-hopf2} holds,\nregardless of $q$.\n\\end{proposition}\n\\begin{proof}\nSuppose such $\\thh$ exists; let $a_n$ be its dimension sequence.\nThen $b_n$ is the Boolean transform of $a_n$, and~\\eqref{e:bool-tran3}\nimplies that\n\\[\na_n:= \\begin{cases}\n 1 & \\text{ if $n$ is even,} \\\\\n 0        & \\text{ if $n$ is odd.}\n\\end{cases}\n\\]\n\nRecall from Section~\\ref{ss:con-hopf} that for any decomposition $I=S\\sqcup T$,\nthe composite $\\Delta_{S,T}\\mu_{S,T}$ is the identity. \nIt follows in the present situation that\n$\\mu_{S,T}$ and $\\Delta_{S,T}$ are inverse \nwhenever $S$ and $T$ are of even cardinality.\nNow let\n\\[\nI=\\{a,b,c,d\\},\\quad S=\\{a,b\\},\\quad T=\\{c,d\\},\\quad S'=\\{a,c\\},\\quad\\text{and}\\quad T'=\\{b,d\\}\n\\]\nand consider the commutative diagram~\\eqref{e:compr}.\nThe bottom horizontal composite in this diagram is an isomorphism between one-dimensional vector spaces,\nwhile the composite obtained by going up, across and down is the zero map.\nThis is a contradiction.\n\\end{proof}\n\nLet $k$ be a positive integer and define, for $n\\geq 1$,\n\\[\nb^{(k)}_n:= \\begin{cases}\n 1 & \\text{ if $n=k$,} \\\\\n 0        & \\text{ otherwise.}\n\\end{cases}\n\\]\nThe inverse Boolean transform of $b^{(k)}_n$ is\n\\[\na^{(k)}_n:= \\begin{cases}\n 1 & \\text{ if $n\\equiv 0\\!\\!\\mod k$,} \\\\\n 0        & \\text{ otherwise.}\n\\end{cases}\n\\]\nAn argument similar to that in Proposition~\\ref{p:ordi-hopf} shows that, if $k\\geq 2$,\nthere is no connected $q$-Hopf monoid with dimension sequence $a^{(k)}_n$.\n(The exponential Hopf monoid~\\cite[Example~8.15]{AguMah:2010} has dimension sequence $a^{(1)}_n$.)\n\n\\subsection{Comparison with previously known conditions}\\label{ss:comparison}\n\nThe paper~\\cite{AL:2012} provides various sets of conditions that the dimension sequence $a_n$\nof a connected Hopf monoid must satisfy. For instance,~\\cite[Proposition~4.1]{AL:2012}\nstates that\n\\begin{equation}\\label{e:submult}\na_ia_j \\leq a_n\n\\end{equation}\nfor $n=i+j$ and every $i,j\\geq 1$. In addition, the coefficients of the power series\n\\begin{equation}\\label{e:exp}\n\\frac{1 + \\sum_{n\\geq 1} a_n x^n}{1 + \\sum_{n\\geq 1} \\frac{a_n}{n!} x^n}\n\\end{equation}\nare nonnegative~\\cite[Corollary 3.3]{AL:2012}, and~\\cite[Equation~(3.2)]{AL:2012} states that\n\\begin{equation}\\label{e:addcond}\na_3\\geq 3a_2a_1-2a_1^3.\n\\end{equation}\nWe proceed to compare these conditions with those imposed by Theorem~\\ref{t:ordi-hopf}.\n\nThe inequalities~\\eqref{e:submult} are implied by Theorem~\\ref{t:ordi-hopf}, \nin view of Lemma~\\ref{l:submul}. \nAn example of a sequence that satisfies~\\eqref{e:submult} but whose  Boolean transform fails to be nonnegative is the following:\n\\[\na_n := \\begin{cases}\nn & \\text{ if $n\\leq 4$,} \\\\\n2^n         & \\text{ if $n\\geq 5$.}\n\\end{cases}\n\\]\n(The first terms of the Boolean transform are $b_1=1$, $b_2=1$, $b_3=0$, $b_4=-1$.)\nThus, the conditions imposed by Theorem~\\ref{t:ordi-hopf} are\nstrictly stronger than~\\eqref{e:submult}.\n\nCondition~\\eqref{e:exp} is also implied by Theorem~\\ref{t:ordi-hopf}, \nin view of Lemma~\\ref{l:increasing} (with $w_n=\\frac{1}{n!}$). \n\nOn the other hand, condition~\\eqref{e:addcond} is \\emph{not} implied by \nTheorem~\\ref{t:ordi-hopf}. To see this, let $a_n$ be the sequence of Fibonacci numbers,\ndefined by $a_0=a_1=1$ and\n\\[\na_n = a_{n-1}+a_{n-2}\n\\]\nfor $n\\geq 2$. The Boolean transform is nonnegative; indeed, it is simply given by\n\\[\nb_n = \\begin{cases}\n1 & \\text{ if $n\\leq 2$,} \\\\\n0 & \\text{ otherwise.}\n\\end{cases}\n\\]\nHowever, condition~\\eqref{e:addcond} is not satisfied.\n\nThe previous example shows that there is no connected Hopf monoid with\ndimensions given by the Fibonacci sequence. It also provides another example\nfor which the answer to question~\\eqref{e:ordi-hopf2} is negative.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nIn 1976 Michael Atiyah\n\\cite{Atiyah:1976:elliptic}\nintroduced $L^2$-cohomology and $L^2$-Betti numbers\nof manifolds with non-trivial fundamental group\nand asked whether these numbers can be irrational.\nFor background on these concepts we refer to\nthe book~\\cite{Lueck:2002:invariants}.\n\nIt was shown later that \nAtiyah's question is equivalent to a purely analytic problem\non group von Neumann algebras\nand we will exclusively work in this context.\nTherefore no knowledge of cohomology theory and geometry is required\nin the present paper, and in the remainder of this section\nwe recall a few basic facts from von Neumann algebra theory\nwhich are necessary to state Atiyah's question.\nLet $\\Gamma$ be a finitely presented discrete group and $\\IQ\\Gamma$ its rational group\nring.\nAn element of this ring can be represented as a bounded left convolution\noperator on the space $\\ell_2(\\Gamma)$ of square summable functions\non $\\Gamma$. Denote this representation by $\\lambda$.\nThe equivalent formulation of Atiyah's question in this\nsetting is:\n\n\\emph{\nLet $\\opT\\in M_n(\\IQ\\Gamma)$ be a symmetric element. Is it possible that \nthe von Neumann dimension $\\dim_{L(\\Gamma)} \\ker \\lambda(\\opT)$ is an irrational\nnumber?\n}\n\nHere the \\emph{von Neumann dimension} is to be understood in the setting\nof the group von Neumann algebra, denoted $L(\\Gamma)$,\nwhich is the completion of the rational convolution operator algebra in the\nweak operator topology. More precisely, one looks for matrices\nof convolution operators, but for simplicity, in the present paper we will\nwork in $L(\\Gamma)$ exclusively.\nA symmetric element $\\opT\\in\\IQ\\Gamma$ gives rise to a selfadjoint\nconvolution operator $\\lambda(\\opT)$, all of whose spectral projections\nlie in the von Neumann algebra $L(\\Gamma)$.\nOn $L(\\Gamma)$ there is a normal\nfaithful trace $\\tau(\\opT) = \\langle \\opT\\delta_e,\\delta_e\\rangle$\nand the value of $\\tau(p)$ at projections $p\\in L(\\Gamma)$\nis called the \\emph{von Neumann dimension function}.\nThe von Neumann dimension of a closed subspace of $\\ell_2(\\Gamma)$\nis the von Neumann dimension of the corresponding orthogonal\nprojection, if the latter happens to be an element of $L(\\Gamma)$.\n\nWhile rationality of kernel dimensions has been shown for many\nexamples, (see, e.g., \\cite {Linnell:1993:division,LinnellSchick:2007:finite}),\nrecently Atiyah's question was answered affirmatively\nby Tim Austin~\\cite{Austin:2009:rational}\nwho constructed an uncountable family of groups and rational\nconvolution operators with distinct kernel dimensions,\nwhich a fortiori must contain irrational numbers, even transcendental ones.\nSubsequently more constructive answers were given in\n\\cite{Grabowski:2010:turing,PichotSchickZuk:2010:closed}.\nA stronger variant of Atiyah's question had been solved earlier\n\\cite{GrigorchukLinnellSchickZuk:2000:Atiyah,\nGrigorchukZuk:2001:lamplighter,DicksSchick:2002:spectral}. \nIn  these examples, so-called lamplighter groups play a central role.\nAlthough these are not finitely presented,\nthey are recursively presentable (see, e.g., \\cite{Baumslag:2005:embedding})\nand therefore by a theorem of Higman \\cite{Higman:1961:subgroups}\ncan be embedded into finitely presented groups.\n\n\nIn the present paper, complementing the above results,\nwe pursue the ideas\nof~\\cite{DicksSchick:2002:spectral}\nand\ncompute explicitly the von Neumann dimension of the kernel of the\n``switch-walk-switch'' adjacency \noperators on the free lamplighter groups $\\CG_m\\wr\\IF_\\f$ \nwith respect to the canonical generators\nand show that\nthey are irrational for any $\\f\\geq 2$ and $m>2\\f-1$. \nThis provides another elementary explicit example of a rational convolution\noperator with irrational kernel dimension.\n\nThe basic ingredient is Theorem~\\ref{thm:lamplighterpercolation},\nwhich generalises the methods of Dicks and\nSchick~\\cite{DicksSchick:2002:spectral} from the infinite cyclic group\nto arbitrary discrete groups and makes a link to percolation theory,\nthus providing a quite explicit description of the spectrum of\nswitch-walk-switch\ntransition operators on lamplighter groups as the union of the spectra of\nall finite connected subgraphs of the Cayley graph.\nIn particular, the lamplighter kernel dimension equals the expected\nnormalised\nkernel dimension of the percolation cluster.\n\nThe paper is organised as follows.\nIn Section~\\ref{sec:lamplighter} we review\nthe necessary prerequisites about lamplighter groups and percolation\nand state the main result.\n\nIn Section~\\ref{sec:matchings} we recall the connection between\nspectra and matchings of finite trees and compute the generating\nfunction of the kernel dimensions of finite subtrees of\nthe Cayley graph of the free group. As a final step\nwe integrate this generating function\nin Section~\\ref{sec:parametrisation} and thus obtain the\ndimensions we are interested in. Another example of a free\nproduct of groups whose Cayley graph is not a tree is discussed in\nSection~\\ref{sec:freeproduct}.\n\n\n\n\\emph{Acknowledgements.}\nWe thank Slava Grigorchuk for explaining Atiyah's question,\nMartin Widmer and Christiaan van de Woestijne  for discussions about transcendental numbers,\nand Mark van Hoeij for a hint to compute a certain abelian integral,\nwhich ultimately led to the discovery of\nthe parametrisation \\eqref{eq:parametrisation}.\nLast but not least  we thank three anonymous referees\nfor numerous remarks which helped to improve the presentation.\n\n\\section{Lamplighter groups and percolation}\n\\label{sec:lamplighter}\nLet $G$ be a discrete group and fix a symmetic generating set $S$.\nWe denote by $\\CX=\\CX(G,S)$ the Cayley graph\nof $G$ with respect to $S$ and in the rest of the paper\nidentify the rational group algebra\nelement $\\opT=\\sum_{s\\in S}s$ with the corresponding convolution operator,\nwhich coincides with the adjacency operator on $\\CX$.\n\n\\subsection{Lamplighter groups}\nThe name \\emph{lamplighter group} has been coined in  recent years\nto denote   wreath products of the form $\\Gamma=\\CG_m\\wr G$.\nThis is the semidirect product\n$\\LL\\rtimes G$, where $\\CG_m$ is the cyclic group of order $m$\nand $\\LL=\\bigoplus_G \\CG_m$ is the group\nof \\emph{configurations} $\\eta:G\\to \\CG_m$ with finite support,\nwhere we define $\\supp \\eta = \\{x\\in G : \\eta(x) \\ne \\eCm\\}$.\nThe group operation on $\\LL$ is pointwise multiplication in $\\CG_m$\nand the natural\nleft action of $G$ on $\\LL$ given by $L_g\\eta(x) = \\eta(g^{-1}x)$\ninduces the twisted group law on $\\CG_m\\wr G$ \n$$\n(\\eta,g)(\\eta',g') = (\\eta\\cdot L_g\\eta',gg')\n$$\nCertain random walks on $\\CG_m\\wr G$ can be interpreted as\na lamplighter walking around on $G$ and turning\non and off lamps. A pair $(\\eta,g)$ encodes both\nthe position of the lamplighter as an element $g\\in G$ \nand the states of the lamps as a function $\\eta\\in\\LL$.\n\nWe will consider here the ``switch-walk-switch'' lamplighter adjacency operator\n$$\n\\tilde{\\opT} = \\sum_{s\\in S} EsE\n$$\non the lamplighter group $\\Gamma$ where $E=\\frac{1}{m}\\sum_{h\\in\\CG_m} h$\nis the idempotent corresponding to the uniform distribution on the lamp group\n$\\CG_m$. \nThe \\emph{underlying} convolution operator on $G$ is $\\opT=\\sum_{s\\in S} s$.\nHere we identify $\\CG_m$ and $G$ with subgroups of $\\Gamma$ via\nthe respective embeddings\n$$\n\\begin{aligned}\n\\CG_m&\\to \\Gamma & \\qquad\\qquad G &\\to\\Gamma \\\\\n h &\\mapsto (\\delta_e^h,e)   & g &\\mapsto (\\iota, g)\n\\end{aligned}\n$$\nwhere $\\iota$ is the neutral element of $\\LL$ and \n$$\n\\delta_g^h(x) =\n\\begin{cases}\n  h & x=g\\\\\n  \\eCm & x\\ne g\n\\end{cases}\n.\n$$\n\\subsection{Percolation clusters}\n\nLet $\\CX=(V,E)$ be a graph. We use the standard notation\n``$x\\in\\CX$'' for vertices and $x\\sim y$ for the neighbour relation.\nFix a parameter $0 < \\mathsf{p} < 1$. In \\emph{Bernoulli\nsite percolation} with parameter $\\mathsf{p}$ on $\\CX$, we have i.i.d.~Bernoulli random\nvariables $Y_x\\,$, $x \\in \\CX\\,$, sitting at the vertices of $\\CX$, with\n$$\n\\Prob_{\\mathsf{p}}[Y_x=1] = \\mathsf{p} ,\\qquad \\Prob_{\\mathsf{p}}[Y_x=0] = q := 1-\\mathsf{p}\\,.\n$$\nWe can realise those random variables on the probability space \n$\\Omega=\\{0,1\\}^\\CX$ with a suitable probability measure $\\Prob$.\nGiven $\\omega\\in\\Omega$, denote by $\\CX(\\omega)$  the full subgraph\nof $\\CX$ induced on $\\{x : Y_x(\\omega)=1\\}$ and for any vertex $x\\in \\CX$, denote by\n$C_x(\\omega)$ the connected component of $\\CX(\\omega)$ containing the vertex $x$,\nwhich is called the \\emph{percolation cluster} at $x$.\nIt is well known that for every connected graph\nthere is a critical parameter $\\mathsf{p}_c$\nsuch that for any vertex $x$ a phase transition occurs in the sense that\nfor $\\mathsf{p}<\\mathsf{p}_c$ the cluster $C_x$ is almost surely finite and for\n$\\mathsf{p}>\\mathsf{p}_c$ it is infinite with positive probability.\nIn order to make use of this fact we recall a combinatorial interpretation\nof criticality.\n\\begin{definition}\n  For a subset $\\clA\\subseteq \\CX$ we denote its \\emph{vertex boundary}\n  $$\n  d\\clA = \\{y\\in \\CX : y\\not\\in \\clA, y\\sim x \\text{ for some $x\\in \\clA$}\\}\n  .\n  $$\n  For $x\\in \\CX$, we denote\n  $$\n  \\mathcal{A}_x=\\{\\clA\\subseteq \\CX: x\\in \\clA, \\text{ $\\clA$ finite and connected}\\}\n  \\cup\\{\\emptyset\\}\n  $$\n  the set of finite, possibly empty, path-connected neighbourhoods of $x$.\n  These sets are sometimes called \\emph{lattice animals}.\n  The boundary of the empty animal \n  is defined to be \n  the set $\\{x\\}$. We denote by $\\mathcal{A}_x^*$ the set of animals at $x$ without\n  the empty animal.\n\\end{definition}\nThe probability of a fixed $\\clA\\in\\mathcal{A}_x$ to occur as  percolation\ncluster at $x$ is\n$$\n\\IP[C_x=\\clA] = \\mathsf{p}^{\\abs{\\clA}} q^{\\abs{d\\clA}}\n;\n$$\nthus for $\\mathsf{p}<\\mathsf{p}_c$ we have\n\\begin{equation}\n\\label{equ:sump1-p=1}\n\\sum_{\\clA\\in \\mathcal{A}_x} \\mathsf{p}^{\\abs{\\clA}} q^{\\abs{d\\clA}} = 1\n\\end{equation}\nbecause some $\\clA\\in \\mathcal{A}_x$ occurs almost surely.\n\nNow for a fixed animal $\\clA$ consider the truncated operator \n$$\n\\opT_\\clA = P_\\clA\\opT P_\\clA\n$$\nwhere $P_\\clA$ is the orthogonal projection onto the finite dimensional\nsubspace $\\{f\\in \\ell_2(\\CX) : \\supp f\\subseteq \\clA\\}$.\nWe denote the random percolation  adjacency operator by\n$$\n\\opT_\\omega = \\opT_{C_e(\\omega)},\n$$\n\nand by $\\dim\\ker \\opT_\\clA$ the dimension of the kernel of $\\opT_F$\nas a finite matrix, while $\\frac{\\dim\\ker\\opT_\\clA}{\\abs{\\clA}}$ will\nbe  the\nvon Neumann dimension of the kernel of $\\opT_\\clA$ regarded as an element\nof the finite von Neumann algebra $M_{\\abs{\\clA}}(\\IC)$ with\nvon Neumann trace $\\frac{1}{\\abs{\\clA}}\\Tr$.\nSpecial care is needed for the  empty animal, for which we define both\n  the cardinality of the boundary \n  and the von Neumann kernel dimension to be $1$. \n\n\nThen we have the following relation between the spectrum of the lamplighter operator\n$\\tilde{\\opT}$ and the spectra of $\\opT_\\clA$.\n\n\n\\begin{theorem}[{\\cite{LehnerNeuhauserWoess:spectrum,Lehner:2009:eigenspaces}}]\n  \\label{thm:lamplighterpercolation}\n  The spectral measure of the lamplighter adjacency operator $\\tilde{\\opT}$ of\n  order $m$ on a Cayley graph\n  $\\CX$ is equal to the expected spectral measure of the random truncated\n  adjacency operator $\\opT_\\omega$ on the percolation clusters of $\\CX$ with\n  percolation parameter $p=1\/m$.\n  In addition, if $p<p_c$, then there is a one-to-one correspondence\n  between the eigenspaces of $\\tilde{\\opT}$ and the collection of eigenspaces\n  of $\\opT_\\omega$ and we have the formula\n  \\begin{equation}\n    \\label{eq:dimensionformula}\n  \\dim_{L(\\Gamma)} \\ker \\tilde{\\opT} = \\IE \\frac{\\dim \\ker \\opT_\\omega}{\\abs{C_x(\\omega)}}\n = \\sum_{\\clA\\in\\mathcal{A}_e} \\frac{\\dim \\ker \\opT_\\clA}{\\abs{\\clA}}\n p^{\\abs{\\clA}}q^{\\abs{d\\clA}}\n .\n  \\end{equation}\n\\end{theorem}\nIn general it is hard to evaluate formula~\\eqref{eq:dimensionformula}\nbecause one has to compute the kernels of the adjacency matrices of\nall finite\nclusters. Due to the recursive structure of the Cayley tree however\nit is possible to compute an algebraic equation for the generating function\n$$\n\\sum_{T\\in \\mathcal{A}_e^*} (\\dim\\ker \\opT_\\clA) x^{\\abs{\\clA}}\n$$\non free groups and\nto evaluate~\\eqref{eq:dimensionformula} by integrating this function,\nthus obtaining our main result, which concludes this section.\n\\begin{theorem}\n  Denote $g_1,g_2,\\dots,g_\\f$ the canonical generators\n  of the free group  $\\IF_\\f$ and\n  consider the adjacency operator $\\opT=\\sum g_i+g_i^{-1}$ \n  on its Cayley graph. Then the von Neumann dimension of the kernel of\n  the corresponding lamplighter operator\n  $\\tilde{\\opT}=E\\opT E$ on $\\CG_m\\wr\\IF_\\f$\n  is the number\n  $$\n  \\dim_{L(\\Gamma)}\\ker \\tilde{\\opT}\n  = 1-2p+\\frac{(\\tau(p)-1)(2-2\\f+2\\f{}\\tau(p))}{\\tau(p)^2}\n  $$\n  where $p=1\/m$ and $\\tau(p)$ is the unique positive solution of the equation\n  $t^{2\\f-1}-t^{2\\f-2}=p$. For $\\f>1$, this is an irrational algebraic number,\n  e.g., for $\\f=2$ and $m=4$, the dimension is\n  $$ -\\frac56 - \\frac{400}{3(766+258\\sqrt{129})^{1\/3}} +\n  \\frac{2(766+258\\sqrt{129})^{1\/3}}{3} \\approx 0.850971. \n  $$\n\\end{theorem}\n\\begin{remark}\n  Similar computations are possible in more general free product groups\n  $G_1*G_2*\\dots*G_n$, where each factor $G_i$ is a finite group\n  whose Cayley graph possesses only cycles of length $\\equiv 2\\mod 6$,\n  like the cyclic groups $\\IZ_2$,  $\\IZ_6$,  $\\IZ_{10}$, etc.\n  An example is briefly discussed in Section~\\ref{sec:freeproduct}.\n  It should be noted however that our technique does not work\n  for nonzero eigenvalues, because in this case \n  it is more complicated to obtain the multiplicity of the eigenvalue.\n  Moreover, \n  in contrast to other approaches\n  (\\cite{DicksSchick:2002:spectral,GrigorchukLinnellSchickZuk:2000:Atiyah}),\n  it is restricted to adjacency operators,\n  i.e., all group elements get the same weight.\n\\end{remark}\n\n\n\n\n\\section{Matchings, rooted trees, and generating functions}\n\\label{sec:matchings}\nIn this section we  prepare the evaluation of the series\n\\eqref{eq:dimensionformula} by computing a generating function.\nTo this end let us recall some notations.\n\nLet $G=(V,E)$ be a finite graph. \nBy \\emph{characteristic polynomial} $\\chi(G,x)$ \n(resp., \\emph{spectrum, kernel dimension}) \n\\emph{of a graph} we mean the characteristic polynomial (resp., spectrum,\nkernel dimension) of its adjacency matrix.\nA \\emph{matching} of a finite graph is a set of disjoint edges,\ni.e., every vertex occurs as an end point of at most one edge.\nA \\emph{perfect matching} is a matching which covers all the vertices\nof the graph.\nThe \\emph{matching polynomial} of a graph on $n$ vertices is the polynomial\n$$\n\\sum_{j \\geq 0} (-1)^j m(G,j)\\, x^{n-2j},\n$$\nwhere $m(G,j)$ is the number of matchings of cardinality $j$.\n\nIt is well known (see, e.g., \\cite{Godsil:1984:spectra,Cvetkovic:1988:recent})\nthat the characteristic polynomial of a finite tree\ncoincides with its matching polynomial.\nSince the kernel dimension equals the multiplicity of eigenvalue zero,\nwhich in turn is the degree of the polynomial $x^n\\chi(G,x^{-1})$,\nit follows immediately that the dimension of the kernel of a tree is given by\n\\begin{equation}\\label{eq:kerdim}\n\\nu(T) = \\dim \\ker T = n - 2\\mu(T),\n\\end{equation}\nwhere $\\mu(T)$ denotes the size of a matching of maximal cardinality in $T$. \nIn particular, $\\dim \\ker T = 0$ if and only if $T$ has a perfect matching. \n\n\nAs a first step to evaluate \\eqref{eq:dimensionformula}\nwe have to determine the generating function\n$$G(x) = \\sum_{T\\in \\mathcal{A}_x^*} (\\dim \\ker T)\\, x^{|T|} \n   = \\sum_{T\\in \\mathcal{A}_x^*} (|T| - 2 \\mu(T))\\, x^{|T|},$$\nwhere the sum is taken over all nonempty animals $T$, i.e., \nconnected subgraphs of the\nCayley graph of the free group $\\IF_\\f{}$ that contain the unit element $e$. \nTo this end, we regard animals as rooted trees, with the root at $e$. \n\n\\begin{definition}\n  A \\emph{ $k$-ary tree} is a planar rooted tree such that every vertex\n  has at most $k$ children. Hence every vertex has degree at most $k+1$, and the root has degree at most $k$.\n  A \\emph{branch} of a $k$-ary tree is a rooted tree obtained\n  by splitting off a neighbor of the root together with its offspring.\n  Thus a $k$-ary tree can be defined recursively as\n  a rooted tree with an ordered collection of $k$ possibly empty \n branches.\n\\end{definition}\nThus our animals are $k$-ary trees with $k=2\\f-1$, with the single exception that the root vertex may have degree $k+1$ (but all branches are $k$-ary trees according to the above definition).\nFor reasons which will become apparent soon\nwe split the family of rooted trees into two groups, \nfollowing ideas similar to those employed in \\cite{Wagner:2007:number}: \n\\begin{definition}\n  We say that a rooted tree is of \\emph{type} $A$ if it has a maximum matching\n  that leaves the root uncovered. Otherwise $T$ is of type $B$.\n\\end{definition}\n\nSuppose that $T$ is of type $A$. Then it has a maximum matching that does not\ncover the root and is therefore a union of maximum matchings in the various\nbranches of $T$. Hence if $S_1,S_2,\\cdots,S_k$ are the branches of $T$, we have \n$$\\mu(T) = \\mu(S_1) + \\mu(S_2) + \\cdots + \\mu(S_k).$$\nWe claim that in this case all $S_j$ are of type $B$.\nFor, suppose on the contrary that one of the branches, say $S_j$, is of type\n$A$. Then we can choose a maximum matching in $S_j$ that does not cover the\nroot. Choose maximum matchings in all the other branches as well, and add the\nedge between the roots of $T$ and $S_j$ to obtain a matching of cardinality\n$$\\mu(S_1) + \\mu(S_2) + \\cdots + \\mu(S_k) + 1,$$\ncontradiction.\nConversely, if all branches of $T$ are of type $B$, \nthen $T$ is of type $A$: clearly, the\nmaximum cardinality of a matching that does not cover the root is $\\mu(S_1) +\n\\mu(S_2) + \\cdots + \\mu(S_k)$, so it remains to show that there are no\nmatchings of greater cardinality that cover the root. Suppose that such a\nmatching contains the edge between the roots of $T$ and $S_j$. Then, since\n$S_j$ is of type $B$, the remaining matching, restricted to $S_j$, can only\ncontain at most $\\mu(S_j) - 1$ edges. Each of the other branches $S_i$ can only\ncontribute $\\mu(S_i)$ edges, so that we obtain a total of\n$$\\mu(S_1) + \\mu(S_2) + \\cdots + \\mu(S_k)$$\nedges, as claimed. This proves the following fact:\n\\begin{lemma}\\label{lem:rec}\n  \\begin{enumerate}\n   \\item Let $T$ be a rooted tree and $S_1,\\ldots,S_k$ its branches,\n    then we have\n$$\\mu(T) = \\sum_{i=1}^k \\mu(S_i) + \\begin{cases} 0 & \\text{ if $T$ is of type $A$,} \\\\ 1 & \\text{ otherwise.} \\end{cases}$$\n\\item \nA rooted tree $T$ is of type $A$ if and only if all its branches are of type $B$. \n  \\end{enumerate}\n\\end{lemma}\nThe only part that was not explicitly proven above is the formula for $\\mu(T)$\nin the case that $T$ is of type $B$. This, however, is easy as well: Clearly\nthe cardinality of a matching is at most $\\mu(S_1) + \\cdots + \\mu(S_k) + 1$\n(the summand $1$ accounting for the edge that covers the root). On the other\nhand, $\\mu(T)$ must be strictly greater than $\\mu(S_1) + \\cdots + \\mu(S_k)$,\nsince there are matchings of this cardinality that do not cover the root.\n\n\\begin{remark}\n  Consistently with Lemma~\\ref{lem:rec}\n  we define the tree $T_1$ that only consists of a single vertex to be of\n  type $A$ with $\\mu(T_1) = 0$ and \n  the empty tree $T_0$  to be of type $B$ with $\\mu(T_0)= 0$.\n  This is important for the generating functions constructed below.\n\\end{remark}\n\nSince we are interested in the parameter $\\nu(T) = \\dim \\ker T = |T| - 2\\mu(T)$\nrather than $\\mu(T)$ itself, we first translate the above formula to a\nrecursion for $\\nu(T)$: since $|T| = |S_1| + \\cdots + |S_k| + 1$, we have\n$$\\nu(T) = \\sum_{i=1}^k \\nu(S_i) + \\begin{cases} 1 & \\text{ if $T$ is of type\n    $A$,} \\\\ -1 & \\text{ otherwise.} \\end{cases}$$\n\nNow let $\\trees_k$, $\\trees_{k,A}$, $\\trees_{k,B}$ denote the set of all $k$-ary trees, $k$-ary trees of type $A$ and $k$-ary trees of type $B$ respectively.\nWe define the bivariate generating functions\n$$\nA:=A(u,x) = \\sum_{T \\in \\trees_{k,A}} u^{\\nu(T)} x^{|T|}\n\\qquad \\text{and}\n\\qquad B:= B(u,x) = \\sum_{T \\in \\trees_{k,B}} u^{\\nu(T)} x^{|T|},\n$$ \nthe\nsummation being over $k$-ary trees in both cases (including the empty tree in\nthe case of $B$, and the one-vertex tree in the case of $A$).\nSince any tree $T$ of type $A$ is a grafting of $k$ (possibly empty) branches $S_1,\\ldots,S_k$ of type $B$ (which we write as $T = \\bigvee_{i=1}^k S_i$),\nwe obtain\n\\begin{align*}\n  A(u,x)\n  &= \\sum_{T\\in \\trees_{k,A}} u^{\\nu(T)}\\,x^{\\abs{T}}\\\\\n  &= \\sum_{S_1,\\dots,S_k\\in \\trees_{k,B}} u^{\\nu(\\bigvee_{i=1}^k\n    S_i)}\\,x^{\\abs{\\bigvee_{i=1}^k S_i}}\\\\\n  &= \\sum_{S_1,\\dots,S_k\\in \\trees_{k,B}} u^{1+\\sum_{i=1}^k\\nu(S_i)}\\,x^{1+\\sum_{i=1}^k\\abs{S_i}}\\\\\n  &= ux \\prod_{i=1}^k \\sum_{S_i\\in \\trees_{k,B}} u^{\\nu(S_i)}\\,x^{\\abs{S_i}}\\\\\n  &= ux  B(u,x)^k\n  .\n\\end{align*}\n\nSimilarly, in the case of type $B$, we get the following equation\n\\begin{align*}\n  B(u,x)\n  &= \\sum_{T\\in \\trees_{k,B}} u^{\\nu(T)}\\,x^{\\abs{T}}\\\\\n  &= 1 + \\sideset{}{^\\prime}\\sum_{S_1,\\dots,S_k\\in \\trees_{k}} u^{\\nu(\\bigvee_{i=1}^k\n    S_i)}\\,x^{\\abs{\\bigvee_{i=1}^k S_i}}\\\\\n  &= 1 + \\sideset{}{^\\prime}\\sum_{S_1,\\dots,S_k\\in \\trees_{k}} u^{-1+\\sum_{i=1}^k\\nu(S_i)}\\,x^{\\abs{\\bigvee_{i=1}^k S_i}},\\\\\n\\intertext{where we took special care of the empty tree \n  and the remaining sum indicated by $\\sum'$ runs over all $k$-tuples of trees such that\n  at least one of them is not of type $B$; \n this means that we have to subtract the sum over $k$-tuples of type B trees\n from the sum over $k$-tuples of arbitrary trees:}\nB(u,x)  &= 1 + \\sum_{S_1,\\dots,S_k\\in \\trees_{k}} \n      u^{-1+\\sum_{i=1}^k\\nu(S_i)}\n      \\,\n      x^{1+\\sum_{i=1}^k\\abs{S_i}}\n    -\n    \\sum_{S_1,\\dots,S_k\\in \\trees_{k,B}} \n      u^{-1+\\sum_{i=1}^k\\nu(S_i)}\n      \\,\n      x^{1+\\sum_{i=1}^k\\abs{S_i}}\n    \\\\\n  &= 1 + \\frac{x}{u}\n     \\biggl(\n       \\prod_{i=1}^k \\sum_{S_i\\in \\trees_{k}}  u^{\\nu(S_i)}\\,x^{\\abs{S_i}} \n       -\n       \\prod_{i=1}^k \\sum_{S_i\\in \\trees_{k,B}}  u^{\\nu(S_i)}\\,x^{\\abs{S_i}} \n     \\biggr)\\\\\n  &= 1 + \\frac{x}{u}\n     \\biggl(\n       \\bigl(A(u,x)+B(u,x)\\bigr)^k - B(u,x)^k\n     \\biggr)\n     .\n\\end{align*}\n\nIn conclusion, we have translated the recursive description into\nthe following two functional equations for $A(u,x)$ and $B(u,x)$:\n\\begin{equation}\n  \\label{eq:ABequations}\n  \\begin{aligned}\nA(u,x) &= ux B(u,x)^k, \\\\\nB(u,x) &= 1 + \\frac{x}{u} ((A(u,x)+B(u,x))^k - B(u,x)^k). \\\\\n  \\end{aligned}\n\\end{equation}\nFinally, we obtain the following generating function for animals (the only\ndifference lying in the possibility that the root is allowed to have degree\n$k+1 = 2\\f{}$ as well and the empty tree is excluded this time):\n$$F(u,x) = \\sum_{\\substack{ T \\in\\mathcal{A}_e^*}} u^{\\nu(T)} x^{|T|} = ux B(u,x)^{k+1} + \\frac{x}{u} ((A(u,x)+B(u,x))^{k+1} - (B(u,x))^{k+1}).$$\nWe are mainly interested in the derivative with respect to $u$, since\n$$G(x) = \\frac{\\partial}{\\partial u} F(u,x) \\Big|_{u=1} = \\sum_{T \\in \\mathcal{A}_e^*} \\nu(T) x^{|T|}.$$\nTo save space, we will use the customary abbreviation $F_u$ etc.\\ to denote\npartial derivatives with respect to $u$. First note that\n$$\nA(u,x)+u^2B(u,x) = u^2+ux(A(u,x)+B(u,x))^k,\n$$\nand we can rewrite the identities~\\eqref{eq:ABequations} as\n\\begin{equation}\n  \\label{eq:BkABkequations}\n  \\begin{aligned}\n    B(u,x)^k &= \\frac{A(u,x)}{ux}, \\\\\n    (A(u,x)+B(u,x))^k &= \\frac{A(u,x)+u^2(B(u,x)-1)}{ux}.\n  \\end{aligned}\n\\end{equation}\nTaking the derivative of the second identity at $u=1$ we obtain\n$$\nA_u(1,x) + B_u(1,x)  = \\frac{2(1-B(1,x))+x (A(1,x)+B(1,x))^k}{1-kx(A(1,x)+B(1,x))^{k-1}}.\n$$\nUsing the identities~\\eqref{eq:BkABkequations} we can express $F$ as\n$$\nF(u,x) = A(u,x)B(u,x)(1-\\frac{1}{u^2}) + (A(u,x)+B(u,x))(\\frac{A(u,x)}{u^2} + B(u,x)-1)\n$$\nand the derivative at $u=1$ is\n\\begin{align*}\nF_u(1,x)\n  &= 2A(1,x)B(1,x) + (A_u(1,x)+B_u(1,x))(A(1,x)+B(1,x)-1) \\\\\n  &\\phantom{=}+ (A(1,x)+B(1,x))(-2A(1,x)     +A_u(1,x)+B_u(1,x)) \\\\\n  &= -2A(1,x)^2+(A_u(1,x)+B_u(1,x))(2A(1,x)+2B(1,x)-1)\\\\\n  &= -2A(1,x)^2\n     + \\frac{2(1-B(1,x))+x(A(1,x)+B(1,x))^k}{1-kx(A(1,x)+B(1,x))^{k-1}}\n      (2(A(1,x)+B(1,x))-1)\\\\\n  &= -2A(1,x)^2\n     + \\frac{(2(1-B(1,x))+A(1,x)+B(1,x)-1)(2(A(1,x)+B(1,x))-1)}{1-k(A(1,x)+B(1,x)-1)\/(A(1,x)+B(1,x))},\n\\end{align*}\nmaking use of~\\eqref{eq:BkABkequations} in the last step once again. So we finally obtain\n\\begin{multline}\\label{eq:G_in_terms_of_B}\nG(x) = F_u(1,x) \\\\\n= -2A(1,x)^2 + \\frac{(A(1,x)+B(1,x))(A(1,x)-B(1,x)+1)(2A(1,x)+2B(1,x)-1)}{k-(k-1)(A(1,x)+B(1,x))}.$$\n\\end{multline}\n\n\\section{Parametrisation}\n\\label{sec:parametrisation}\n\nRecall that we are considering percolation on a $(k+1)$-regular tree, where $p = \\frac{1}{m} < \\frac{1}{k}$ is the percolation probability, and $q = 1-p$. For an animal $T$ (i.e., a potential percolation cluster), the size of the boundary is $|dT| = 2 + (k-1)|T|$, as can be seen immediately by induction on $|T|$. In view of the identity~\\eqref{eq:dimensionformula}, we are interested in the expression\n\\begin{equation}\\label{eq:our_constant}\n\\begin{split}\nC(p) &= q + \\sum_{T\\in\\mathcal{A}_e^*} \\frac{\\dim \\ker T}{|T|} p^{|T|}q^{2+(k-1)|T|} =\nq + q^2 \\sum_{T\\in\\mathcal{A}_e^*} \\frac{\\dim \\ker T}{|T|} (pq^{k-1})^{|T|} \\\\\n&= q + q^2 \\int_0^{pq^{k-1}} \\frac{G(x)}{x}\\,dx,\n\\end{split}\n\\end{equation}\nsince it gives the von Neumann dimension of the kernel of the lamplighter operator $\\tilde{\\opT}$ on $\\CG_m\\wr\\IF_\\f$. The summand $q$ takes care of the ``empty'' animal, i.e., the possibility that the vertex $x$ is not actually in $\\CX(\\omega)$ (which happens with probability $q$).\n\nIn order to compute this integral, we determine a parametrisation of $G$; since\n$G$ is a rational function of $x$, $A$ and $B$, we first find such a parametrisation\nfor the functions $A$ and $B$. \nThis is possible because the implicit equation~\\eqref{eq:ABequations}\nfor $B$ defines an algebraic curve of genus zero.\nRecall that $A = A(1,x)$ and $B = B(1,x)$ satisfy the equations\n\\begin{align*}\nA &= xB^k, \\\\\nB &= 1+x((A+B)^k-B^k).\n\\end{align*}\nIt turns out that the following parametrisation satisfies these two equations:\n\\begin{equation}\n  \\label{eq:parametrisation}\n  \\begin{aligned}\nx &= (t-1)t^{k-1}(1+t^{k-1}-t^k)^{k-1}, \\\\\nA &= \\frac{t-1}{t(1+t^{k-1}-t^k)}, \\\\\nB &= \\frac{1}{t(1+t^{k-1}-t^k)}.\n  \\end{aligned}\n\\end{equation}\nThis parametrisation was essentially obtained by ``guessing'', i.e., finding\nthe parametrisation in special cases, which was done with an algorithm\nby M.~van~Hoeij~\\cite {vanHoeij:1994:algorithm} in the\n\\verb|algcurves| package of the computer algebra system\n\\verb|Maple|${}^\\mathrm{TM}$~\\cite{Maple10},\nand extrapolating to the general case. Once the parametrisation has\nbeen found, however, it is easy to verify it directly.\n\nThe two equations determine the coefficients of the expansions of $A$ and $B$ at $x = 0$ uniquely, hence they define unique functions $A$ and $B$ that are analytic at $0$. The above parametrisation provides such an analytic solution in which $t = 1$ corresponds to $x = 0$. Furthermore, the interval $[1,t_0]$, where $t_0$ is the solution of $t^k-t^{k-1} = \\frac{1}{k}$, maps to the interval $[0,x_0]$ with \n$$x_0 = \\frac1k \\left(1 - \\frac1k \\right)^{k-1},$$\nand the parametrisation is monotone on this interval. At $t = t_0$, it has a singularity (of square root type), which corresponds to the fact that $p = \\frac1k$ is the critical percolation parameter and that $x_0$ is the radius of convergence and the smallest singularity of $A$ and $B$ (and thus in turn $G$). Therefore, the computation of~\\eqref{eq:our_constant} amounts to integrating a rational function between $0$ and the unique solution $\\tau(p)$ of\n$$(t-1)t^{k-1}(1+t^{k-1}-t^k)^{k-1} = pq^{k-1}$$\ninside the interval $[1,t_0]$. To show existence and uniqueness of $\\tau(p)$, note again that the function $x(1-x)^{k-1}$ is strictly increasing on $[0,\\frac{1}{k}]$ and thus maps this interval bijectively to $[0,x_0]$. Moreover, it follows that $\\tau(p)$ is the unique positive solution of\n$$\\tau(p)^k-\\tau(p)^{k-1} = p.$$\nPlugging the parametrisations of $A$ and $B$ into~\\eqref{eq:G_in_terms_of_B} yields\n$$G(x) =\\frac{(t-1)(t^{2k}(1-t)+t^{k-1}((2k+1)t^2-(4k+2)t+2k)+2)}{t^2(1+t^{k-1}-t^k)^2(1+kt^{k-1}-kt^k)}.$$\nTogether with\n$$\\frac{dx}{x} = \\frac{(kt-k+1)(1+kt^{k-1}-kt^k)}{t(t-1)(1+t^{k-1}-t^k)}\\,dt,$$\nwe finally end up with an integral which has a surprisingly simple antiderivative for arbitrary $k$. This antiderivative was also found by means of computer algebra, but can of course be checked directly to be an antiderivative:\n\\begin{align*}\nC(p) &= q + q^2 \\int_1^{\\tau(p)} \\frac{(kt-k+1)(t^{2k}(1-t)+t^{k-1}((2k+1)t^2-(4k+2)t+2k)+2)}{t^3(1+t^{k-1}-t^k)^3}\\,dt \\\\\n&= q + q^2 \\frac{(t-1)(1-k+(k+1)t-t^{k+1})}{t^2(1+t^{k-1}-t^k)^2} \\Big|_{t = \\tau(p)} \\\\\n&= q - p + \\frac{(\\tau(p)-1)(1-k+(k+1)\\tau(p))}{\\tau(p)^2},\n\\end{align*}\nwhich shows that the constant $C(p)$ is always algebraic, since $p = \\frac{1}{m}$ is rational in our context and $\\tau(p)$ is a solution to an algebraic equation. In particular, for $k = 1$, one has $\\tau(p) = 1 + p$, which yields\n$$C(p) = 3 - 2p - \\frac{2}{1+p}.$$\nIn general, however, $C(p)$ is not rational: take, for instance, $k = 3$ and $p = \\frac14$, to obtain\n$$C(p) = -\\frac56 - \\frac{400}{3(766+258\\sqrt{129})^{1\/3}} + \\frac{2(766+258\\sqrt{129})^{1\/3}}{3} \\approx 0.850971.$$\nOne can even easily prove the following:\n\\begin{proposition}\nIf $p = \\frac1m$ for $m > k \\geq 3$, then $C(p)$ is an irrational algebraic number.\n\\end{proposition}\n\\begin{proof}\nSuppose that $C(p)$ is rational. Then \n$$\\frac{(\\tau(p)-1)(1-k+(k+1)\\tau(p))}{\\tau(p)^2} = \\frac{a}{b}$$\nfor some coprime integers $a,b$ with $b > 0$. Hence $\\tau = \\tau(p)$ is a root of the polynomial\n$$P(t) = (b(k+1)-a)t^2-2bkt+b(k-1).$$\nDivide by $g = \\gcd(b(k+1)-a,-2bk,b(k-1))$ to obtain a primitive polynomial $\\tilde{P}(t)$ (in the ring-theoretic sense, i.e., a polynomial whose coefficients have greatest common divisor $1$). It is easy to see that $g \\leq 2$. Now note that $\\tau$ is also a zero of\n$$Q(t) = mt^k-mt^{k-1}-1.$$\nIf $\\tau$ was rational, it would have to be of the form $\\pm \\frac{1}{r}$ (since the denominator has to divide the leading coefficient, while the numerator has to divide the constant coefficient), contradicting the fact that $\\tau(p) > 1$. Hence $\\tilde{P}$ is the minimal polynomial of $\\tau$, and $Q(t)$ must be divisible by $\\tilde{P}(t)$ in $\\IZ[t]$ (by Gauss' lemma), which implies that the constant coefficient of $\\tilde{P}$ must be $\\pm 1$. But this is only possible if $b(k-1) = g = 2$, i.e., $k = 3$, $b = 1$, and $a$ must be even. But then\n$$C(p) = q-p + \\frac{a}{b} \\geq q-p+2 = 1+2q > 1,$$\nand we reach a contradiction.\n\\end{proof}\n\n\\section{A free product}\\label{sec:freeproduct}\n\nThe method of the preceding sections is generally not applicable if the Cayley graph is not a tree; however, \\eqref{eq:kerdim} remains true if all cycles of $T$ have length $\\equiv 2 \\mod 4$ (see for instance \\cite[Theorem 2]{Borovicanin:2009:nullity}). Hence it is possible to apply the same techique if the free group $\\IF_\\f{}$ is replaced by special free products such as $\\IZ_6 \\ast \\IZ_6$; the Cayley graph of this group has hexagons as its only cycles and therefore satisfies the aforementioned condition. Once again, one can distinguish between (rooted) animals with the property that there exists a maximum matching that does not cover the root (type A) and (rooted) animals for which this is not the case (type B) and derive recursions. In addition, one needs to take the size of the boundary of an animal into account, which is no longer uniquely determined by the size of an animal. Hence we consider the trivariate generating functions\n$$A = A(u,x,y) = \\sum_{\\clA \\text{ of type $A$}} u^{\\nu(\\clA)}\nx^{|\\clA|}y^{|d\\clA|} \\quad \\text{and} \\quad B = B(u,x,y) = \\sum_{\\clA \\text{ of type $B$}} u^{\\nu(\\clA)} x^{|\\clA|}y^{|d\\clA|},$$\nfor which one obtains, after some lengthy calculations, functional equations in analogy to those in~\\eqref{eq:ABequations} as well as an integral representation analogous to~\\ref{eq:our_constant} for the von Neumann dimension of the kernel of the lamplighter operator $\\tilde{\\opT}$ on $\\CG_m\\wr(\\IZ_6 \\ast \\IZ_6)$. \n\nIn order to determine the resulting integral, one can use the Risch-Trager algorithm, as implemented for example \nin the computer algebra system\n\\texttt{FriCAS}, a fork of ~\\cite{axiom},\nOnce again, we found that there exists an algebraic antiderivative, so that we obtain an algebraic von Neumann dimension for any $m \\geq 3$ (the critical\npercolation parameter is $p = 0.339303$ in this case, which is a zero of the polynomial $3p^5-2p^4-2p^3-2p^2-2p+1$). It is likely that it is also irrational for all $m \\geq 3$, although we do not have a proof for this conjecture. Moreover, we conjecture that in fact the kernel dimension of the adjacency operator of an arbitrary free product of\ncyclic groups $\\IZ_{4k+2}$ is algebraic (and probably irrational), but the computations outlined above quickly become intractable by\nthe present method if more complicated examples are studied.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nGeneralized eigenvalue problems (GEVP) appear in various fields and are often attributed to\nstandard eigenvalue problems (EVP). If the matrix $B$ of the matrix pencil $(A,B)$ is regular,\nthen it is possible to consider its inverse and solve it as an EVP for the matrix $B^{-1}A$.\nHowever, even when both $A$ and $B$ are sparse matrices, the matrix $B^{-1}A$ will lose\nthe sparsity of $A$ and $B$. This will lead to computational complexity and numerical instability.\nHence it is useful to consider a transformation from GEVP to EVP preserving the sparsity of\nthe original matrix without using subtractions, which can cause numerical instability,\nif possible.\n\nThe simplest GEVP treated here is a matrix pencil consisting of a tridiagonal matrix\n$A$ and an almost diagonal matrix $B$, in which a nonzero component appears in the subdiagonal\nin one place:\n\\begin{gather*}\n  A=\n  \\begin{pmatrix}\n    a_{0, 0} & a_{0, 1}\\\\\n    a_{1, 0} & \\ddots & \\ddots\\\\\n    & \\ddots & a_{j, j} & a_{j, j+1}\\\\\n    && a_{j+1, j} & a_{j+1, j+1} & \\ddots\\\\\n    &&& \\ddots & \\ddots & a_{N-2, N-1}\\\\\n    &&&& a_{N-1, N-2} & a_{N-1, N-1}\n  \\end{pmatrix},\\quad\n  B=\n  \\begin{pmatrix}\n    \\alpha_0\\\\\n    & \\ddots\\\\\n    && \\alpha_j\\\\\n    && \\beta & \\alpha_{j+1}\\\\\n    &&&& \\ddots \\\\\n    &&&&& \\alpha_{N-1}\n  \\end{pmatrix},\n\\end{gather*}\nwhere $\\beta\\ne 0$.\nIn this case, we first convert the GEVP $A\\bm\\phi = \\lambda B\\bm\\phi$ to the following GEVP\n\\begin{align*}\n \\left(A-\\frac{a_{j+1,j}}{\\beta}B\\right)\\bm\\phi = \\lambda' B\\bm\\phi,\n\\end{align*}\nwhere $\\lambda'=\\lambda-\\frac{a_{j+1,j}}{\\beta}$.\nWe note here that in the tridiagonal matrix $A-\\frac{a_{j+1,j}}{\\beta}B$,\none of the subdiagonal components is a zero component.\nWe will show that GEVP in the above form can be rewritten as an EVP without subtraction\nby using the theory of Laurent biorthogonal polynomials~\\cite{kharchev1997frt,zhedanov1998clb}.\nFurthermore, the same procedure can be introduced for the generalization of the above form.\nThe following is an outline of the concrete forms and procedures of the GEVP treated in this paper.\n\nIn Section~\\ref{sec:transf-betw-trid},\nfirst, we consider a simple GEVP\n\\begin{equation*}\n R \\bm p = x L \\bm p\n\\end{equation*}\nof the pencil $(R, L)$ defined by the upper bidiagonal matrix $R$\nand the lower bidiagonal matrix $L$ as\n\\begin{equation*}\n  R\\coloneq\n  \\begin{pmatrix}\n    q_{0} & 1\\\\\n    & q_{1} & 1\\\\\n    && \\ddots & \\ddots\\\\\n    &&& \\ddots & 1\\\\\n    &&&& q_{N-1}\n  \\end{pmatrix}, \\quad\n   L\\coloneq\n  \\begin{pmatrix}\n    1 \\\\\n    -e_0 & 1\\\\\n    & -e_1 & \\ddots\\\\\n    && \\ddots & \\ddots\\\\\n    &&& -e_{N-2} & 1\n \\end{pmatrix}.\n\\end{equation*}\nWe can see that each element of the vector $\\bm p$ is a polynomial of\n$x$ defined by a three-term recurrence relation for\nthe Laurent biorthogonal polynomials.\nThe followings will be shown:\n\\begin{itemize}\n\\item LU factorization of the matrix $RL^{-1}$\n  yields new bidiagonal matrices $R^*$, $L^*$ and the associated vector (polynomials) $\\bm p^*$,\n  where the new GEVP $R^*\\bm p^*=x L^*\\bm p^*$ has the same eigenvalues as\n  the original GEVP.\n\\item Setting $(R^{(0)}, L^{(0)})\\coloneq (R, L)$, $\\bm p^{(0)}\\coloneq \\bm p$ and iterating the procedure above,\n  we obtain the matrix pencil sequence $(R^{(0)}, L^{(0)})$, $(R^{(1)}, L^{(1)})$,\n  $(R^{(2)}, L^{(2)})$, ...,\n  and the corresponding vector sequence $\\bm p^{(0)}$, $\\bm p^{(1)}$, $\\bm p^{(2)}$, ....\n  Then, we can construct a tridiagonal matrix $\\hat T$\n  from the matrix pencil sequence and another vector $\\hat{\\bm p}$ from the vector sequence,\n  where $\\hat{\\bm p}$ is the eigenvector of the EVP\n  $\\hat T\\hat{\\bm p}=x\\hat{\\bm p}$\n  which has the same eigenvalues as the original GEVP.\n  Elements of the vector $\\hat{\\bm p}$ are orthogonal polynomials~\\cite{chihara1978iop} defined by\n  the three-term recurrence relation whose coefficients are given by $\\hat T$.\n  Note that, since $L^{-1}R$ is not a tridiagonal matrix,\n  this is not a trivial isospectral transformation.\n\\end{itemize}\n\nIn Section~\\ref{sec:transf-betw-trid-1},\nthe transformation in Section~\\ref{sec:transf-betw-trid} is generalized to\na matrix pencil $(T, L)$, where $T$ is the tridiagonal matrix\nand $L$ is the lower bidiagonal matrix of the form\n\\begin{gather*}\n  T=\n  \\begin{pmatrix}\n    a_0 & 1\\\\\n    (1-\\epsilon_0) b_0 & a_1 & 1\\\\\n    & (1-\\epsilon_1) b_1 & \\ddots & \\ddots\\\\\n    && \\ddots & \\ddots & 1\\\\\n    &&& (1-\\epsilon_{N-2}) b_{N-2} & a_{N-1}\n  \\end{pmatrix},\\\\\n  L=\n  \\begin{pmatrix}\n    1\\\\\n    -\\epsilon_0 e_0 & 1\\\\\n    & -\\epsilon_1 e_1 & \\ddots\\\\\n    && \\ddots & \\ddots\\\\\n    &&& -\\epsilon_{N-2} e_{N-2} & 1\n  \\end{pmatrix},\\quad\n  \\epsilon_0, \\dots, \\epsilon_{N-2} \\in \\{0, 1\\}.\n\\end{gather*}\nIt will be shown that\nthere is also a transformation to a tridiagonal matrix with the same eigenvalues.\n\nIn Section~\\ref{sec:transf-betw-hess}, further,\nthe transformation in Section~\\ref{sec:transf-betw-trid-1}\nis generalized to a matrix pencil $(H, L)$, where\n$H$ is the upper Hessenberg matrix and $L$ is the lower bidiagonal matrix\nof the form\n\\begin{gather*}\n  H=\n  \\begin{pmatrix}\n    a_{0, 0} & a_{0, 1} & \\dots & a_{0, M-1} & 1\\\\\n    (1-\\epsilon_0) b_0 & a_{1, 0} & a_{1, 1} & \\dots & a_{1, M-1} & 1\\\\\n    & (1-\\epsilon_1) b_1 & \\ddots & \\ddots & \\dots & \\ddots & \\ddots\\\\\n    && \\ddots & \\ddots & \\ddots & \\dots & \\ddots \\\\\n    &&& \\ddots & \\ddots & \\ddots & \\dots\\\\\n    &&&& \\ddots & \\ddots & a_{N-2, 1}\\\\\n    &&&&& (1-\\epsilon_{N-2}) b_{N-2} & a_{N-1, 0}\n  \\end{pmatrix},\\\\\n  L=\n  \\begin{pmatrix}\n    1\\\\\n    -\\epsilon_0 e_0 & 1\\\\\n    & -\\epsilon_1 e_1 & \\ddots\\\\\n    && \\ddots & \\ddots\\\\\n    &&& -\\epsilon_{N-2} e_{N-2} & 1\n  \\end{pmatrix},\\quad\n  \\epsilon_0, \\dots, \\epsilon_{N-2} \\in \\{0, 1\\}.\n\\end{gather*}\nIt will be shown that\nthere is a transformation to an upper Hessenberg matrix with the same eigenvalues.\n\n\n\\section{Isospectral transformation between tridiagonal matrix and bidiagonal--bidiagonal matrix pencil}\n\\label{sec:transf-betw-trid}\n\n\\subsection{Sequence of bidiagonal--bidiagonal GEVPs and eigenvectors}\nLet $N$ be a positive integer and $\\{p_n(x)\\}_{n=0}^{N}$ be monic polynomials of degree $n$\ndefined by the three-term recurrence relation\n\\begin{gather}\n  p_{-1}(x)\\coloneq 0,\\quad p_0(x)\\coloneq 1,\\quad\n  p_{n+1}(x) \\coloneq (x-q_n)p_n(x)-x e_{n-1} p_{n-1}(x),\\quad n=0, 1, \\dots, N-1,\\label{eq:trr-LBP}\n\\end{gather}\nwhere $q_n, e_n\\in \\mathbb C$ and $e_n\\ne 0$.\nWe can rewrite the three-term recurrence relation~\\eqref{eq:trr-LBP} as\n\\begin{equation}\\label{eq:trr-LBP-vec}\n  R\\bm p(x)+\\bm p_N(x)=xL\\bm p(x),\n\\end{equation}\nwhere $R$ and $L$ are bidiagonal matrices\n\\begin{equation*}\n  R\\coloneq\n  \\begin{pmatrix}\n    q_0 & 1\\\\\n    & q_1 & 1\\\\\n    && \\ddots & \\ddots\\\\\n    &&& \\ddots & 1\\\\\n    &&&& q_{N-1}\n  \\end{pmatrix},\\quad\n  L\\coloneq\n  \\begin{pmatrix}\n    1 \\\\\n    -e_0 & 1\\\\\n    & -e_1 & \\ddots\\\\\n    && \\ddots & \\ddots\\\\\n    &&& -e_{N-2} & 1\n  \\end{pmatrix},\n\\end{equation*}\nand $\\bm p(x)$, $\\bm p_N(x)$ are vectors of polynomials\n\\begin{equation}\\label{eq:def-vec-p}\n  \\bm p(x)\\coloneq\n  \\begin{pmatrix}\n    p_0(x)\\\\\n    p_1(x)\\\\\n    \\vdots\\\\\n    p_{N-1}(x)\n  \\end{pmatrix},\\quad\n  \\bm p_N(x)\\coloneq\n  \\begin{pmatrix}\n    0\\\\\n    \\vdots\\\\\n    0\\\\\n    p_N(x)\n  \\end{pmatrix}.\n\\end{equation}\nLet $x_0, x_1, \\dots, x_{N-1}$ be the zeros of $p_N(x)$, then we have\n\\begin{equation*}\n  R\\bm p(x_i)=x_i L\\bm p(x_i),\\quad i=0, 1, \\dots, N-1,\n\\end{equation*}\ni.e., $x_i$ and $\\bm p(x_i)$ are a generalized eigenvalue and eigenvector\nof the matrix pencil $(R, L)$.\n\nNext, let us introduce new monic polynomials $\\{p_n^*(x)\\}_{n=0}^N$ of degree $n$ generated from\nthe monic polynomials $\\{p_n(x)\\}_{n=0}^N$ as\n\\begin{gather}\n  p_0^*(x)\\coloneq 1,\\quad p_N^*(x)\\coloneq p_N(x),\\quad\n  p_n^*(x)\\coloneq p_n(x)-e_{n-1}p_{n-1}(x),\\quad n=1, 2, \\dots, N-1.\\label{eq:GT-LBP}\n\\end{gather}\nThen the three-term recurrence relation of $\\{p_n(x)\\}_{n=0}^N$ \\eqref{eq:trr-LBP} is\nrewritten as\n\\begin{equation}\n  xp_n^*(x)=p_{n+1}(x)+q_n p_n(x),\\quad n=0, 1, \\dots, N-1.\\label{eq:CT-LBP}\n\\end{equation}\n\nUsing the bidiagonal matrices and vectors of polynomials,\nthe relations \\eqref{eq:GT-LBP} and \\eqref{eq:CT-LBP} are rewritten as\n\\begin{equation}\\label{eq:GT-CT-LBP}\n  \\bm p^*(x)=L\\bm p(x),\\quad\n  x\\bm p^*(x)=R\\bm p(x)+\\bm p_N(x),\n\\end{equation}\nwhere\n\\begin{equation*}\n  \\bm p^*(x)=\n  \\begin{pmatrix}\n    p_0^*(x)\\\\\n    p_1^*(x)\\\\\n    \\vdots\\\\\n    p_{N-1}^*(x)\n  \\end{pmatrix}.\n\\end{equation*}\nSince $L$ is a regular matrix, we obtain\n\\begin{equation}\\label{eq:next-LBP}\n  x\\bm p^*(x)=RL^{-1}\\bm p^*(x)+\\bm p_N(x).\n\\end{equation}\nThe LU factorization of $RL^{-1}$ generates bidiagonal matrices\n\\begin{equation*}\n  R^*\\coloneq\n  \\begin{pmatrix}\n    q_0^* & 1\\\\\n    & q_1^* & 1\\\\\n    && \\ddots & \\ddots\\\\\n    &&& \\ddots & 1\\\\\n    &&&& q_{N-1}^*\n  \\end{pmatrix},\\quad\n  L^*\\coloneq\n  \\begin{pmatrix}\n    1 \\\\\n    -e_0^* & 1\\\\\n    & -e_1^* & \\ddots\\\\\n    && \\ddots & \\ddots\\\\\n    &&& -e_{N-2}^* & 1\n  \\end{pmatrix},\n\\end{equation*}\nsatisfying\n\\begin{equation*}\n  RL^{-1}=(L^*)^{-1}R^*.\n\\end{equation*}\nThen, substituting this relation to \\eqref{eq:next-LBP},\nwe obtain\n\\begin{equation*}\n  R^*\\bm p^*(x)+\\bm p_N(x)=x L^* \\bm p^*(x).\n\\end{equation*}\nThis means that the monic polynomials $\\{p_n^*(x)\\}_{n=0}^N$ also satisfy\nthree-term recurrence relation of the same form as $\\{p_n(x)\\}_{n=0}^N$.\nTherefore $x_i$ and $\\bm p^*(x_i)$ are a generalized eigenvalue and eigenvector\nof the matrix pencil $(R^*, L^*)$.\nSuch $R^*$ and $L^*$ are computed by the relation\n\\begin{equation*}\n  L^*R=R^*L.\n\\end{equation*}\nEach element above gives\n\\begin{gather}\n  q_n-e_{n-1}^*=q_n^*-e_{n},\\quad\n  -q_ne_n^*=-q_{n+1}^*e_n\\label{eq:drToda}\n\\end{gather}\nfor $n=0, 1, \\dots, N-1$, where $e_{-1}=e_{-1}^*=e_{N-1}=e_{N-1}^*=0$.\nLet us introduce\n\\begin{equation*}\n  f_n\\coloneq q_n+e_n,\\quad n=0, 1, \\dots, N-2,\\quad\n  f_{N-1}\\coloneq q_{N-1}.\n\\end{equation*}\nThen the relations \\eqref{eq:drToda} yield\n\\begin{align*}\n  f_n\n  =q_n^*+e_{n-1}^*\n  =q_n^*+e_{n-1}\\frac{q_n^*}{q_{n-1}}\n  =\\frac{q_n^*}{q_{n-1}}(q_{n-1}+e_{n-1})\n  =f_{n-1}\\frac{q_n^*}{q_{n-1}}.\n\\end{align*}\nHence, we obtain\n\\begin{gather*}\n  f_n=q_n+e_n,\\quad n=0, 1, \\dots, N-2,\\quad f_{N-1}=q_{N-1},\\\\\n  q_0^*=f_0,\\quad q_n^*=q_{n-1}\\frac{f_n}{f_{n-1}},\\quad n=1, 2, \\dots, N-1,\\\\\n  e_n^*=e_n\\frac{f_{n+1}}{f_n},\\quad n=0, 1, \\dots, N-2.\n\\end{gather*}\n\nLet us iterate the procedure above.\nSet $p^{(0)}_n(x)\\coloneq p_n(x)$, $q_n^{(0)}\\coloneq q_n$ and $e_n^{(0)}\\coloneq e_n$.\nGenerate new monic polynomials $\\{p_n^{(k)}(x)\\}_{n=0}^N$, $k=1, 2, 3, \\dots$, by\n\\begin{gather}\n  p_0^{(k+1)}(x)\\coloneq 1,\\quad p^{(k+1)}_N(x)\\coloneq p_N(x),\\quad\n  p_n^{(k+1)}(x)\\coloneq p_n^{(k)}(x)-e_{n-1}^{(k)}p_{n-1}^{(k)}(x),\\quad\n  n=1, 2, \\dots, N-1,\\label{eq:GT-k-LBP}\n\\end{gather}\nwhere $\\{q_n^{(k)}\\}_{n=0}^{N-1}$ and $\\{e_n^{(k)}\\}_{n=0}^{N-2}$ are computed by\n\\begin{gather}\n  f_n^{(k)}=q_n^{(k)}+e_n^{(k)},\\quad n=0, 1, \\dots, N-2,\\quad f_{N-1}^{(k)}=q_{N-1}^{(k)},\\label{eq:drToda-d}\\\\\n  q_0^{(k+1)}=f_0^{(k)},\\quad q_n^{(k+1)}=q_{n-1}^{(k)}\\frac{f_n^{(k)}}{f_{n-1}^{(k)}},\\quad n=1, 2, \\dots, N-1,\\\\\n  e_n^{(k+1)}=e_n^{(k)}\\frac{f_{n+1}^{(k)}}{f_n^{(k)}},\\quad n=0, 1, \\dots, N-2.\\label{eq:drToda-e}\n\\end{gather}\nThe equations~\\eqref{eq:drToda-d}--\\eqref{eq:drToda-e} are called\nthe \\emph{discrete relativistic Toda lattice}.\nOne can see that this system is a variation of the differential qd (dqd) algorithm~\\cite{fernando1994asv,rutishauser1990lnm},\nwhich is the subtraction-free version of the quotient-difference algorithm used to\ncompute the eigenvalues of a tridiagonal matrix.\n\nThen $\\{p_n^{(k)}(x)\\}_{n=0}^N$ satisfy the relation\n\\begin{equation}\\label{eq:CT-k-LBP}\n  x p_n^{(k+1)}(x)=p_{n+1}^{(k)}(x)+q_n^{(k)}p_n^{(k)}(x),\\quad\n  n=0, 1, \\dots, N-1,\n\\end{equation}\nand the three-term recurrence relation\n\\begin{equation*}\n  p^{(k)}_{n+1}(x)=(x-q_n^{(k)})p_n^{(k)}(x)-x e_{n-1}^{(k)} p_{n-1}^{(k)}(x),\\quad\n  n=0, 1, \\dots, N-1.\n\\end{equation*}\n\n\\subsection{Isospectral transformation to a tridiagonal matrix}\nSubtraction of \\eqref{eq:GT-k-LBP} with $n\\to n+1$ from \\eqref{eq:CT-k-LBP} and $k\\to k+n-1$ yield\n\\begin{equation}\\label{eq:CT-k-OPS}\n  x p_{n}^{(k+n)}(x)=p_{n+1}^{(k+n)}(x)+f_n^{(k+n-1)} p_n^{(k+n-1)}(x),\\quad\n  n=0, 1, \\dots, N-1.\n\\end{equation}\nLet us introduce variables\n\\begin{equation*}\n  \\hat q_n^{(k)}\\coloneq f_n^{(k+n-1)},\\quad\n  \\hat e_n^{(k)}\\coloneq e_n^{(k+n)},\n\\end{equation*}\nbidiagonal matrices\n\\begin{equation}\\label{eq:bidiagonal-OPS}\n  \\hat R^{(k)}\\coloneq\n  \\begin{pmatrix}\n    \\hat q_0^{(k)} & 1\\\\\n    & \\hat q_1^{(k)} & 1\\\\\n    && \\ddots & \\ddots\\\\\n    &&& \\ddots & 1\\\\\n    &&&& \\hat q_{N-1}^{(k)}\n  \\end{pmatrix},\\quad\n  \\hat L^{(k)}\\coloneq\n  \\begin{pmatrix}\n    1\\\\\n    \\hat e_0^{(k)} & 1\\\\\n    & \\hat e_1^{(k)} & \\ddots\\\\\n    && \\ddots & \\ddots \\\\\n    &&& \\hat e_{N-2}^{(k)} & 1\n  \\end{pmatrix},\n\\end{equation}\nand vectors of polynomials\n\\begin{equation}\\label{eq:vector-OPS}\n  \\hat{\\bm p}^{(k)}(x)\\coloneq\n  \\begin{pmatrix}\n    \\hat p_0^{(k)}(x)\\\\\n    \\hat p_1^{(k)}(x)\\\\\n    \\vdots\\\\\n    \\hat p_{N-1}^{(k)}(x)\n  \\end{pmatrix},\n\\end{equation}\nwhere\n\\begin{equation*}\n  \\hat p_n^{(k)}(x)\\coloneq p_n^{(k+n-1)}(x).\n\\end{equation*}\nThen the relations~\\eqref{eq:GT-k-LBP} and \\eqref{eq:CT-k-OPS} yield\n\\begin{equation}\\label{eq:GT-CT-OPS}\n  \\hat{\\bm p}^{(k-1)}(x)=\\hat L^{(k-1)}\\hat{\\bm p}^{(k)}(x),\\quad\n  x\\hat{\\bm p}^{(k+1)}(x)=\\hat R^{(k)}\\hat{\\bm p}^{(k)}(x)+\\bm p_N(x).\n\\end{equation}\nHence, $\\hat{\\bm p}^{(k)}(x)$ satisfies\n\\begin{equation}\\label{eq:trr-OPS-vec}\n  \\hat T^{(k)}\\hat{\\bm p}^{(k)}(x)+\\bm p_N(x)=x\\hat{\\bm p}^{(k)}(x),\n\\end{equation}\nwhere $\\hat T^{(k)}$ is a tridiagonal matrix\n\\begin{equation*}\n  \\hat T^{(k)}\\coloneq \\hat L^{(k)}\\hat R^{(k)}=\\hat R^{(k-1)}\\hat L^{(k-1)}.\n\\end{equation*}\nTherefore, Favard's theorem says that there exists a linear functional\n$\\mathcal L^{(k)}\\colon \\mathbb C[x]\\to \\mathbb C$ satisfying\nthe orthogonality condition\n\\begin{equation*}\n  \\mathcal L^{(k)}[\\hat p_m^{(k)}(x)\\hat p_n^{(k)}(x)]=\\hat h_n^{(k)} \\delta_{m, n},\\quad\n  \\hat h_n^{(k)}\\ne 0,\\quad m, n=0, 1, \\dots, N-1,\n\\end{equation*}\nwhere $\\delta_{m, n}$ is the Kronecker delta, and the terminating condition\n\\begin{equation*}\n  \\mathcal L^{(k)}[p_N(x)\\pi(x)]=0 \\quad \\text{for all $\\pi(x)\\in\\mathbb C[x]$}.\n\\end{equation*}\nNote that the orthogonality relation is equivalent to\n\\begin{equation}\\label{eq:orthogonality-OPS}\n  \\mathcal L^{(k)}[x^m \\hat p_n^{(k)}(x)]=h_n^{(k)}\\delta_{m, n},\\quad\n  h_n^{(k)}\\ne 0,\\quad\n  n=0, 1, \\dots, N-1,\\quad m=0, 1, \\dots, n.\n\\end{equation}\nThe linear functional $\\mathcal L^{(k)}$ is concretely given by~\\cite{akhiezer1965cmp}\n\\begin{equation}\\label{eq:def-lf-tridiagonal-OPS}\n  \\mathcal L^{(k)}[\\pi(x)]=\\bm e_0^{\\mathrm T}\\pi(\\hat T^{(k)})\\bm e_0\\quad\n  \\text{for all $\\pi(x)\\in\\mathbb C[x]$},\n\\end{equation}\nwhere\n\\begin{equation*}\n  \\bm e_0\\coloneq\n  \\begin{pmatrix}\n    1\\\\\n    0\\\\\n    \\vdots\\\\\n    0\n  \\end{pmatrix}\\in \\mathbb C^N.\n\\end{equation*}\nHence, if the eigenvalues $x_0, x_1, \\dots, x_{N-1}$ of $\\hat T^{(k)}$,\nwhich are the zeros of $p_N(x)$,\nare all simple, then there exist some constants $w_0^{(k)}, w_1^{(k)}, \\dots, w_{N-1}^{(k)} \\in \\mathbb C$\nsuch that\n\\begin{equation*}\n  \\mathcal L^{(k)}[\\pi(x)]=\\sum_{i=0}^{N-1} \\pi(x_i)w_i^{(k)}\\quad \\text{for all $\\pi(x)\\in\\mathbb C[x]$}.\n\\end{equation*}\nThis means that $\\{\\hat p_n^{(k)}(x)\\}_{n=0}^N$ is\nthe \\emph{monic discrete orthogonal polynomial sequence} with respect to $\\mathcal L^{(k)}$.\n\nIn the theory of orthogonal polynomials, the relations~\\eqref{eq:CT-k-OPS} and \\eqref{eq:GT-k-LBP}\nare called \\emph{Christoffel transformation} and \\emph{Geronimus transformation}, respectively~\\cite{spiridonov1995ddt},\nand the relation\n\\begin{equation*}\n  \\mathcal L^{(k+1)}[\\pi(x)]=\\mathcal L^{(k)}[x\\pi(x)]\\quad \\text{for all $\\pi(x)\\in\\mathbb C[x]$,}\n\\end{equation*}\nis shown. Let us introduce the moment of the linear functional $\\mathcal L^{(k)}$ as\n\\begin{equation}\\label{eq:def-moment}\n  \\mu_{k+m}\\coloneq \\mathcal L^{(k)}[x^m]=\\mathcal L^{(0)}[x^{k+m}].\n\\end{equation}\nThen, from the orthogonality relation~\\eqref{eq:orthogonality-OPS},\nthe determinant expression of the orthogonal polynomials $\\{\\hat p_n^{(k)}(x)\\}_{n=0}^N$\nis given by\n\\begin{equation*}\n  \\hat p_n^{(k)}(x)=\\frac{1}{\\tau^{(k)}_n}\n  \\begin{vmatrix}\n    \\mu_{k} & \\mu_{k+1} & \\dots & \\mu_{k+n-1} & \\mu_{k+n}\\\\\n    \\mu_{k+1} & \\mu_{k+2} & \\dots & \\mu_{k+n} & \\mu_{k+n+1}\\\\\n    \\vdots & \\vdots & & \\vdots & \\vdots\\\\\n    \\mu_{k+n-1} & \\mu_{k+n} & \\dots & \\mu_{k+2n-2} & \\mu_{k+2n-1}\\\\\n    1 & x & \\dots & x^{n-1} & x^n\n  \\end{vmatrix}, \\quad n=1, 2, \\dots, N,\n\\end{equation*}\nwhere $\\tau_n^{(k)}$ is the Hankel determinant of the moments\n\\begin{equation*}\n  \\tau_0^{(k)}\\coloneq 1,\\quad\n  \\tau_n^{(k)}\\coloneq |\\mu_{k+i+j}|_{i, j=0}^{n-1},\\quad\n  n=1, 2, 3, \\dots.\n\\end{equation*}\n\nNext, we consider the determinant expression of $\\{p_n^{(k)}(x)\\}_{n=0}^N$:\n\\begin{equation*}\n  p_n^{(k)}(x)=\\hat p_n^{(k-n+1)}(x)=\\frac{1}{\\tau^{(k-n+1)}_n}\n  \\begin{vmatrix}\n    \\mu_{k-n+1} & \\mu_{k-n+2} & \\dots & \\mu_{k} & \\mu_{k+1}\\\\\n    \\mu_{k-n+2} & \\mu_{k-n+3} & \\dots & \\mu_{k+1} & \\mu_{k+2}\\\\\n    \\vdots & \\vdots & & \\vdots & \\vdots\\\\\n    \\mu_{k} & \\mu_{k+1} & \\dots & \\mu_{k+n-1} & \\mu_{k+n}\\\\\n    1 & x & \\dots & x^{n-1} & x^n\n  \\end{vmatrix}, \\quad n=1, 2, \\dots, N.\n\\end{equation*}\nSince we can extend the domain of the linear functional $\\mathcal L^{(k)}$ for\n$\\mathbb C[x, x^{-1}]$ by \\eqref{eq:def-lf-tridiagonal-OPS},\nthis determinant expression leads to the biorthogonality relation\n\\begin{equation*}\n  \\mathcal L^{(k)}[x^{-m} p_n^{(k)}(x)]=(-1)^n\\frac{\\tau_{n+1}^{(k-n)}}{\\tau_n^{(k-n+1)}} \\delta_{m, n},\\quad\n  n=0, 1, \\dots, N-1,\\quad m=0, 1, \\dots, n.\n\\end{equation*}\nThe monic polynomials $\\{p_n^{(k)}(x)\\}_{n=0}^N$ are known as\nthe \\emph{Laurent biorthogonal polynomials} with respect to $\\mathcal L^{(k)}$.\nThen, the relations \\eqref{eq:CT-k-LBP}, \\eqref{eq:GT-k-LBP}, \\eqref{eq:CT-k-OPS} and\n\\begin{equation*}\n  p_n^{(k)}(0)=(-1)^n \\frac{\\tau_n^{(k-n+2)}}{\\tau_n^{(k-n+1)}}\n\\end{equation*}\ngive\n\\begin{gather*}\n  q_n^{(k)}\n  =-\\frac{p_{n+1}^{(k)}(0)}{p_{n}^{(k)}(0)}\n  =\\frac{\\tau_{n}^{(k-n+1)}\\tau_{n+1}^{(k-n+1)}}{\\tau_{n}^{(k-n+2)}\\tau_{n+1}^{(k-n)}},\\\\\n  e_n^{(k)}\n  =-\\frac{\\mathcal L^{(k+1)}[x^{-n-1}p_{n+1}^{(k+1)}(x)]}{\\mathcal L^{(k)}[x^{-n}p_{n}^{(k)}(x)]}\n  =\\frac{\\tau_n^{(k-n+1)}\\tau_{n+2}^{(k-n)}}{\\tau_{n+1}^{(k-n)}\\tau_{n+1}^{(k-n+1)}},\\\\\n  f_n^{(k)}\n  =-\\frac{p_{n+1}^{(k+1)}(0)}{p_n^{(k)}(0)}\n  =\\frac{\\tau_{n}^{(k-n+1)}\\tau_{n+1}^{(k-n+2)}}{\\tau_{n}^{(k-n+2)}\\tau_{n+1}^{(k-n+1)}},\n\\end{gather*}\nand\n\\begin{equation*}\n  \\hat q_n^{(k)}=f_n^{(k+n-1)}=\\frac{\\tau_{n}^{(k)}\\tau_{n+1}^{(k+1)}}{\\tau_{n}^{(k+1)}\\tau_{n+1}^{(k)}},\\quad\n  \\hat e_n^{(k)}=e_n^{(k+n)}=\\frac{\\tau_n^{(k+1)}\\tau_{n+2}^{(k)}}{\\tau_{n+1}^{(k)}\\tau_{n+1}^{(k+1)}}.\n\\end{equation*}\n\nSummarizing the above, we obtain Algorithm~\\ref{alg:bi-bi-to-tri} of the isospectral transformation.\n\n\\begin{algorithm}[t]\n  \\caption{Isospectral transformation from bidiagonal--bidiagonal matrix pencil to tridiagonal matrix}\n  \\label{alg:bi-bi-to-tri}\n\\begin{algorithmic}\n  \\REQUIRE $\\{q_n^{(0)}\\}_{n=0}^{N-1}$ and $\\{e_n^{(0)}\\}_{n=0}^{N-2}$ (or bidiagonal matrices $R^{(0)}$ and $L^{(0)}$)\n  \\FOR{$k=0$ to $N-1$}\n  \\FOR{$n=0$ to $N-1$}\n  \\STATE\\IfThenElse{$n<N-1$}\n                   {$f_n^{(k)} \\leftarrow q_n^{(k)}+e_n^{(k)}$}\n                   {$f_n^{(k)} \\leftarrow q_n^{(k)}$}\n  \\ENDFOR\n  \\FOR{$n=0$ to $N-1$}\n  \\STATE\\IfThenElse{$n=0$}\n                   {$q_n^{(k+1)} \\leftarrow f_n^{(k)}$}\n                   {$q_n^{(k+1)} \\leftarrow q_{n-1}^{(k)}\\frac{f_n^{(k)}}{f_{n-1}^{(k)}}$}\n  \\STATE\\IfThen{$n<N-1$}\n               {$e_n^{(k+1)} \\leftarrow e_n^{(k)}\\frac{f_{n+1}^{(k)}}{f_n^{(k)}}$}\n  \\ENDFOR\n  \\ENDFOR\n  \\FOR{$n=0$ to $N-1$}\n  \\STATE $\\hat q_n^{(1)} \\leftarrow f_n^{(n)}$\n  \\STATE\\IfThen{$n<N-1$}\n               {$\\hat e_n^{(1)} \\leftarrow e_n^{(n+1)}$}\n  \\ENDFOR\n  \\ENSURE $\\{\\hat q_n^{(1)}\\}_{n=0}^{N-1}$ and $\\{\\hat e_n^{(1)}\\}_{n=0}^{N-2}$ (or tridiagonal matrix $\\hat T^{(1)}=\\hat L^{(1)}\\hat R^{(1)}$)\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Numerical example}\nLet us consider\n\\begin{equation*}\n  R^{(0)}=\n  \\begin{pmatrix}\n    1 & 1 & 0 & 0 & 0\\\\\n    0 & 2 & 1 & 0 & 0\\\\\n    0 & 0 & 3 & 1 & 0\\\\\n    0 & 0 & 0 & 4 & 1\\\\\n    0 & 0 & 0 & 0 & 5\n  \\end{pmatrix},\\quad\n  L^{(0)}=\n  \\begin{pmatrix}\n    1 & 0 & 0 & 0 & 0\\\\\n    -6 & 1 & 0 & 0 & 0\\\\\n    0 & -7 & 1 & 0 & 0\\\\\n    0 & 0 & -8 & 1 & 0\\\\\n    0 & 0 & 0 & -9 & 1\n  \\end{pmatrix}.\n\\end{equation*}\nThen, Algorithm~\\ref{alg:bi-bi-to-tri} yields\n\\begin{gather*}\n  \\hat R^{(1)}=\n  \\begin{pmatrix}\n    7& 1& 0& 0& 0\\\\\n    0& \\frac{620}{63}& 1& 0& 0\\\\\n    0& 0& \\frac{41949}{5890}& 1& 0\\\\\n    0& 0& 0& \\frac{5722439}{7639379}& 1\\\\\n    0& 0& 0& 0& \\frac{98340}{301181}\n  \\end{pmatrix},\\quad\n  \\hat L^{(1)}=\n  \\begin{pmatrix}\n    1& 0& 0& 0& 0\\\\\n    \\frac{54}{7}& 1& 0& 0& 0\\\\\n    0& \\frac{931}{90}& 1& 0& 0\\\\\n    0& 0& \\frac{4720320}{2745329}& 1& 0\\\\\n    0& 0& 0& \\frac{90306875}{493635659}& 1\n  \\end{pmatrix},\n\\end{gather*}\nand\n\\begin{gather*}\n  \\hat T^{(1)}\n  =\\hat L^{(1)}\\hat R^{(1)}\n  =\n  \\begin{pmatrix}\n    7& 1& 0& 0& 0\\\\[.5em]\n    54& \\frac{158}{9}& 1& 0& 0\\\\[.5em]\n    0& \\frac{8246}{81}& \\frac{92590}{5301}& 1& 0\\\\[.5em]\n    0& 0& \\frac{4248288}{346921}& \\frac{2382991}{965371} & 1\\\\[.5em]\n    0& 0& 0& \\frac{368125}{2686321} & \\frac{835}{1639}\n  \\end{pmatrix}\n\\end{gather*}\nThe computation above was executed by SymPy (a symbolic computation library for Python).\nThe eigenvalues of both $(R^{(0)}, L^{(0)})$ and $\\hat T^{(1)}$ are\n29.10515103, 12.22484344, 2.82190399, 0.17848385 and 0.66961769,\nwhich are verified by \\texttt{numpy.linalg.eig()}.\n\n\\section{Isospectral transformation between tridiagonal matrix and tridiagonal--bidiagonal matrix pencil}\n\\label{sec:transf-betw-trid-1}\n\n\\subsection{Sequence of tridiagonal--bidiagonal GEVPs and eigenvectors}\nWe first fix a vector\n$\\bm{\\varepsilon} = (\\varepsilon_0,\\varepsilon_1,\\ldots,\\varepsilon_{N-2}) \\in \\{0,1\\}^{N-1}$.\nLet $\\{p_n^{(k)}(x)\\}_{n=0}^N$ be monic polynomials of degree $n$ defined by\nthe three-term recurrence relation\n\\begin{gather}\n  p_{-1}^{(k)}(x)\\coloneq 0,\\quad p_0^{(k)}(x)\\coloneq 1,\\nonumber\\\\\n  \\begin{multlined}[b]\n    p_{n+1}^{(k)}(x)\\coloneq (x-q_n^{(k)}-(1-\\epsilon_{n-1})e_{n-1}^{(k)})p_n^{(k)}(x)-(\\epsilon_{n-1}x+(1-\\epsilon_{n-1})q_{n-1}^{(k)})e_{n-1}^{(k)}p_{n-1}^{(k)}(x),\\\\\n    n=0, 1, \\dots, N-1,\n  \\end{multlined}\\label{eq:trr-e-BLP}\n\\end{gather}\nwhere $e_{-1}\\coloneq 0$.\nLet us consider lower bidiagonal matrices\n\\begin{equation}\\label{eq:def-L-epsilon}\n  L_{(\\xi_0,\\ldots,\\xi_{N-2})}^{(k)}\\coloneq\n  \\begin{pmatrix}\n    1\\\\\n    -\\xi_0 e_0^{(k)} & 1\\\\\n    & -\\xi_1 e_1^{(k)} & \\ddots\\\\\n    && \\ddots & \\ddots\\\\\n    &&& -\\xi_{N-2} e_{N-2}^{(k)} & 1\n  \\end{pmatrix},\n\\end{equation}\nupper bidiagonal matrices\n\\begin{equation*}\n  R^{(k)}\\coloneq\n  \\begin{pmatrix}\n    q_0^{(k)} & 1\\\\\n    & q_1^{(k)} & 1\\\\\n    && \\ddots & \\ddots\\\\\n    &&& \\ddots & 1\\\\\n    &&&& q_{N-1}^{(k)}\n  \\end{pmatrix},\n\\end{equation*}\nand vectors of polynomials which are the same as \\eqref{eq:def-vec-p}.\nThen we can rewrite the three-term recurrence relation~\\eqref{eq:trr-e-BLP} as\n\\begin{equation}\\label{eq:trr-e-BLP-vec}\n  L_{\\bm{\\varepsilon}^*}^{(k)}R^{(k)}\\bm p^{(k)}(x)+\\bm p_N(x)=x L_{\\bm\\varepsilon}^{(k)}\\bm p^{(k)}(x),\n\\end{equation}\nwhere $\\bm{\\varepsilon}^*\\coloneq\\bm{\\varepsilon}-(1,1,\\ldots,1)$.\n\n\\begin{example} When $N=5, \\bm\\varepsilon=(1,1,0,1)$,\n  we obtain\n  \\begin{equation*}\n    L_{\\bm{\\varepsilon}^*}^{(k)} =\n    \\begin{pmatrix}\n      1 \\\\\n      0 & 1\\\\\n      & 0 & 1\\\\\\\n      && e_2^{(k)} & 1\\\\\n      &&& 0 & 1\n    \\end{pmatrix}, \\quad\n    L_{\\bm{\\varepsilon}}^{(k)} =\n    \\begin{pmatrix}\n      1 \\\\\n      -e_0^{(k)} & 1\\\\\n      & -e_1^{(k)} & 1\\\\\\\n      && 0 & 1\\\\\n      &&& -e_3^{(k)} & 1\n    \\end{pmatrix}.\n  \\end{equation*}\n\\end{example}\n\nNote that\n\\begin{itemize}\n\\item if $\\bm\\varepsilon=(0, 0, \\dots, 0)$, then \\eqref{eq:trr-e-BLP-vec} becomes\n  the three-term recurrence relation of monic orthogonal polynomials (see \\eqref{eq:trr-OPS-vec});\n\\item if $\\bm\\varepsilon=(1, 1, \\dots, 1)$, then \\eqref{eq:trr-e-BLP-vec} becomes\n  the three-term recurrence relation of monic Laurent biorthogonal polynomials (see \\eqref{eq:trr-LBP-vec}).\n\\end{itemize}\nHence the monic polynomial sequence $\\{p_n^{(k)}(x)\\}_{n=0}^N$ is a generalization\nof both the orthogonal polynomials and the Laurent biorthogonal polynomials.\nWe therefore use the notation that is almost the same as that of the previous section.\n\nLet us introduce new monic polynomials $\\{p_n^{(k+1)}(x)\\}_{n=0}^N$ of degree $n$\ngenerated from the monic polynomials $\\{p_n^{(k)}(x)\\}_{n=0}^N$ by\n\\begin{equation*}\n  \\bm p^{(k+1)}(x)\\coloneq \\left(L_{\\bm{\\varepsilon}^*}^{(k)}\\right)^{-1}L_{\\bm{\\varepsilon}}^{(k)}\\bm p^{(k)}(x).\n\\end{equation*}\nSubstituting $\\bm p^{(k)}(x)=\\left(L_{\\bm{\\varepsilon}}^{(k)}\\right)^{-1}L_{\\bm{\\varepsilon}^*}^{(k)}\\bm p^{(k+1)}(x)$ into\n\\eqref{eq:trr-e-BLP-vec}, we obtain\n\\begin{equation}\\label{eq:GT-CT-BLP}\n  L_{\\bm{\\varepsilon}^*}^{(k)}\\bm p^{(k+1)}(x)=L_{\\bm\\varepsilon}^{(k)}\\bm p^{(k)}(x),\\quad\n  R^{(k)}\\bm p^{(k)}(x)+\\bm p_N(x)=x\\bm p^{(k+1)}(x).\n\\end{equation}\nThis is a generalization of \\eqref{eq:GT-CT-LBP} and \\eqref{eq:GT-CT-OPS}.\nElimination of $\\bm p(x)$ yields\n\\begin{equation*}\n  R^{(k)}\\left(L_{\\bm{\\varepsilon}}^{(k)}\\right)^{-1}L_{\\bm{\\varepsilon}^*}^{(k)}\\bm p^{(k+1)}(x)+\\bm p_N(x)=x\\bm p^{(k+1)}(x).\n\\end{equation*}\nThe LU factorization generates bidiagonal matrices\n$L_{\\bm{\\varepsilon}}^{(k+1)}$, $L_{\\bm{\\varepsilon}^*}^{(k+1)}$, $\\tilde R^{(k)}$ and $R^{(k+1)}$ satisfying\n\\begin{equation}\\label{eq:de-Toda}\n  L_{\\bm{\\varepsilon}}^{(k+1)}R^{(k)}=\\tilde R^{(k)} L_{\\bm{\\varepsilon}}^{(k)},\\quad\n  \\tilde R^{(k)} L_{\\bm{\\varepsilon}^*}^{(k)}=L_{\\bm{\\varepsilon}^*}^{(k+1)}R^{(k+1)}.\n\\end{equation}\nThen, we obtain\n\\begin{equation*}\n  L_{\\bm{\\varepsilon}^*}^{(k+1)}R^{(k+1)}\\bm p^{(k+1)}(x)+\\bm p_N(x)=x L_{\\bm{\\varepsilon}}^{(k+1)}\\bm p^{(k+1)}(x).\n\\end{equation*}\nEach element of \\eqref{eq:de-Toda} give the relations\n\\begin{gather*}\n  q_n^{(k)}-\\epsilon_{n-1}e_{n-1}^{(k+1)}=\\tilde q_n^{(k)}-\\epsilon_n e_n^{(k)},\\quad\n  \\tilde q_n^{(k)}+(1-\\epsilon_n) e_n^{(k)}=q_n^{(k+1)}+(1-\\epsilon_{n-1})e_{n-1}^{(k+1)}\n\\end{gather*}\nfor $n=0, 1, \\dots, N-1$, and\n\\begin{gather*}\n  q_n^{(k)} e_n^{(k+1)}=\\tilde q_{n+1}^{(k)} e_n^{(k)} \\quad \\text{if $\\epsilon_n=1$},\\\\\n  \\tilde q_{n+1}^{(k)} e_n^{(k)}=q_n^{(k+1)} e_n^{(k+1)} \\quad \\text{if $\\epsilon_n=0$}\n\\end{gather*}\nfor $n=0, 1, \\dots, N-2$.\nElimination of $\\tilde q_n^{(k)}$ yields\n\\begin{gather}\n  q_n^{(k)}+e_n^{(k)}=q_n^{(k+1)}+e_{n-1}^{(k+1)},\\label{eq:de-Toda-p}\\\\\n  q_n^{(k)}e_n^{(k+1)}=(q_{n+1}^{(k+1)}-(1-\\epsilon_{n+1})e_{n+1}^{(k)})e_n^{(k)}\\quad \\text{if $\\epsilon_n=1$}\\label{eq:de-Toda-m1},\\\\\n  (q_{n+1}^{(k)}+\\epsilon_{n+1}e_{n+1}^{(k)})e_n^{(k)}=q_n^{(k+1)}e_n^{(k+1)}\\quad \\text{if $\\epsilon_n=0$}.\\label{eq:de-Toda-m2}\n\\end{gather}\nFurther, substituting \\eqref{eq:de-Toda-p} to \\eqref{eq:de-Toda-m1} and \\eqref{eq:de-Toda-m2},\nwe can see that\n\\begin{gather}\n  (q_n^{(k)}-(1-\\epsilon_n)e_{n-1}^{(k+1)})e_n^{(k+1)}=(q_{n+1}^{(k+1)}-(1-\\epsilon_{n+1})e_{n+1}^{(k)})e_n^{(k)},\\label{eq:de-Toda-m3}\\\\\n  (q_n^{(k+1)}+\\epsilon_{n}e_{n-1}^{(k+1)})e_n^{(k+1)}=(q_{n+1}^{(k)}+\\epsilon_{n+1}e_{n+1}^{(k)})e_n^{(k)}\\label{eq:de-Toda-m4}\n\\end{gather}\nhold in any case.\nLet us introduce\n\\begin{gather*}\n  d_0^{(k+1)}\\coloneq q_0^{(k)}+\\epsilon_0 e_0^{(k)},\\quad\n  d_n^{(k+1)}\\coloneq q_n^{(k+1)}-(1-\\epsilon_n) e_{n}^{(k)},\\quad n=1, \\dots, N-1,\\\\\n  f_n^{(k)}\\coloneq q_n^{(k)}+\\epsilon_n e_n^{(k)},\\quad n=0, 1, \\dots, N-2,\\quad\n  f_{N-1}^{(k)}\\coloneq q_{N-1}^{(k)}.\n\\end{gather*}\nThen, from \\eqref{eq:de-Toda-p}, \\eqref{eq:de-Toda-m3} and \\eqref{eq:de-Toda-m4}, we obtain\n\\begin{align*}\n  d_n^{(k+1)}\n  &=q_n^{(k)}+\\epsilon_n e_n^{(k)}-e_{n-1}^{(k+1)}\\\\\n  &=\\frac{f_n^{(k)}}{q_{n-1}^{(k+1)}+\\epsilon_{n-1}e_{n-2}^{(k+1)}}(q_{n-1}^{(k+1)}+\\epsilon_{n-1}e_{n-2}^{(k+1)}-e_{n-1}^{(k)})\\\\\n  &=\\frac{f_n^{(k)}}{q_{n-1}^{(k+1)}+\\epsilon_{n-1}e_{n-2}^{(k+1)}}(d_{n-1}^{(k+1)}+\\epsilon_{n-1}(e_{n-2}^{(k+1)}-e_{n-1}^{(k)})),\\\\\n  f_n^{(k)}\n  &=q_n^{(k+1)}-(1-\\epsilon_n)e_n^{(k)}+e_{n-1}^{(k+1)}\\\\\n  &=\\frac{d_n^{(k+1)}}{q_{n-1}^{(k)}-(1-\\epsilon_{n-1})e_{n-2}^{(k+1)}}(q_{n-1}^{(k)}-(1-\\epsilon_{n-1})e_{n-2}^{(k+1)}+e_{n-1}^{(k)})\\\\\n  &=\\frac{d_n^{(k+1)}}{q_{n-1}^{(k)}-(1-\\epsilon_{n-1})e_{n-2}^{(k+1)}}(f_{n-1}^{(k)}-(1-\\epsilon_{n-1})(e_{n-2}^{(k+1)}-e_{n-1}^{(k)})).\n\\end{align*}\nHence, we can compute $\\{q_n^{(k+1)}\\}_{n=0}^{N-1}$ and $\\{e_n^{(k+1)}\\}_{n=0}^{N-2}$\nfrom $\\{q_n^{(k)}\\}_{n=0}^{N-1}$ and $\\{e_n^{(k)}\\}_{n=0}^{N-2}$ by\n\\begin{gather*}\n  f_n^{(k)}=q_n^{(k)}+\\epsilon_n e_n^{(k)},\\quad n=0, 1, \\dots, N-2,\\quad\n  f_{N-1}^{(k)}=q_{N-1}^{(k)},\\\\\n  d_0^{(k+1)}=f_0^{(k)},\\quad\n  d_n^{(k+1)}=\n  \\begin{dcases*}\n    d_{n-1}^{(k+1)}\\frac{f_n^{(k)}}{q_{n-1}^{(k+1)}} & if $\\epsilon_{n-1}=0$,\\\\\n    q_{n-1}^{(k)}\\frac{f_n^{(k)}}{f_{n-1}^{(k)}} & if $\\epsilon_{n-1}=1$,\n  \\end{dcases*}\\quad\n  n=1, 2, \\dots, N-1,\\\\\n  q_n^{(k+1)}=d_n^{(k+1)}+(1-\\epsilon_n)e_n^{(k)},\\quad n=0, 1, \\dots, N-1,\\\\\n  e_n^{(k+1)}=e_n^{(k)}\\frac{f_{n+1}^{(k)}}{q_n^{(k+1)}+\\epsilon_n e_{n-1}^{(k+1)}},\\quad\n  n=0, 1, \\dots, N-2.\n\\end{gather*}\nWe call this system the \\emph{discrete elementary Toda orbits}~\\cite{kobayashi2021nde}.\n\n\\subsection{Isospectral transformation to a tridiagonal matrix}\nEach element of \\eqref{eq:GT-CT-BLP} gives\n\\begin{alignat}{2}\n  xp_{n}^{(k+1)}(x)&=p_{n+1}^{(k)}(x)+q_n^{(k)}p_n^{(k)}(x),\\label{eq:e-BLP-st-q}\\\\\n  p_{n+1}^{(k-1)}(x)&=p_{n+1}^{(k)}(x)+e_n^{(k-1)}p_n^{(k)}(x) &\\quad& \\text{if $\\epsilon_n=0$},\\label{eq:e-BLP-st-e0}\\\\\n  p_{n+1}^{(k+1)}(x)&=p_{n+1}^{(k)}(x)-e_n^{(k)}p_n^{(k)}(x) &\\quad& \\text{if $\\epsilon_n=1$}.\\label{eq:e-BLP-st-e1}\\\\\n\\intertext{Therefore, we obtain}\n  xp_{n}^{(k+1)}(x)&=p_{n+1}^{(k-1)}(x)+d_n^{(k)}p_n^{(k)}(x) &\\quad& \\text{if $\\epsilon_n=0$},\\label{eq:e-BLP-st-d0}\\\\\n  xp_{n}^{(k+1)}(x)&=p_{n+1}^{(k+1)}(x)+f_n^{(k)}p_n^{(k)}(x) && \\text{if $\\epsilon_n=1$}.\\label{eq:e-BLP-st-f1}\n\\end{alignat}\nSince $d_n^{(k)}=q_n^{(k)}$ if $\\epsilon_n=1$ and $f_n^{(k)}=q_n^{(k)}$ if $\\epsilon_n=0$,\nwe can see that\n\\begin{gather*}\n  xp_n^{(k+1)}(x)=p_{n+1}^{(k+\\epsilon_n-1)}(x)+d_n^{(k)}p_n^{(k)}(x),\\\\\n  xp_n^{(k+1)}(x)=p_{n+1}^{(k+\\epsilon_n)}(x)+f_n^{(k)}p_n^{(k)}(x)\n\\end{gather*}\nhold in any case.\nIt also holds in any case that\n\\begin{equation*}\n  p_n^{(k)}(x)=p_n^{(k+1)}(x)+e_{n-1}^{(k)} p_{n-1}^{(k+1-\\epsilon_{n-1})}(x).\n\\end{equation*}\nThese relations lead to the three-term recurrence relation\n\\begin{equation}\\label{eq:trr-e-BLP-to-OPS-pre}\n  xp_n^{(k)}(x)=p_{n+1}^{(k+\\epsilon_n)}(x)+(f_n^{(k)}+e_{n-1}^{(k)})p_n^{(k)}(x)+f_{n-1}^{(k-\\epsilon_{n-1})}e_{n-1}^{(k)}p_{n-1}^{(k-\\epsilon_{n-1})}(x).\n\\end{equation}\n\nLet us introduce $\\bm{\\eta}=(\\eta_0,\\eta_1,\\ldots,\\eta_{N-1})$ by\n\\begin{equation*}\n  \\eta_0\\coloneq 0,\\quad\n  \\eta_n\\coloneq \\sum_{j=0}^{n-1}\\epsilon_j,\\quad\n  n=1, 2, \\dots, N-1,\n\\end{equation*}\nvariables\n\\begin{equation*}\n  \\hat q_n^{(k)}\\coloneq f_n^{(k+\\eta_n)},\\quad\n  \\hat e_n^{(k)}\\coloneq e_n^{(k+\\eta_{n+1})},\n\\end{equation*}\nand polynomials\n\\begin{equation*}\n  \\hat p_n^{(k)}(x)\\coloneq p_n^{(k+\\eta_n)}(x).\n\\end{equation*}\nThen the relation \\eqref{eq:trr-e-BLP-to-OPS-pre} yields\n\\begin{equation*}\n  \\hat T^{(k)}\\hat{\\bm p}^{(k)}(x)+\\bm p_N(x)=x\\hat{\\bm p}^{(k)}(x),\n\\end{equation*}\nthat is the same as \\eqref{eq:trr-OPS-vec};\nthe monic polynomial sequence $\\{\\hat p_n^{(k)}(x)\\}_{n=0}^N$ is\nthe orthogonal polynomial sequence with respect to the linear functional\n$\\mathcal L^{(k)}$ defined by \\eqref{eq:def-lf-tridiagonal-OPS}.\nThis fact gives the determinant expression of $\\{p_n^{(k)}(x)\\}_{n=0}^N$:\n\\begin{equation*}\n  p_n^{(k)}(x)=\\frac{1}{\\tau_n^{(k-\\eta_n)}}\n  \\begin{vmatrix}\n    \\mu_{k-\\eta_n} & \\mu_{k-\\eta_n+1} & \\dots & \\mu_{k-\\eta_n+n-1} & \\mu_{k-\\eta_n+n}\\\\\n    \\mu_{k-\\eta_n+1} & \\mu_{k-\\eta_n+2} & \\dots & \\mu_{k-\\eta_n+n} & \\mu_{k-\\eta_n+n+1}\\\\\n    \\vdots & \\vdots & & \\vdots & \\vdots\\\\\n    \\mu_{k-\\eta_n+n-1} & \\mu_{k-\\eta_n+n} & \\dots & \\mu_{k-\\eta_n+2n-2} & \\mu_{k-\\eta_n+2n-1}\\\\\n    1 & x & \\dots & x^{n-1} & x^n\n  \\end{vmatrix},\\quad\n  n=1, 2, \\dots, N.\n\\end{equation*}\nHence, $\\{p_n^{(k)}(x)\\}_{n=0}^N$ satisfies the $\\epsilon$-biorthogonality relation\nfor $n=0, 1, \\dots, N-1$:\n\\begin{equation*}\n  \\mathcal L^{(k)}[x^m p_n^{(k)}(x)]=0,\\quad  m=-\\eta_n, -\\eta_n+1, \\dots, -\\eta_n+n-1,\n\\end{equation*}\nand\n\\begin{equation*}\n  \\mathcal L^{(k)}[x^{-\\eta_n+n-\\epsilon_n(n+1)} p_n^{(k)}(x)]=(1-2\\epsilon_n)^n\\frac{\\tau_{n+1}^{(k-\\eta_{n+1})}}{\\tau_n^{(k-\\eta_n)}},\n\\end{equation*}\nor\n\\begin{gather*}\n  \\mathcal L^{(k)}[x^{-\\eta_n+n} p_n^{(k)}(x)]=\\frac{\\tau_{n+1}^{(k-\\eta_{n})}}{\\tau_n^{(k-\\eta_n)}},\\quad\n  \\mathcal L^{(k)}[x^{-\\eta_n-1} p_n^{(k)}(x)]=(-1)^n\\frac{\\tau_{n+1}^{(k-\\eta_{n}-1)}}{\\tau_n^{(k-\\eta_n)}}.\n\\end{gather*}\nThe monic polynomial sequence $\\{p_n^{(k)}(x)\\}_{n=0}^N$ is called\nthe \\emph{$\\epsilon$-biorthogonal Laurent polynomials}~\\cite{faybusovich2001imp}.\nThen, the relations \\eqref{eq:e-BLP-st-q}--\\eqref{eq:e-BLP-st-f1} and\n\\begin{equation*}\n  p_n^{(k)}(0)=(-1)^n\\frac{\\tau_n^{(k-\\eta_n+1)}}{\\tau_n^{(k-\\eta_n)}},\\quad\n  p_{n+1}^{(k+\\epsilon_{n})}(0)=(-1)^{n+1}\\frac{\\tau_{n+1}^{(k-\\eta_{n+1}+\\epsilon_n+1)}}{\\tau_{n+1}^{(k-\\eta_{n+1}+\\epsilon_n)}}=(-1)^{n+1}\\frac{\\tau_{n+1}^{(k-\\eta_{n}+1)}}{\\tau_{n+1}^{(k-\\eta_{n})}}\n\\end{equation*}\ngive\n\\begin{gather*}\n  q_n^{(k)}\n  =-\\frac{p_{n+1}^{(k)}(0)}{p_{n}^{(k)}(0)}\n  =\\frac{\\tau_n^{(k-\\eta_n)}\\tau_{n+1}^{(k-\\eta_{n+1}+1)}}{\\tau_{n}^{(k-\\eta_n+1)}\\tau_{n+1}^{(k-\\eta_{n+1})}},\\\\\n  d_n^{(k)}\n  =-\\frac{p_{n+1}^{(k+\\epsilon_n-1)}(0)}{p_{n}^{(k)}(0)}\n  =\\frac{\\tau_n^{(k-\\eta_n)}\\tau_{n+1}^{(k-\\eta_{n})}}{\\tau_{n}^{(k-\\eta_n+1)}\\tau_{n+1}^{(k-\\eta_{n}-1)}},\\\\\n  f_n^{(k)}\n  =-\\frac{p_{n+1}^{(k+\\epsilon_n)}(0)}{p_{n}^{(k)}(0)}\n  =\\frac{\\tau_n^{(k-\\eta_n)}\\tau_{n+1}^{(k-\\eta_{n}+1)}}{\\tau_{n}^{(k-\\eta_n+1)}\\tau_{n+1}^{(k-\\eta_{n})}}\n\\end{gather*}\nand, since $\\mathcal L^{(k+1)}[x^{-\\eta_{n+1}+n-\\epsilon_n(n+1)}\\pi(x)]=\\mathcal L^{(k+1-\\epsilon_n)}[x^{-\\eta_{n}+n-\\epsilon_n(n+1)}\\pi(x)]=\\mathcal L^{(k)}[x^{-\\eta_{n+1}+n+1-\\epsilon_n(n+1)}\\pi(x)]$\nfor all $\\pi(x) \\in \\mathbb C[x]$ and\n\\begin{gather*}\n  \\mathcal L^{(k)}[x^{-\\eta_{n+1}+n+1-\\epsilon_n(n+1)}p_{n+1}^{(k)}(x)]=\n  \\begin{dcases*}\n    \\frac{\\tau_{n+2}^{(k-\\eta_{n+1})}}{\\tau_{n+1}^{(k-\\eta_{n+1})}} & if $\\epsilon_n=0$,\\\\\n    0 & if $\\epsilon_n=1$,\n  \\end{dcases*}\\\\\n  \\mathcal L^{(k+1)}[x^{-\\eta_{n+1}+n-\\epsilon_n(n+1)}p_{n+1}^{(k+1)}(x)]=\n  \\begin{dcases*}\n    0 & if $\\epsilon_n=0$,\\\\\n    (-1)^{n+1}\\frac{\\tau_{n+2}^{(k-\\eta_{n+1})}}{\\tau_{n+1}^{(k-\\eta_{n+1}+1)}} & if $\\epsilon_n=1$,\n  \\end{dcases*}\\\\\n  \\mathcal L^{(k+1-\\epsilon_n)}[x^{-\\eta_n+n-\\epsilon_n(n+1)}p_n^{(k+1-\\epsilon_n)}(x)]=\n  \\begin{dcases*}\n    \\frac{\\tau_{n+1}^{(k-\\eta_n+1)}}{\\tau_n^{(k-\\eta_n+1)}}=\\frac{\\tau_{n+1}^{(k-\\eta_{n+1}+1)}}{\\tau_n^{(k-\\eta_{n+1}+1)}} & if $\\epsilon_n=0$,\\\\\n    (-1)^n\\frac{\\tau_{n+1}^{(k-\\eta_n-1)}}{\\tau_n^{(k-\\eta_n)}}=(-1)^n\\frac{\\tau_{n+1}^{(k-\\eta_{n+1})}}{\\tau_n^{(k-\\eta_{n+1}+1)}} & if $\\epsilon_n=1$\n  \\end{dcases*}\n\\end{gather*}\nhold,\n\\begin{equation*}\n  e_n^{(k)}\n  =\\frac{\\mathcal L^{(k)}[x^{-\\eta_{n+1}+n+1-\\epsilon_n(n+1)}p_{n+1}^{(k)}(x)]-\\mathcal L^{(k+1)}[x^{-\\eta_{n+1}+n-\\epsilon_n(n+1)}p_{n+1}^{(k+1)}(x)]}{\\mathcal L^{(k+1-\\epsilon_n)}[x^{-\\eta_n+n-\\epsilon_n(n+1)}p_n^{(k+1-\\epsilon_n)}(x)]}\n  =\\frac{\\tau_n^{(k-\\eta_{n+1}+1)}\\tau_{n+2}^{(k-\\eta_{n+1})}}{\\tau_{n+1}^{(k-\\eta_{n+1}+1)}\\tau_{n+1}^{(k-\\eta_{n+1})}}.\n\\end{equation*}\n\nSummarizing the above, we obtain Algorithm~\\ref{alg:tri-bi-to-tri} of\nthe isospectral transformation.\n\n\\begin{algorithm}[t]\n  \\caption{Isospectral transformation from tridiagonal--bidiagonal matrix pencil to tridiagonal matrix}\n  \\label{alg:tri-bi-to-tri}\n\\begin{algorithmic}\n  \\REQUIRE $\\{q_n^{(0)}\\}_{n=0}^{N-1}$, $\\{e_n^{(0)}\\}_{n=0}^{N-2}$ and $\\bm\\epsilon\\in\\{0, 1\\}^{N-1}$ (or bidiagonal matrices $R^{(0)}$, $L_{\\bm\\epsilon^*}^{(0)}$ and $L_{\\bm\\epsilon}^{(0)}$)\n  \\STATE $\\eta_0 \\leftarrow 0$\n  \\FOR{$n=1$ to $N-1$}\n  \\STATE $\\eta_n \\leftarrow \\eta_{n-1}+\\epsilon_{n-1}$\n  \\ENDFOR\n  \\FOR{$k=0$ to $\\eta_{N-1}$}\n  \\FOR{$n=0$ to $N-1$}\n  \\STATE\\IfThenElse{$n<N-1$}\n                   {$f_n^{(k)} \\leftarrow q_n^{(k)}+\\epsilon_n e_n^{(k)}$}\n                   {$f_n^{(k)} \\leftarrow q_n^{(k)}$}\n  \\ENDFOR\n  \\FOR{$n=0$ to $N-1$}\n  \\STATE\\IfThenElse{$n=0$}\n                   {$d_n^{(k+1)}\\leftarrow f_n^{(k)}$}\n                   {\\IfThenElse{$\\epsilon_{n-1}=0$}\n                               {$d_n^{(k+1)}\\leftarrow d_{n-1}^{(k+1)}\\frac{f_n^{(k)}}{q_{n-1}^{(k+1)}}$}\n                               {$d_n^{(k+1)}\\leftarrow q_{n-1}^{(k)}\\frac{f_n^{(k)}}{f_{n-1}^{(k)}}$}\n                   }\n  \\STATE $q^{(k+1)}_n \\leftarrow d_n^{(k+1)}+(1-\\epsilon_n)e_n^{(k)}$\n  \\STATE\\IfThen{$n<N-1$}\n               {$e_n^{(k+1)} \\leftarrow e_n^{(k)}\\frac{f_{n+1}^{(k)}}{q_n^{(k+1)}+\\epsilon_n e_{n-1}^{(k+1)}}$}\n  \\ENDFOR\n  \\ENDFOR\n  \\FOR{$n=0$ to $N-1$}\n  \\STATE $\\hat q_n^{(0)} \\leftarrow f_n^{(\\eta_n)}$\n  \\STATE\\IfThen{$n<N-1$}\n               {$\\hat e_n^{(0)} \\leftarrow e_n^{(\\eta_{n+1})}$}\n  \\ENDFOR\n  \\ENSURE $\\{\\hat q_n^{(0)}\\}_{n=0}^{N-1}$ and $\\{\\hat e_n^{(0)}\\}_{n=0}^{N-2}$ (or tridiagonal matrix $\\hat T^{(0)}=\\hat L^{(0)}\\hat R^{(0)}$)\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Numerical example}\nLet us consider\n\\begin{gather*}\n  L_{\\bm\\epsilon^*}^{(0)}=\n  \\begin{pmatrix}\n    1 & 0 & 0 & 0 & 0 & 0\\\\\n    0 & 1 & 0 & 0 & 0 & 0\\\\\n    0 & 0 & 1 & 0 & 0 & 0\\\\\n    0 & 0 & 0 & 1 & 0 & 0\\\\\n    0 & 0 & 0 & 10 & 1 & 0\\\\\n    0 & 0 & 0 & 0 & 11 & 1\n  \\end{pmatrix},\\quad\n  R^{(0)}=\n  \\begin{pmatrix}\n    1 & 1 & 0 & 0 & 0 & 0\\\\\n    0 & 2 & 1 & 0 & 0 & 0\\\\\n    0 & 0 & 3 & 1 & 0 & 0\\\\\n    0 & 0 & 0 & 4 & 1 & 0\\\\\n    0 & 0 & 0 & 0 & 5 & 1\\\\\n    0 & 0 & 0 & 0 & 0 & 6\n  \\end{pmatrix},\n\\end{gather*}\ni.e.,\n\\begin{gather*}\n  L_{\\bm\\epsilon^*}^{(0)}R^{(0)}\n  =\n  \\begin{pmatrix}\n    1 & 1 & 0 & 0 & 0 & 0\\\\\n    0 & 2 & 1 & 0 & 0 & 0\\\\\n    0 & 0 & 3 & 1 & 0 & 0\\\\\n    0 & 0 & 0 & 4 & 1 & 0\\\\\n    0 & 0 & 0 & 40 & 15 & 1\\\\\n    0 & 0 & 0 & 0 & 55 & 17\n  \\end{pmatrix},\n\\end{gather*}\nand\n\\begin{gather*}\n  L_{\\bm\\epsilon}^{(0)}=\n  \\begin{pmatrix}\n    1 & 0 & 0 & 0 & 0 & 0\\\\\n    -7 & 1 & 0 & 0 & 0 & 0\\\\\n    0 & -8 & 1 & 0 & 0 & 0\\\\\n    0 & 0 & -9 & 1 & 0 & 0\\\\\n    0 & 0 & 0 & 0 & 1 & 0\\\\\n    0 & 0 & 0 & 0 & 0 & 1\n  \\end{pmatrix}.\n\\end{gather*}\nThen, Algorithm~\\ref{alg:tri-bi-to-tri} yields\n\\begin{gather*}\n  \\hat R^{(0)}=\n  \\begin{pmatrix}\n    8& 1& 0& 0& 0& 0\\\\\n    0& \\frac{217}{20}& 1& 0& 0& 0\\\\\n    0& 0& \\frac{13150}{1953}& 1& 0& 0\\\\\n    0& 0& 0& \\frac{3924423}{614105}& 1& 0\\\\\n    0& 0& 0& 0& \\frac{2596480772}{1515844721}& 1\\\\\n    0& 0& 0& 0& 0 & \\frac{156435}{1389979}\n  \\end{pmatrix},\\quad\n  \\hat L^{(0)}=\n  \\begin{pmatrix}\n    1& 0& 0& 0& 0 &0\\\\\n    \\frac{35}{4} & 1& 0& 0& 0 &0\\\\\n    0& \\frac{5184}{1085}& 1& 0& 0 &0\\\\\n    0& 0& \\frac{101339}{11835}& 1& 0 &0\\\\\n    0& 0& 0& \\frac{685706750}{67877983}& 1& 0\\\\\n    0& 0& 0& 0& \\frac{119912925}{14496090991} & 1\n  \\end{pmatrix},\n\\end{gather*}\nand\n\\begin{gather*}\n  \\hat T^{(0)}=\\hat L^{(0)}\\hat R^{(0)}=\n  \\begin{pmatrix}\n    8& 1& 0& 0& 0 &0\\\\[0.5em]\n    70& \\frac{98}{5}& 1& 0& 0&0\\\\[0.5em]\n    0& \\frac{1296}{25}& \\frac{518}{45}& 1& 0&0\\\\[0.5em]\n    0& 0& \\frac{4670}{81}& \\frac{62848}{4203}& 1&0\\\\[0.5em]\n    0& 0& 0& \\frac{14079150}{218089}& \\frac{57542826}{4870343}&1\\\\[0.5em]\n    0 & 0& 0& 0& \\frac{1541100}{108764041}& \\frac{1260}{10429}\\\\[0.5em]\n  \\end{pmatrix}.\n\\end{gather*}\nThe eigenvalues of both $(L_{\\bm\\epsilon^*}^{(0)}R^{(0)}, L_{\\bm\\epsilon}^{(0)})$ and $\\hat T^{(0)}$ are 28.1051142, 22.50730913, 10.85981143,\n4.18583949, 0.23568694 and 0.10623881,\nwhich are verified by \\texttt{numpy.linalg.eig()}.\n\n\\section{Isospectral transformation between Hessenberg matrix and Hessenberg--bidiagonal matrix pencil}\n\\label{sec:transf-betw-hess}\n\n\\subsection{Sequence of Hessenberg--bidiagonal GEVPs and eigenvectors}\nAs a generalization, let us consider a Hessenberg--bidiagonal matrix pencil,\ni.e., the Hessenberg matrix of the form\n\\begin{equation*}\n  H^{(k)}\\coloneq L_{\\bm{\\varepsilon}^*}^{(k)}R^{(k+M-1)}R^{(k+M-2)}\\dots R^{(k)},\n\\end{equation*}\nwhere $M$ is a positive integer,\nand the bidiagonal matrix $L_{\\bm\\varepsilon}^{(k)}$ defined by \\eqref{eq:def-L-epsilon}.\nTo obtain the polynomial sequence related to the matrix pencil,\nlet us consider polynomial sequences $\\{p_n^{(k)}(x)\\}_{n=0}^N$ satisfying the relations\n\\begin{equation*}\n  L_{\\bm{\\varepsilon}^*}^{(k)}\\bm p^{(k+M)}(x)=L_{\\bm\\varepsilon}^{(k)}\\bm p^{(k)}(x),\\quad\n  R^{(k)}\\bm p^{(k)}(x)+\\bm p_N^{(k)}(x)=x\\bm p^{(k+1)}(x),\n\\end{equation*}\nand\n\\begin{equation*}\n  p_N^{(k+M)}(x)=p_N^{(k)}(x)\n\\end{equation*}\nfor all integer $k$.\nNote that $p_N^{(k+1)}(x)=p_N^{(k)}(x)$ does not hold in general.\nThen the matrix equation~\\eqref{eq:de-Toda} is generalized as\n\\begin{equation*}\n  L_{\\bm{\\varepsilon}}^{(k+1)}R^{(k)}=\\tilde R^{(k)} L_{\\bm{\\varepsilon}}^{(k)},\\quad\n  \\tilde R^{(k)} L_{\\bm{\\varepsilon}^*}^{(k)}=L_{\\bm{\\varepsilon}^*}^{(k+1)}R^{(k+M)},\n\\end{equation*}\nand it is shown that $\\{p_n^{(k)}(x)\\}_{n=0}^N$ satisfies\nthe $(M+2)$-terms recurrence relation\n\\begin{equation*\n  H^{(k)}\\bm p^{(k)}(x)+\\sum_{j=0}^{M-1}L_{\\bm{\\varepsilon}^*}^{(k)}R^{(k+M-1)}R^{(k+M-2)}\\dots R^{(k+j+1)}x^j \\bm p_N^{(k+j)}(x)\n  =x^M L_{\\bm\\varepsilon}^{(k)}\\bm p^{(k)}(x),\n\\end{equation*}\nSubstituting $x=x\\mathrm{e}^{-2\\pi \\mathrm{i} \\nu\/M}$, $\\nu=0, 1, \\dots, M-1$,\nand taking a linear combination, we obtain\n\\begin{multline}\\label{eq:trr-eM-BLP-2}\n  H^{(k)}\\sum_{\\nu=0}^{M-1} w_\\nu\\bm p^{(k)}(x\\mathrm{e}^{-2\\pi \\mathrm{i} \\nu\/M})\n  +\\sum_{j=0}^{M-1}L_{\\bm{\\varepsilon}^*}^{(k)}R^{(k+M-1)}R^{(k+M-2)}\\dots R^{(k+j+1)}x^j\\sum_{\\nu=0}^{M-1} w_\\nu \\mathrm{e}^{-2\\pi \\mathrm{i} \\nu j\/M} \\bm p_N^{(k+j)}(x\\mathrm{e}^{-2\\pi \\mathrm{i} \\nu\/M})\\\\\n  =x^ML_{\\bm\\varepsilon}^{(k)}\\sum_{\\nu=0}^M w_\\nu \\bm p^{(k)}(x\\mathrm{e}^{-2\\pi \\mathrm{i} \\nu\/M}),\n\\end{multline}\nwhere $w_0, w_1, \\dots, w_{M-1}$ are some constants.\nIf there exist some values $x_0, x_1, \\dots, x_{N-1}$ and\nconstants $w_{r, 0}^{(k)}, w_{r, 1}^{(k)}, \\dots, w_{r, M-1}^{(k)}$ for $r=0, 1, \\dots, N-1$\nsatisfying\n\\begin{equation*}\n  \\sum_{\\nu=0}^{M-1} w_{r, \\nu}^{(k)}\\mathrm{e}^{-2\\pi\\mathrm{i} \\nu j\/M}p_N^{(k+j)}(x_r \\mathrm{e}^{-2\\pi\\mathrm{i} \\nu\/M})=0,\\quad\n  j=0, 1, \\dots, M-1,\n\\end{equation*}\nthen \\eqref{eq:trr-eM-BLP-2} yields\n\\begin{equation*}\n  H^{(k)}\\sum_{\\nu=0}^{M-1} w_{r, \\nu}^{(k)}\\bm p^{(k)}(x_r \\mathrm{e}^{-2\\pi\\mathrm{i}\\nu\/M})\n  =x_r^M L_{\\bm\\varepsilon}^{(k)}\\sum_{\\nu=0}^{M-1} w_{r, \\nu}^{(k)}\\bm p^{(k)}(x_r \\mathrm{e}^{-2\\pi\\mathrm{i}\\nu\/M}),\n\\end{equation*}\ni.e., $x_r^M$ and $\\sum_{\\nu=0}^{M-1} w_{r, \\nu}^{(k)}\\bm p^{(k)}(x_r \\mathrm{e}^{-2\\pi\\mathrm{i}\\nu\/M})$\nare an eigenvalue and an eigenvector of $(H^{(k)}, L_{\\bm\\epsilon}^{(k)})$.\n\nLet us introduce the variables\n\\begin{gather*}\n  d_0^{(k+M)}\\coloneq q_0^{(k)}+\\epsilon_0 e_0^{(k)},\\quad\n  d_n^{(k+M)}\\coloneq q_n^{(k+M)}-(1-\\epsilon_n) e_{n}^{(k)},\\quad n=1, \\dots, N-1,\\\\\n  f_n^{(k)}\\coloneq q_n^{(k)}+\\epsilon_n e_n^{(k)},\\quad n=0, 1, \\dots, N-2,\\quad\n  f_{N-1}^{(k)}\\coloneq q_{N-1}^{(k)}.\n\\end{gather*}\nThen, in the same way as in the previous section,\nwe obtain\n\\begin{gather}\n  f_n^{(k)}=q_n^{(k)}+\\epsilon_n e_n^{(k)},\\quad n=0, 1, \\dots, N-2,\\quad\n  f_{N-1}^{(k)}=q_{N-1}^{(k)},\\label{eq:d-heToda-f}\\\\\n  d_0^{(k+M)}=f_0^{(k)},\\quad\n  d_n^{(k+M)}=\n  \\begin{dcases*}\n    d_{n-1}^{(k+M)}\\frac{f_n^{(k)}}{q_{n-1}^{(k+M)}} & if $\\epsilon_{n-1}=0$,\\\\\n    q_{n-1}^{(k)}\\frac{f_n^{(k)}}{f_{n-1}^{(k)}} & if $\\epsilon_{n-1}=1$,\n  \\end{dcases*}\\quad\n  n=1, 2, \\dots, N-1,\\\\\n  q_n^{(k+M)}=d_n^{(k+M)}+(1-\\epsilon_n)e_n^{(k)},\\quad n=0, 1, \\dots, N-1,\\\\\n  e_n^{(k+1)}=e_n^{(k)}\\frac{f_{n+1}^{(k)}}{q_n^{(k+M)}+\\epsilon_n e_{n-1}^{(k+1)}},\\quad\n  n=0, 1, \\dots, N-2.\\label{eq:d-heToda-e}\n\\end{gather}\nWe call this system the \\emph{discrete hungry elementary Toda orbits}~\\cite{kobayashi2022geb}.\n\n\\subsection{Isospectral transformation to an upper Hessenberg matrix}\nIn the same manner as in the previous section,\nwe can derive the relations\n\\begin{gather}\n  xp_n^{(k+1)}(x)=p_{n+1}^{(k)}(x)+q_n^{(k)}p_n^{(k)}(x),\\label{eq:CT-eM-biorth}\\\\\n  xp_n^{(k+1)}(x)=p_{n+1}^{(k+(\\epsilon_n-1)M)}(x)+d_n^{(k)}p_n^{(k)}(x),\\\\\n  xp_n^{(k+1)}(x)=p_{n+1}^{(k+\\epsilon_n M)}(x)+f_n^{(k)}p_n^{(k)}(x),\\label{eq:CT-M1-biorth}\\\\\n  p_n^{(k)}(x)=p_n^{(k+M)}(x)+e_{n-1}^{(k)}p_{n-1}^{(k+(1-\\epsilon_{n-1})M)}(x).\\label{eq:GT-M1-biorth}\n\\end{gather}\nLet us introduce the variables\n\\begin{equation*}\n  \\hat q_n^{(k)}\\coloneq f_n^{(k+\\eta_n M)},\\quad\n  \\hat e_n^{(k)}\\coloneq e_n^{(k+\\eta_{n+1} M)},\n\\end{equation*}\npolynomials\n\\begin{equation*}\n  \\hat p_n^{(k)}(x)\\coloneq p_n^{(k+\\eta_n M)}(x),\n\\end{equation*}\nand the bidiagonal matrices and the vectors of polynomials\nsame as \\eqref{eq:bidiagonal-OPS} and \\eqref{eq:vector-OPS}.\nThen, the relations \\eqref{eq:CT-M1-biorth} and \\eqref{eq:GT-M1-biorth} are rewritten as\n\\begin{equation*}\n  \\hat{\\bm p}^{(k-M)}(x)=\\hat L^{(k-M)}\\hat{\\bm p}^{(k)}(x),\\quad\n  x\\hat{\\bm p}^{(k+1)}(x)=\\hat R^{(k)}\\hat{\\bm p}^{(k)}(x)+\\bm p_N^{(k)}(x).\n\\end{equation*}\nHence, $\\{\\hat{p}_n^{(k)}(x)\\}_{n=0}^N$ satisfies the $(M+2)$-terms recurrence relation\n\\begin{equation*}\n  \\hat H^{(k)}\\hat{\\bm p}^{(k)}(x)+\\sum_{j=0}^{M-1}\\hat L^{(k)}\\hat R^{(k+M-1)}\\hat R^{(k+M-2)}\\dots \\hat R^{(k+j+1)}x^j \\bm p_N^{(k+j)}(x)=x^M \\hat{\\bm p}^{(k)}(x),\n\\end{equation*}\nwhere\n\\begin{equation*}\n  \\hat H^{(k)}\\coloneq \\hat L^{(k)}\\hat R^{(k+M-1)}\\hat R^{(k+M-2)}\\dots \\hat R^{(k)}.\n\\end{equation*}\nIt is known that the polynomial sequence satisfying this relation\nbecomes the $(M, 1)$-biorthogonal polynomials~\\cite{maeda2017nuh};\nthere exists a linear functional $\\mathcal L^{(k)}\\colon \\mathbb C[x]\\to \\mathbb C$\nsatisfying the $(M, 1)$-biorthogonality condition\n\\begin{equation*}\n  \\mathcal L^{(k)}[x^{mM} \\hat p_n^{(k)}(x)]=h_n^{(k)} \\delta_{m, n} ,\\quad\n  h_n^{(k)}\\ne 0,\\quad\n  n=0, 1, \\dots, N-1, \\quad\n  m=0, 1, \\dots, n.\n\\end{equation*}\nLet $\\mu_{k+m}$ be the moment of this linear functional which is the same definition\nas \\eqref{eq:def-moment}.\nThen, the determinant expression of $\\{\\hat p_n^{(k)}(x)\\}_{n=0}^N$ is given by\n\\begin{equation*}\n  \\hat p_n^{(k)}(x)=\\frac{1}{\\tau_n^{(k)}}\n  \\begin{vmatrix}\n    \\mu_k & \\mu_{k+1} & \\dots & \\mu_{k+n-1} & \\mu_{k+n}\\\\\n    \\mu_{k+M} & \\mu_{k+M+1} & \\dots & \\mu_{k+M+n-1} & \\mu_{k+M+n}\\\\\n    \\vdots & \\vdots & & \\vdots & \\vdots\\\\\n    \\mu_{k+(n-1)M} & \\mu_{k+(n-1)M+1} & \\dots & \\mu_{k+(n-1)M+n-1} & \\mu_{k+(n-1)M+n}\\\\\n    1 & x & \\dots & x^{n-1} & x^n\n  \\end{vmatrix},\\quad\n  n=1, 2, \\dots, N,\n\\end{equation*}\nwhere $\\tau_n^{(k)}$ is the block Hankel determinant of the moments\n\\begin{equation*}\n  \\tau_0^{(k)}\\coloneq 1,\\quad\n  \\tau_n^{(k)}\\coloneq |\\mu_{k+iM+j}|_{i, j=0}^{n-1},\\quad\n  n=1, 2, 3, \\dots.\n\\end{equation*}\n\nNext, we consider the determinant expression of $\\{p_n^{(k)}(x)\\}_{n=0}^N$:\n\\begin{multline*}\n  p_n^{(k)}(x)=\\frac{1}{\\tau_n^{(k-\\eta_n M)}}\n  \\begin{vmatrix}\n    \\mu_{k-\\eta_n M} & \\mu_{k-\\eta_n M+1} & \\dots & \\mu_{k-\\eta_n M+n-1} & \\mu_{k-\\eta_n M+n}\\\\\n    \\mu_{k+(-\\eta_n+1) M} & \\mu_{k-(-\\eta_n+1)M+1} & \\dots & \\mu_{k+(-\\eta_n+1)M+n-1} & \\mu_{k+(-\\eta_n+1)M+n}\\\\\n    \\vdots & \\vdots & & \\vdots & \\vdots\\\\\n    \\mu_{k+(-\\eta_n+n-1)M} & \\mu_{k+(-\\eta_n+n-1)M+1} & \\dots & \\mu_{k+(-\\eta_n+n-1)M+n-1} & \\mu_{k+(-\\eta_n+n-1)M+n}\\\\\n    1 & x & \\dots & x^{n-1} & x^n\n  \\end{vmatrix},\\\\\n  n=1, 2, \\dots, N.\n\\end{multline*}\nHence, $\\{p_n^{(k)}(x)\\}_{n=0}^N$ satisfies the $(\\epsilon, M)$-biorthogonality relation\nfor $n=0, 1, \\dots, N-1$:\n\\begin{equation*}\n  \\mathcal L^{(k)}[x^{mM} p_n^{(k)}(x)]=0,\\quad\n  m=-\\eta_n , -\\eta_n+1, \\dots, -\\eta_n+n-1,\n\\end{equation*}\nand\n\\begin{equation*}\n  \\mathcal L^{(k)}[x^{(-\\eta_n+n-\\epsilon_n(n+1))M} p_n^{(k)}(x)]=(1-2\\epsilon_n)^n \\frac{\\tau_{n+1}^{(k-\\eta_{n+1} M)}}{\\tau_n^{(k-\\eta_n M)}},\n\\end{equation*}\nor\n\\begin{equation*}\n  \\mathcal L^{(k)}[x^{(-\\eta_n+n) M} p_n^{(k)}(x)]=\\frac{\\tau_{n+1}^{(k-\\eta_{n} M)}}{\\tau_n^{(k-\\eta_n M)}},\\quad\n  \\mathcal L^{(k)}[x^{(-\\eta_n-1) M} p_n^{(k)}(x)]=(-1)^n\\frac{\\tau_{n+1}^{(k+(-\\eta_{n}-1) M)}}{\\tau_n^{(k-\\eta_n M)}}.\n\\end{equation*}\nThe monic polynomial sequence $\\{p_n^{(k)}(x)\\}_{n=0}^N$ is called\nthe \\emph{$(\\epsilon, M)$-biorthogonal Laurent polynomials}.\nThen, the relations~\\eqref{eq:CT-eM-biorth}--\\eqref{eq:GT-M1-biorth} and\n\\begin{equation*}\n  p_n^{(k)}(0)=(-1)^n\\frac{\\tau_n^{(k-\\eta_n M+1)}}{\\tau_n^{(k-\\eta_n M)}},\\quad\n  p_{n+1}^{(k+\\epsilon_n M)}=(-1)^{n+1} \\frac{\\tau_{n+1}^{(k+(-\\eta_{n+1}+\\epsilon_n)M+1)}}{\\tau_{n+1}^{(k+(-\\eta_{n+1}+\\epsilon_n) M)}}=(-1)^{n+1} \\frac{\\tau_{n+1}^{(k-\\eta_{n}M+1)}}{\\tau_{n+1}^{(k-\\eta_{n} M)}}\n\\end{equation*}\ngive\n\\begin{gather*}\n  q_n^{(k)}\n  =-\\frac{p_{n+1}^{(k)}(0)}{p_{n}^{(k)}(0)}\n  =\\frac{\\tau_n^{(k-\\eta_n M)}\\tau_{n+1}^{(k-\\eta_{n+1}M+1)}}{\\tau_{n}^{(k-\\eta_n M+1)}\\tau_{n+1}^{(k-\\eta_{n+1}M)}},\\\\\n  d_n^{(k)}\n  =-\\frac{p_{n+1}^{(k+(\\epsilon_n-1)M)}(0)}{p_{n}^{(k)}(0)}\n  =\\frac{\\tau_n^{(k-\\eta_n M)}\\tau_{n+1}^{(k+(-\\eta_{n}-1)M+1)}}{\\tau_{n}^{(k-\\eta_n M+1)}\\tau_{n+1}^{(k+(-\\eta_{n}-1)M)}},\\\\\n  f_n^{(k)}\n  =-\\frac{p_{n+1}^{(k+\\epsilon_n M)}(0)}{p_{n}^{(k)}(0)}\n  =\\frac{\\tau_n^{(k-\\eta_n M)}\\tau_{n+1}^{(k-\\eta_{n}M+1)}}{\\tau_{n}^{(k-\\eta_nM+1)}\\tau_{n+1}^{(k-\\eta_{n}M)}},\n\\end{gather*}\nand, since\n\\begin{gather*}\n  \\mathcal L^{(k)}[x^{(-\\eta_{n+1}+n+1-\\epsilon_n(n+1))M}p_{n+1}^{(k)}(x)]=\n  \\begin{dcases*}\n    \\frac{\\tau_{n+2}^{(k-\\eta_{n+1}M)}}{\\tau_{n+1}^{(k-\\eta_{n+1}M)}} & if $\\epsilon_n=0$,\\\\\n    0 & if $\\epsilon_n=1$,\n  \\end{dcases*}\\\\\n  \\mathcal L^{(k+M)}[x^{(-\\eta_{n+1}+n-\\epsilon_n(n+1))M}p_{n+1}^{(k+M)}(x)]=\n  \\begin{dcases*}\n    0 & if $\\epsilon_n=0$,\\\\\n    (-1)^{n+1}\\frac{\\tau_{n+2}^{(k-\\eta_{n+1}M)}}{\\tau_{n+1}^{(k+(-\\eta_{n+1}+1)M)}} & if $\\epsilon_n=1$,\n  \\end{dcases*}\\\\\n  \\mathcal L^{(k+(1-\\epsilon_n)M)}[x^{(-\\eta_n+n-\\epsilon_n(n+1))M}p_n^{(k+(1-\\epsilon_n)M)}(x)]=\n  \\begin{dcases*}\n    \\frac{\\tau_{n+1}^{(k+(-\\eta_n+1) M)}}{\\tau_n^{(k+(-\\eta_n+1) M)}}=\\frac{\\tau_{n+1}^{(k+(-\\eta_{n+1}+1) M)}}{\\tau_n^{(k+(-\\eta_{n+1}+1) M)}} & if $\\epsilon_n=0$,\\\\\n    (-1)^n \\frac{\\tau_{n+1}^{(k+(-\\eta_n-1)M)}}{\\tau_{n}^{(k-\\eta_n M)}}=(-1)^n \\frac{\\tau_{n+1}^{(k-\\eta_{n+1}M)}}{\\tau_{n}^{(k+(-\\eta_{n+1}+1) M)}}& if $\\epsilon_n=1$\n  \\end{dcases*}\n\\end{gather*}\nhold,\n\\begin{equation*}\n  e_n^{(k)}\n  =\\frac{\\mathcal L^{(k)}[x^{(-\\eta_{n+1}+n+1-\\epsilon_n(n+1))M}p_{n+1}^{(k)}(x)]-\\mathcal L^{(k+M)}[x^{(-\\eta_{n+1}+n-\\epsilon_n(n+1))M}p_{n+1}^{(k+M)}(x)]}{\\mathcal L^{(k+(1-\\epsilon_n)M)}[x^{(-\\eta_n+n-\\epsilon_n(n+1))M}p_n^{(k+(1-\\epsilon_n)M)}(x)]}\n  =\\frac{\\tau_n^{(k+(-\\eta_{n+1}+1) M)}\\tau_{n+2}^{(k-\\eta_{n+1}M)}}{\\tau_{n+1}^{(k+(-\\eta_{n+1}+1)M)}\\tau_{n+1}^{(k-\\eta_{n+1}M)}}.\n\\end{equation*}\nFurther, we have\n\\begin{equation*}\n  \\hat q_n^{(k)}=f_n^{(k+\\eta_n M)}=\\frac{\\tau_n^{(k)}\\tau_{n+1}^{(k+1)}}{\\tau_{n}^{(k+1)}\\tau_{n+1}^{(k)}},\\quad\n  \\hat e_n^{(k)}=e_n^{(k+\\eta_{n+1} M)}=\\frac{\\tau_n^{(k+M)}\\tau_{n+2}^{(k)}}{\\tau_{n+1}^{(k+M)}\\tau_{n+1}^{(k)}}.\n\\end{equation*}\n\nSummarizing the above, we obtain Algorithm~\\ref{alg:Hes-bi-to-Hes} of\nthe isospectral transformation.\n\n\\begin{algorithm}[t]\n  \\caption{Isospectral transformation from Hessenberg--bidiagonal matrix pencil to Hessenberg matrix}\n  \\label{alg:Hes-bi-to-Hes}\n\\begin{algorithmic}\n  \\REQUIRE $\\{q_n^{(0)}\\}_{n=0}^{N-1}$, $\\{q_n^{(1)}\\}_{n=0}^{N-1}$, $\\dots$, $\\{q_n^{(M-1)}\\}_{n=0}^{N-1}$,  $\\{e_n^{(0)}\\}_{n=0}^{N-2}$ and $\\bm\\epsilon\\in\\{0, 1\\}^{N-1}$ (or bidiagonal matrices $R^{(0)}$, $R^{(1)}$, $\\dots$, $R^{(M-1)}$, $L_{\\bm\\epsilon^*}^{(0)}$ and $L_{\\bm\\epsilon}^{(0)}$)\n  \\STATE $\\eta_0 \\leftarrow 0$\n  \\FOR{$n=1$ to $N-1$}\n  \\STATE $\\eta_n \\leftarrow \\eta_{n-1}+\\epsilon_{n-1}$\n  \\ENDFOR\n  \\FOR{$k=0$ to $(\\eta_{N-1}+1)M-1$}\n  \\FOR{$n=0$ to $N-1$}\n  \\STATE\\IfThenElse{$n<N-1$}\n                   {$f_n^{(k)} \\leftarrow q_n^{(k)}+\\epsilon_n e_n^{(k)}$}\n                   {$f_n^{(k)} \\leftarrow q_n^{(k)}$}\n  \\ENDFOR\n  \\FOR{$n=0$ to $N-1$}\n  \\STATE\\IfThenElse{$n=0$}\n                   {$d_n^{(k+M)}\\leftarrow f_n^{(k)}$}\n                   {\\IfThenElse{$\\epsilon_{n-1}=0$}\n                               {$d_n^{(k+M)}\\leftarrow d_{n-1}^{(k+M)}\\frac{f_n^{(k)}}{q_{n-1}^{(k+M)}}$}\n                               {$d_n^{(k+M)}\\leftarrow q_{n-1}^{(k)}\\frac{f_n^{(k)}}{f_{n-1}^{(k)}}$}\n                   }\n  \\STATE $q^{(k+M)}_n \\leftarrow d_n^{(k+M)}+(1-\\epsilon_n)e_n^{(k)}$\n  \\STATE\\IfThen{$n<N-1$}\n               {$e_n^{(k+1)} \\leftarrow e_n^{(k)}\\frac{f_{n+1}^{(k)}}{q_n^{(k+M)}+\\epsilon_n e_{n-1}^{(k+1)}}$}\n  \\ENDFOR\n  \\ENDFOR\n  \\FOR{$n=0$ to $N-1$}\n  \\FOR{$k=0$ to $M-1$}\n  \\STATE $\\hat q_n^{(k)} \\leftarrow f_n^{(k+\\eta_n M)}$\n  \\ENDFOR\n  \\STATE\\IfThen{$n<N-1$}\n               {$\\hat e_n^{(0)} \\leftarrow e_n^{(\\eta_{n+1}M)}$}\n  \\ENDFOR\n  \\ENSURE $\\{\\hat q_n^{(0)}\\}_{n=0}^{N-1}$, $\\{\\hat q_n^{(1)}\\}_{n=0}^{N-1}$, $\\dots$, $\\{\\hat q_n^{(M-1)}\\}_{n=0}^{N-1}$ and $\\{\\hat e_n^{(0)}\\}_{n=0}^{N-2}$ (or Hessenberg matrix $\\hat H^{(0)}=\\hat L^{(0)}\\hat R^{(M-1)}\\dots\\hat R^{(0)}$)\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsection{Numerical example}\nLet us consider\n\\begin{gather*}\n  L_{\\bm\\epsilon^*}^{(0)}\n  =\n  \\begin{pmatrix}\n    1 & 0 & 0 & 0 & 0 & 0\\\\\n    0 & 1 & 0 & 0 & 0 & 0\\\\\n    0 & 0 & 1 & 0 & 0 & 0\\\\\n    0 & 0 & 0 & 1 & 0 & 0\\\\\n    0 & 0 & 0 & 10 & 1 & 0\\\\\n    0 & 0 & 0 & 0 & 11 & 1\n  \\end{pmatrix},\\\\\n  R^{(0)}=\n  \\begin{pmatrix}\n    1 & 1 & 0 & 0 & 0 & 0\\\\\n    0 & 2 & 1 & 0 & 0 & 0\\\\\n    0 & 0 & 3 & 1 & 0 & 0\\\\\n    0 & 0 & 0 & 4 & 1 & 0\\\\\n    0 & 0 & 0 & 0 & 5 & 1\\\\\n    0 & 0 & 0 & 0 & 0 & 6\n  \\end{pmatrix},\\quad\n  R^{(1)}=\n  \\begin{pmatrix}\n    2 & 1 & 0 & 0 & 0 & 0\\\\\n    0 & 3 & 1 & 0 & 0 & 0\\\\\n    0 & 0 & 4 & 1 & 0 & 0\\\\\n    0 & 0 & 0 & 5 & 1 & 0\\\\\n    0 & 0 & 0 & 0 & 6 & 1\\\\\n    0 & 0 & 0 & 0 & 0 & 7\n  \\end{pmatrix},\\quad\n  R^{(2)}=\n  \\begin{pmatrix}\n    3 & 1 & 0 & 0 & 0 & 0\\\\\n    0 & 4 & 1 & 0 & 0 & 0\\\\\n    0 & 0 & 5 & 1 & 0 & 0\\\\\n    0 & 0 & 0 & 6 & 1 & 0\\\\\n    0 & 0 & 0 & 0 & 7 & 1\\\\\n    0 & 0 & 0 & 0 & 0 & 8\n  \\end{pmatrix},\n\\end{gather*}\ni.e.,\n\\begin{equation*}\n  L_{\\bm\\epsilon^*}^{(0)}R^{(2)}R^{(1)}R^{(0)}\n  =\n  \\begin{pmatrix}\n    6 & 18 8 & 9 & 1 & 0 & 0\\\\\n    0 & 24 & 36 & 12 & 1 & 0\\\\\n    0 & 0 & 60 & 60 & 15 & 1\\\\\n    0 & 0 & 0 & 120 & 90 & 18\\\\\n    0 & 0 & 0 & 1200 & 1110 & 306\\\\\n    0 & 0 & 0 & 0 & 2310 & 1722\n  \\end{pmatrix},\n\\end{equation*}\nand\n\\begin{equation*}\n  L_{\\bm \\epsilon}^{(0)}\n  =\n  \\begin{pmatrix}\n    1 & 0 & 0 & 0 & 0 & 0\\\\\n    -7 & 1 & 0 & 0 & 0 & 0\\\\\n    0 & -8 & 1 & 0 & 0 & 0\\\\\n    0 & 0 & -9 & 1 & 0 & 0\\\\\n    0 & 0 & 0 & 0 & 1 & 0\\\\\n    0 & 0 & 0 & 0 & 0 & 1\n  \\end{pmatrix}.\n\\end{equation*}\nThen, Algorithm~\\ref{alg:Hes-bi-to-Hes} yields\n\\allowdisplaybreaks\n\\begin{gather*}\n  \\hat R^{(0)}\n  =\n  \\begin{pmatrix}\n    8 & 1 & 0 & 0 & 0 & 0\\\\\n    0 & \\frac{1045}{196} & 1 & 0 & 0 & 0\\\\\n    0 & 0 & \\frac{11783226}{1951015} & 1 & 0 & 0\\\\\n    0 & 0 & 0 & \\frac{11202591839}{1537751072} & 1 & 0\\\\\n    0 & 0 & 0 & 0 & \\frac{1793288934976}{673133562011} & 1\\\\\n    0 & 0 & 0 & 0 & 0 & \\frac{3365490}{23369591}\n  \\end{pmatrix},\\\\\n  \\hat R^{(1)}\n  =\n  \\begin{pmatrix}\n    \\frac{43}{4} & 1 & 0 & 0 & 0 & 0\\\\\n    0 & \\frac{249816}{44935} & 1 & 0 & 0 & 0\\\\\n    0 & 0 & \\frac{4459329545}{417182311} & 1 & 0 & 0\\\\\n    0 & 0 & 0 & \\frac{281563249429787}{25605158734417} & 1 & 0\\\\\n    0 & 0 & 0 & 0 & \\frac{61342417293160530}{164176201497170723} & 1\\\\\n    0 & 0 & 0 & 0 & 0 & \\frac{654348548}{340773203}\n  \\end{pmatrix},\\\\\n  \\hat R^{(2)}\n  =\n  \\begin{pmatrix}\n    \\frac{570}{43} & 1 & 0 & 0 & 0 & 0\\\\\n    0 & \\frac{5738006}{988855} & 1 & 0 & 0 & 0\\\\\n    0 & 0 & \\frac{2131337471900}{284718590021} & 1 & 0 & 0\\\\\n    0 & 0 & 0 & \\frac{417593915190317388}{71923747531523615} & 1 & 0\\\\\n    0 & 0 & 0 & 0 & \\frac{5065558609120017904}{2778977782301483047} & 1\\\\\n    0 & 0 & 0 & 0 & 0 & \\frac{340773203}{103007824}\n  \\end{pmatrix},\\\\\n  \\hat L^{(0)}\n  =\n  \\begin{pmatrix}\n    1 & 0 & 0 & 0 & 0 & 0\\\\\n    \\frac{686}{95} & 1 & 0 & 0 & 0 & 0\\\\\n    0 & \\frac{17736500}{3269329} & 1 & 0 & 0 & 0\\\\\n    0 & 0 & \\frac{92158247808}{19114261985} & 1 & 0 & 0\\\\\n    0 & 0 & 0 & \\frac{393943905477395}{312887922561632} & 1 & 0\\\\\n    0 & 0 & 0 & 0 & \\frac{448520531195}{11555726719792} & 1\n  \\end{pmatrix}\n\\end{gather*}\n\\allowdisplaybreaks[0]\nand\n\\begin{equation*}\n  \\hat H^{(0)}\n  =\\hat L^{(0)}\\hat R^{(2)}\\hat R^{(1)}\\hat R^{(0)}\n  =\n  \\begin{pmatrix}\n    1140 & \\frac{11898}{49} & \\frac{46404}{1867} & 1 & 0 & 0\\\\[0.5em]\n    8232 & \\frac{94344}{49} & \\frac{581274}{1867} & \\frac{1189329}{38368} & 1 & 0\\\\[0.5em]\n    0 & \\frac{2240400}{2401} & \\frac{109654800}{91483} & \\frac{646077099}{1880032} & \\frac{146061709}{5496967} & 1\\\\[0.5em]\n    0 & 0 & \\frac{8121738240}{3485689} & \\frac{26863943637}{17908264} & \\frac{2245552524}{12320333} & \\frac{20808}{1867}\\\\[0.5em]\n    0 & 0 & 0 & \\frac{215519585169}{368025856} & \\frac{110756457399}{1076059336} & \\frac{44037}{4796}\\\\[0.5em]\n    0 & 0 & 0 & 0 & \\frac{886300800}{12585025489} & \\frac{107940}{112183}\n  \\end{pmatrix}.\n\\end{equation*}\nThe eigenvalues of both $(L_{\\bm\\epsilon^*}^{(0)}R^{(2)}R^{(1)}R^{(0)}, L_{\\bm \\epsilon}^{(0)})$\nand $\\hat H^{(0)}$ are 3188.27018, 2167.22313, 485.176601, 25.4359335,\n1.14466512 and 0.749495172,\nwhich are verified by \\texttt{numpy.linalg.eig()}.\n\n\\section{Conclusion}\nIn this paper, we have constructed an isospectral transformation from\na Hessenberg--bidiagonal matrix pencil to a Hessenberg matrix.\nSince the obtained algorithms do not contain any subtractions,\nif all the input values are positive, then all computations can be performed\nwithout loss of digits for floating point arithmetic.\n\nAs a future work, we are interested in an isospectral transformation from\na tridiagonal--tridiagonal matrix pencil to some type of sparse matrix.\nThe GEVP of tridiagonal--tridiagonal matrix pencils has attracted much attention~\\cite{li1994ags,vandebril2009qsa}.\nIt is known that this GEVP defines so-called $\\mathrm{R}_{\\mathrm{II}}$\\  polynomials~\\cite{ismail1995goc}\nand its associated discrete integrable systems are $\\mathrm{R}_{\\mathrm{II}}$\\  chain~\\cite{maeda2016gea,spiridonov2000stc} and FST chain~\\cite{spiridonov2007idt}.\nWe will be able to derive the isospectral transformation by using the relation between these\ntwo discrete integrable systems.\n\n\\section*{Declaration of competing interest}\nThere is no competing interest.\n\n\\section*{Data availability}\nNo data was used for research described in the article.\n\n\\section*{Acknowledgement}\nThis work was supported by JSPS KAKENHI Grant Numbers JP19J23445, JP21K13837 and JP19H01792.\nThis work was partially supported by the joint project\n``Advanced Mathematical Science for Mobility Society''\nof Kyoto University and Toyota Motor Corporation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}