diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzddfg" "b/data_all_eng_slimpj/shuffled/split2/finalzzddfg"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzddfg"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction} \\label{sect:intro}\n\nSpectral measures for the representation graphs for the irreducible representations of the exceptional compact Lie group $G_2$ and its maximal torus $\\mathbb{T}^2$, and for nimrep graphs associated to the $G_2$ modular invariants, were studied in \\cite{evans\/pugh:2012i}. Here we consider the McKay graphs for finite subgroups of $G_2$ and study their spectral measures.\n\nThe spectral measure of a self-adjoint operator $a$ in a unital $C^{\\ast}$-algebra $A$ with state $\\varphi$ is a compactly supported probability measure $\\nu_a$ on the spectrum $\\sigma(a) \\subset \\mathbb{R}$ of $a$, uniquely determined by its moments\n\\begin{equation} \\label{eqn:moments_sa_operator}\n\\varphi(a^m) = \\int_{\\sigma(a)} x^m \\mathrm{d}\\nu_a (x),\n\\end{equation}\nfor all non-negative integers $m$.\n\nThe self-adjoint operators we consider here are the adjacency matrices of the McKay graphs for finite subgroups of $G_2$. The characters of the rank two Lie group $G_2$ are functions on $\\mathbb{T}^2$, and it is convenient to write the spectral measures for these operators as measures $\\varepsilon$ over the torus $\\mathbb{T}^2$. However, $\\mathbb{T}^2$ has dimension one greater than $\\sigma(a) \\subset \\mathbb{R}$, so that there is an infinite family of pullback measures $\\varepsilon$ over $\\mathbb{T}^2$ for any spectral measure $\\nu_a$. The details of the relation between the measures $\\varepsilon$ and $\\nu_a$ are given in Section \\ref{sect:measures-different_domains}.\n\nIn order to remove this ambiguity, we also consider joint spectral measures, that is, measures over the joint spectrum $\\sigma(a,b) \\subset \\sigma(a) \\times \\sigma(b) \\subset \\mathbb{R}^2$ of commuting self-adjoint operators $a$ and $b$. The abelian $C^{\\ast}$-algebra $B$ generated by $a$, $b$ and the identity 1 is isomorphic to $C(X)$, where $X$ is the spectrum of $B$. Then the joint spectrum is defined as $\\sigma(a,b) = \\{ (a(x), b(x)) | \\, x \\in X \\}$. In fact, one can identify the spectrum $X$ with its image $\\sigma(a,b)$ in $\\mathbb{R}^2$, since the map $x \\mapsto (a(x), b(x))$ is continuous and injective, and hence a homomorphism since $X$ is compact \\cite{takesaki:2002}.\nIn the case where the operators $a$, $b$ act on a finite-dimensional Hilbert space, this is the set of all pairs of real numbers $(\\lambda_a,\\lambda_b)$ for which there exists a vector $\\phi$, $||\\phi||=1$, such that $a\\phi = \\lambda_a \\phi$, $b\\phi = \\lambda_b \\phi$.\nThen there exists a compactly supported probability measure $\\widetilde{\\nu}_{a,b}$ on $\\sigma(a,b)$, which is uniquely determined by its cross moments\n\\begin{equation} \\label{eqn:cross_moments_sa_operators}\n\\varphi(a^m b^n) = \\int_{\\sigma(a,b)} x^m y^n \\mathrm{d}\\widetilde{\\nu}_{a,b} (x,y),\n\\end{equation}\nfor all non-negative integers $m$, $n$.\nSuch joint spectral measures specialize to the spectral measures $\\nu_a$ (respectively $\\nu_b$) by integrating over all $y$ for which $(\\lambda_a,y) \\in \\sigma(a,b)$ (respectively all $x$ for which $(x,\\lambda_b) \\in \\sigma(a,b)$).\nAs discussed in Section \\ref{sect:measures-different_domains}, such a measure uniquely defines a measure over $\\mathbb{T}^2$ invariant under an action of the Weyl group of $G_2$. In this paper we determine the joint spectral measure for each finite subgroup $\\Gamma$ of $G_2$ for all non-conjugate embeddings of $\\Gamma$ into the fundamental representations of $G_2$.\n\nThe paper is organised as follows. In Section \\ref{sect:preliminaries} we present some preliminary material, including a discussion on the relation between spectral measures over certain different domains in Section \\ref{sect:measures-different_domains} and a summary of relevant results for $G_2$ and its nimreps from \\cite{evans\/pugh:2012i}.\n\nIn Section \\ref{sect:subgroupsG2} we discuss the finite subgroups of $G_2$, including their embeddings into the fundamental representations of $G_2$. We also give a general discussion of their spectral measures. In Sections \\ref{sect:II1}-\\ref{sect:IP5} we discuss each finite subgroup of $G_2$ individually, including determining all non-conjugate embeddings into the fundamental representations of $G_2$. For each such embedding we construct their McKay graphs, some of which have appeared before in \\cite{he:2003}, and we determine their joint spectral measures.\n\nSpectral measures associated to the compact Lie groups $A_1 = SU(2)$ and $A_2 = SU(3)$ and their maximal tori, nimrep graphs associated to the $SU(2)$ and $SU(3)$ modular invariants, and the McKay graphs for finite subgroups of $SU(2)$ and $SU(3)$ were studied in \\cite{banica\/bisch:2007, evans\/pugh:2009v, evans\/pugh:2010i}.\nSpectral measures associated to the compact Lie group $C_2$ are studied in \\cite{evans\/pugh:2012iii}, whilst spectral measures associated to other compact rank two Lie groups and their maximal tori are studied in \\cite{evans\/pugh:2012iv}.\n\n\n\n\\section{Preliminaries} \\label{sect:preliminaries}\n\\subsection{Spectral measures over different domains} \\label{sect:measures-different_domains}\n\nThe Weyl group of $G_2$ is the dihedral group $D_{12}$ of order 12.\nAs a subgroup of $GL(2,\\mathbb{Z})$, $D_{12}$ is generated by matrices $T_2$, $T_6$, of orders 2, 6 respectively, given by\n\\begin{equation} \\label{T2,T6}\nT_2 = \\left( \\begin{array}{cc} 0 & -1 \\\\ -1 & 0 \\end{array} \\right), \\qquad T_6 = \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 1 \\end{array} \\right),\n\\end{equation}\nwhere the action of $D_{12}$ on $\\mathbb{T}^2$ is given by $T(\\omega_1,\\omega_2) = (\\omega_1^{a_{11}}\\omega_2^{a_{12}},\\omega_1^{a_{21}}\\omega_2^{a_{22}})$, for $T = (a_{il}) \\in D_{12}$. This action leaves $\\chi_{\\mu}(\\omega_1,\\omega_2)$ invariant, for any $\\mu \\in P_{++} = \\{ (\\mu_1,\\mu_2) \\in \\mathbb{N}^2 | \\, \\mu_1 \\geq \\mu_2 \\}$, the interior of the Weyl alcove for $G_2$.\nAny $D_{12}$-invariant measure $\\varepsilon_{\\mu}$ on $\\mathbb{T}^2$ yields a pushforward probability measure $\\nu_{\\mu}$ on $I_{\\mu} = \\chi_{\\mu}( \\mathbb{T}^2)\\subset \\mathbb{R}$ by\n\\begin{equation} \\label{eqn:measures-T2-Ij_G2}\n\\int_{I_{\\mu}} \\psi(x) \\mathrm{d}\\nu_{\\mu}(x) = \\int_{\\mathbb{T}^2} \\psi(\\chi_{\\mu}(\\omega_1,\\omega_2)) \\mathrm{d}\\varepsilon_{\\mu}(\\omega_1,\\omega_2),\n\\end{equation}\nfor any continuous function $\\psi:I_{\\mu} \\rightarrow \\mathbb{C}$, where $\\mathrm{d}\\varepsilon_{\\mu}(\\omega_1,\\omega_2) = \\mathrm{d}\\varepsilon_{\\mu}(g(\\omega_1,\\omega_2))$ for all $g \\in D_{12}$.\nThere is a loss of dimension here, in the sense that the integral on the right hand side is over the two-dimensional torus $\\mathbb{T}^2$, whereas on the right hand side it is over the interval $I_{\\mu}$. Thus there is an infinite family of pullback measures $\\varepsilon_{\\mu}$ over $\\mathbb{T}^2$ for any measure $\\nu_{\\mu}$ on $I_{\\mu}$, that is, any $\\varepsilon_{\\mu}$ such that $\\varepsilon_{\\mu}(I_{\\mu}^{-1}[x]) = \\nu_{\\mu}(x)$ for all $x \\in I_{\\mu}$ will yield the probability measure $\\nu_{\\mu}$ on $I_{\\mu}$ as a pushforward measure by (\\ref{eqn:measures-T2-Ij_G2}).\nAs in \\cite{evans\/pugh:2012i}, we instead work with an intermediate probability measure $\\widetilde{\\nu}_{\\lambda,\\mu}$ which lives over the joint spectrum $\\mathfrak{D}_{\\lambda,\\mu} \\subset I_{\\lambda} \\times I_{\\mu} \\subset \\mathbb{R}^2$, for $\\lambda,\\mu \\in P_{+}$, where there is no loss of dimension.\n\nA fundamental domain $C$ of $\\mathbb{T}^2$ under the action of the dihedral group $D_{12}$ is illustrated in Figure \\ref{fig:fund_domain-G2inT2}, where the axes are labelled by the parameters $\\theta_1$, $\\theta_2$ in $(e^{2 \\pi i \\theta_1},e^{2 \\pi i \\theta_2}) \\in \\mathbb{T}^2$, which is a quotient of the fundamental domain of $\\mathbb{T}^2\/S_3$ illustrated in Figure \\ref{fig:fund_domain-A2inT2} (see \\cite{evans\/pugh:2009v}) by the $\\mathbb{Z}_2$-action given by the matrix -1.\nNote that in Figure \\ref{fig:fund_domain-G2inT2}, the lines $\\theta_1=0$ and $\\theta_2=0$ are also boundaries of copies of the fundamental domain $C$ under the action of $D_{12}$, whereas in Figure \\ref{fig:fund_domain-A2inT2} they are not boundaries of copies of the fundamental domain under the action of $S_3$. The torus $\\mathbb{T}^2$ contains 12 copies of $C$, so that\n\\begin{equation} \\label{eqn:measureT2=12C}\n\\int_{\\mathbb{T}^2} \\phi(\\omega_1,\\omega_2) \\mathrm{d}\\varepsilon(\\omega_1,\\omega_2) = 12 \\int_{C} \\phi(\\omega_1,\\omega_2) \\mathrm{d}\\varepsilon(\\omega_1,\\omega_2),\n\\end{equation}\nfor any $D_{12}$-invariant function $\\phi:\\mathbb{T}^2 \\rightarrow \\mathbb{C}$ and $D_{12}$-invariant measure $\\varepsilon$ over $\\mathbb{T}^2$. The only fixed point of $\\mathbb{T}^2$ under the action of $D_{12}$ is the point $(1,1)$.\n\n\\begin{figure}[tb]\n\\begin{minipage}[t]{7.9cm}\n\\begin{center}\n  \\includegraphics[width=55mm]{Fig-fund_domain-A2inT2.eps}\\\\\n \\caption{\\small A fundamental domain of $\\mathbb{T}^2\/S_3$.} \\label{fig:fund_domain-A2inT2}\n\\end{center}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{7.9cm}\n\\begin{center}\n  \\includegraphics[width=55mm]{Fig-fund_domain-G2inT2.eps}\\\\\n \\caption{\\small A fundamental domain $C$ of $\\mathbb{T}^2\/D_{12}$.} \\label{fig:fund_domain-G2inT2}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\nLet $x_{\\lambda} = \\chi_{\\lambda}(\\omega_1,\\omega_2)$ and let $\\Psi_{\\lambda,\\mu}$ be the map $(\\omega_1,\\omega_2) \\mapsto (x_{\\lambda},x_{\\mu})$. We denote by $\\mathfrak{D}_{\\lambda,\\mu}$ the image of $\\Psi_{\\lambda,\\mu}(C) (= \\Psi_{\\lambda,\\mu}(\\mathbb{T}^2))$ in $\\mathbb{R}^2$.\nNote that we can identify $\\mathfrak{D}_{\\lambda,\\mu}$ with $\\mathfrak{D}_{\\mu,\\lambda}$ by reflecting about the line $x_{\\lambda} = x_{\\mu}$.\nThen the joint spectral measure $\\widetilde{\\nu}_{\\lambda,\\mu}$ is the measure on $\\mathfrak{D}_{\\lambda,\\mu}$ uniquely determined by its cross-moments as in (\\ref{eqn:cross_moments_sa_operators}).\nThen there is a unique $D_{12}$-invariant pullback measure $\\varepsilon$ on $\\mathbb{T}^2$ such that\n\\begin{equation} \\label{eqn:measures-T2-D_G2}\n\\int_{\\mathfrak{D}_{\\lambda,\\mu}} \\psi(x_{\\lambda},x_{\\mu}) \\mathrm{d}\\widetilde{\\nu}_{\\lambda,\\mu}(x_{\\lambda},x_{\\mu}) = \\int_{\\mathbb{T}^2} \\psi(\\chi_{\\lambda}(\\omega_1,\\omega_2),\\chi_{\\mu}(\\omega_1,\\omega_2)) \\mathrm{d}\\varepsilon_{\\lambda,\\mu}(\\omega_1,\\omega_2),\n\\end{equation}\nfor any continuous function $\\psi:\\mathfrak{D}_{\\lambda,\\mu} \\rightarrow \\mathbb{C}$.\n\nAny probability measure on $\\mathfrak{D}_{\\lambda,\\mu}$ yields a probability measure on the interval $I_{\\lambda}$, given by the pushforward $(p_{\\lambda})_{\\ast}(\\widetilde{\\nu}_{\\lambda,\\mu})$ of the joint spectral measure $\\widetilde{\\nu}_{\\lambda,\\mu}$ under the orthogonal projection $p_{\\lambda}$ onto the spectrum $\\sigma(\\lambda)$ (see \\cite{evans\/pugh:2012i} for more details).\nSince the spectral measure $\\nu_{\\lambda}$ over $I_{\\lambda}$ is also uniquely determined by its (one-dimensional) moments $\\widetilde{\\varsigma}_m = \\int_{I_{\\lambda}} x_{\\lambda}^m \\mathrm{d}\\nu_{\\lambda}(x_{\\lambda})$ for all $m \\in \\mathbb{N}$, one could alternatively consider the moments in (\\ref{eqn:cross_moments_sa_operators}) with $n=0$ to determine the measure $\\nu_{\\lambda}$ over $I_{\\lambda}$.\n\nLet\n\\begin{equation} \\label{def:Dl}\nC_k^W = \\{ (e^{2 \\pi i q_1\/3(k+4)}, e^{2 \\pi i q_2\/3(k+4)}) \\in \\mathbb{T}^2 | \\; q_1,q_2 = 0, 1, \\ldots, 3k+11; \\, q_1 + q_2 \\equiv 0 \\textrm{ mod } 3 \\}\n\\end{equation}\nwhich is the support (over $\\mathbb{T}^2$) of the spectral measure of the nimrep graph $\\mathcal{A}_k(G_2)$ associated to the trivial $G_2$ modular invariant at level $k$.\nThe following $G_2$-invariant measures will be useful later, c.f. \\cite{evans\/pugh:2010i}.\n\\begin{Def} \\label{def:4measures}\nLet $\\omega = e^{2 \\pi i\/3}$, $\\tau = e^{2 \\pi i\/n}$. We define the following measures on $\\mathbb{T}^2$:\n\\begin{itemize}\n\\item[(1)] $\\mathrm{d}_m \\times \\mathrm{d}_n$, where $\\mathrm{d}_k$ is the uniform measure on the $k^{\\mathrm{th}}$ roots of unity, for $k \\in \\mathbb{N}$.\n\\item[(2)] $\\mathrm{d}^{(n)}$, the uniform measure on $C_n^W$ for $n \\in \\mathbb{N}$.\n\\item[(3)] $\\mathrm{d}^{((n))}$, the uniform measure on the $S_3$-orbit of the points $(\\tau, \\tau)$, $(\\overline{\\omega} \\, \\overline{\\tau}, \\omega)$, $(\\omega, \\overline{\\omega} \\, \\overline{\\tau})$, for $n \\in \\mathbb{Q}$, $n \\geq 2$.\n\\item[(4)] $\\mathrm{d}^{(n,k)}$, the uniform measure on the $S_3$-orbit of the points $(\\tau \\, e^{2 \\pi i k}, \\tau)$, $(\\tau, \\tau \\, e^{2 \\pi i k})$, $(\\overline{\\omega} \\, \\overline{\\tau}, \\omega \\, e^{2 \\pi i k})$, $(\\omega \\, e^{2 \\pi i k}, \\overline{\\omega} \\, \\overline{\\tau})$, $(\\overline{\\omega} \\, \\overline{\\tau} \\, e^{-2 \\pi i k}, \\omega \\, e^{-2 \\pi i k})$, $(\\omega \\, e^{-2 \\pi i k}, \\overline{\\omega} \\, \\overline{\\tau} \\, e^{-2 \\pi i k})$, for $n,k \\in \\mathbb{Q}$, $n > 2$, $0 \\leq k \\leq 1\/n$.\n\\end{itemize}\n\\end{Def}\n\nThe sets $\\mathrm{Supp}(\\mathrm{d}^{((n))})$, $\\mathrm{Supp}(\\mathrm{d}^{(n,k)})$ are illustrated in Figures \\ref{fig:poly-15}, \\ref{fig:poly-16} respectively, where $\\mathrm{Supp}(\\mathrm{d}\\mu)$ denotes the set of points $(\\theta_1,\\theta_2) \\in [0,1]^2$ such that $(e^{2 \\pi i \\theta_1}, e^{2 \\pi i \\theta_2})$ is in the support of the measure $\\mathrm{d}\\mu$. The white circles in Figure \\ref{fig:poly-16} denote the points given by the measure $\\mathrm{d}^{((n))}$. The cardinality $|\\mathrm{Supp}(\\mathrm{d}_m \\times \\mathrm{d}_n)|$ of $\\mathrm{Supp}(\\mathrm{d}_m \\times \\mathrm{d}_n)$ is $mn$, whilst $|\\mathrm{Supp}(\\mathrm{d}^{(n)})| = |D_n| = 3n^2$ was shown in \\cite[Section 7.1]{evans\/pugh:2009v}. For $n > 2$ and $0 < k < 1\/n$, $|\\mathrm{Supp}(\\mathrm{d}^{((n))})| = 18$, whilst $|\\mathrm{Supp}(\\mathrm{d}^{(n,k)})| = 36$. The cardinalities of the other sets are $|\\mathrm{Supp}(\\mathrm{d}^{(n,0)})| = |\\mathrm{Supp}(\\mathrm{d}^{(n,1\/n)})| = 18$ for $n > 2$, and $|\\mathrm{Supp}(\\mathrm{d}^{((2))})| = 9$.\nSome relations between these measures are given in \\cite[Section 2]{evans\/pugh:2010i}.\n\n\\begin{figure}[tb]\n\\begin{minipage}[t]{7.5cm}\n\\begin{center}\n  \\includegraphics[width=55mm]{fig-poly-15.eps}\\\\\n \\caption{$\\mathrm{Supp}(\\mathrm{d}^{((n))})$} \\label{fig:poly-15}\n\\end{center}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{7.5cm}\n\\begin{center}\n  \\includegraphics[width=55mm]{fig-poly-16.eps}\\\\\n \\caption{$\\mathrm{Supp}(\\mathrm{d}^{(n,k)})$} \\label{fig:poly-16}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\n\n\\subsection{Spectral measures for $G_2$} \\label{sect:spec_measure-G2}\n\nHere we review the results, determined in \\cite{evans\/pugh:2012i}, for the spectral measures for $G_2$.\nLet $\\rho_1$, $\\rho_2$ denote the fundamental representations of $G_2$ of dimensions 7, 14 respectively.\nThe restrictions of the characters $\\chi_{\\rho_j}$ of the fundamental representations of $G_2$ to $\\mathbb{T}^2$ yield maps from the torus to the interval $I_j = \\chi_{\\rho_j}(\\mathbb{T}^2) \\subset \\mathbb{R}$:\n\\begin{align*}\n\\chi_{\\rho_1}(\\omega_1,\\omega_2) & = 1 + \\omega_1 + \\omega_1^{-1} + \\omega_2 + \\omega_2^{-1} + \\omega_1\\omega_2^{-1} + \\omega_1^{-1}\\omega_2 \\\\\n& = 1 + 2\\cos(2\\pi\\theta_1) + 2\\cos(2\\pi\\theta_2) + 2\\cos(2\\pi(\\theta_1-\\theta_2)), \\\\\n\\chi_{\\rho_2}(\\omega_1,\\omega_2) & = \\chi_{\\rho_1}(\\omega_1,\\omega_2) + 1 + \\omega_1\\omega_2 + \\omega_1^{-1}\\omega_2^{-1} + \\omega_1^2\\omega_2^{-1} + \\omega_1^{-2}\\omega_2 + \\omega_1\\omega_2^{-2} + \\omega_1^{-1}\\omega_2^2 \\\\\n= \\chi_{\\rho_1}&(\\omega_1,\\omega_2) + 1 + 2\\cos(2\\pi(\\theta_1+\\theta_2)) + 2\\cos(2\\pi(2\\theta_1-\\theta_2)) + 2\\cos(2\\pi(\\theta_1-2\\theta_2)),\n\\end{align*}\nwhere $\\omega_j = e^{2\\pi i \\theta_j} \\in \\mathbb{T}$ for $\\theta_j \\in [0,1]$, $j=1,2$.\n\nLet\n\\begin{equation} \\label{eqn:x,y-G2}\nx := \\chi_{\\rho_1}(\\omega_1,\\omega_2), \\qquad y := \\chi_{\\rho_2}(\\omega_1,\\omega_2),\n\\end{equation}\nand denote by $\\Psi$ the map $\\Psi_{(1,0),(1,1)}: (\\omega_1,\\omega_2) \\mapsto (x,y)$.\nThe image $\\mathfrak{D} = \\Psi(C)$ of the fundamental domain $C$ of $\\mathbb{T}^2\/D_{12}$ (illustrated in Figure \\ref{fig:fund_domain-G2inT2}) under $\\Psi$ is illustrated in Figure \\ref{fig:DomainD-G2}, where the boundaries of $C$ given by $\\theta_1 = 2\\theta_2$, $\\theta_1 = -\\theta_2$, $\\theta_1 = \\theta_2$ yield the curves $c_1$, $c_2$, $c_3$ respectively. These curves are given by \\cite{uhlmann\/meinel\/wipf:2007}\n\\begin{align*}\nc_1: && y & = -5(x+1)+2(x+2)^{3\/2}, \\qquad x \\in [-2,7], \\\\\nc_2: && y & = -5(x+1)-2(x+2)^{3\/2}, \\qquad x \\in [-2,-1], \\\\\nc_3: && 4y & = x^2+2x-7, \\hspace{28mm} x \\in [-1,7].\n\\end{align*}\nThe fixed point $(1,1)$ of $\\mathbb{T}^2$ under the action of $D_{12}$ maps to 7, 14 in the intervals $I_1$, $I_2$ respectively.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=60mm]{Fig-DomainD2-G2.eps}\\\\\n \\caption{The domain $\\mathfrak{D} = \\Psi(C)$.} \\label{fig:DomainD-G2}\n\\end{center}\n\\end{figure}\n\nUnder the change of variables (\\ref{eqn:x,y-G2}) the Jacobian is given by \\cite{evans\/pugh:2012i}\n\\begin{equation} \\label{eqn:J[theta]-D12}\n\\begin{split}\nJ & = 8 \\pi^2 (\\cos(2 \\pi (2\\theta_1 + \\theta_2)) + \\cos(2 \\pi (\\theta_1 - 3\\theta_2)) + \\cos(2 \\pi (3\\theta_1 - 2\\theta_2)) \\\\\n& \\qquad - \\cos(2 \\pi (\\theta_1 + 2\\theta_2)) - \\cos(2 \\pi (3\\theta_1 - \\theta_2)) - \\cos(2 \\pi (2\\theta_1 - 3\\theta_2))).\n\\end{split}\n\\end{equation}\nThe Jacobian is real and vanishes in $\\mathbb{T}^2$ only on the boundaries of the images of the fundamental domain $C$ under $D_{12}$.\nAgain, $J^2$ can be written in terms of the $D_{12}$-invariant elements $x$, $y$ as $J^2 = (4x^3-x^2-2x-10xy-y^2-10y+7)(x^2+2x-7-4y)$ (see also \\cite{uhlmann\/meinel\/wipf:2007}), which is non-negative since $J$ is real. We write $J$ in terms of $x$ and $y$ as\n\\begin{eqnarray} \\label{eqn:J[x,y]-D12}\n|J| & = & 4 \\pi^2 \\sqrt{(4x^3-x^2-2x-10xy-y^2-10y+7)(x^2+2x-7-4y)}.\n\\end{eqnarray}\n\n\n\n\n\n\n\n\n\\subsection{Spectral measures for nimrep graphs associated to $G_2$ modular invariants} \\label{sect:spec_measure-nimrepsG2}\n\nSuppose $G$ is the nimrep associated to a $G_2$ braided subfactor at some finite level $k$ with vertex set $G_0$.\nWe define a state $\\varphi$ on $\\ell^2(G_0)$ by $\\varphi( \\, \\cdot \\, ) = \\langle \\,\\cdot \\, e_{\\ast}, e_{\\ast} \\rangle$, where $e_{\\ast}$ is the basis vector in $\\ell^2(G_0)$ corresponding to the distinguished vertex $\\ast$ with lowest Perron-Frobenius weight.\n\nIf we consider the nimrep graphs $G_{\\lambda}$, $G_{\\mu}$, which have joint spectrum $\\mathfrak{D}_{\\lambda,\\mu}$, then the $m,n^{\\mathrm{th}}$ cross moment $\\varsigma_{m,n} = \\varphi(G_{\\lambda}^m G_{\\mu}^n) = \\int_{\\mathfrak{D}_{\\lambda,\\mu}} x^m y^n \\mathrm{d}\\widetilde{\\nu}(x,y)$, where $x=x_{\\lambda}$, $y=x_{\\mu}$, is given by $\\langle G_{\\lambda}^m G_{\\mu}^n e_{\\ast}, e_{\\ast} \\rangle$.\nLet $\\beta_{\\lambda}^{(\\nu)} = \\chi_{\\lambda}(t_{\\nu})$ be the eigenvalues of $G_{\\lambda}$, where $t_{\\nu}=(\\exp(\\xi(\\nu_1+1)),\\exp(-3\\xi (\\nu_2+1)))$ for $\\xi = 6\\pi i\/(k+4)$, with corresponding eigenvectors $\\psi^{(\\nu)}$ (note that the eigenvectors of $G_{\\lambda}$ are the same for all $\\lambda$). Each eigenvalue $\\beta_{\\lambda}^{(\\mu)}$ is also given by a ratio of the $S$-matrix for $G_2$ at level $k$, $\\beta_{\\lambda}^{(\\mu)} = S_{\\lambda\\mu}\/S_{0\\mu}$, where $\\mu \\in \\mathrm{Exp}(G) \\subset P^k_{+} = \\{ (\\lambda_1,\\lambda_2) | \\, \\lambda_1,\\lambda_2 \\geq 0; \\lambda_1 + 2\\lambda_2 \\leq k \\}$ are given by the modular invariant $Z$. Then $G_{\\lambda}^m G_{\\mu}^n = \\mathcal{U} \\Lambda_{\\lambda}^m \\Lambda_{\\mu}^n \\mathcal{U}^{\\ast}$, where $\\Lambda_{\\lambda}$ is the diagonal matrix with the eigenvalues $\\beta_{\\lambda}^{(\\nu)}$ on the diagonal, and $\\mathcal{U}$ is the matrix whose columns are given by the eigenvectors $\\psi^{(\\nu)}$, so that\n\\begin{equation}\\label{eqn:moments-nimrep-G2}\n\\varsigma_{m,n} \\;\\; = \\;\\; \\langle \\mathcal{U} \\Lambda_{\\lambda}^m \\Lambda_{\\mu}^n \\mathcal{U}^{\\ast} e_{\\ast}, e_{\\ast} \\rangle \\;\\; = \\;\\; \\langle \\Lambda_{\\lambda}^m \\Lambda_{\\mu}^n \\mathcal{U}^{\\ast} e_{\\ast}, \\mathcal{U}^{\\ast} e_{\\ast} \\rangle \\;\\; = \\;\\; \\sum_{\\nu} (\\beta_{\\lambda}^{(\\nu)})^m (\\beta_{\\mu}^{(\\nu)})^n |\\psi^{(\\nu)}_{\\ast}|^2,\n\\end{equation}\nwhere $\\psi^{(\\nu)}_{\\ast} = \\mathcal{U}^{\\ast} e_{\\ast}$ is the entry of the eigenvector $\\psi^{(\\nu)}$ corresponding to the distinguished vertex $\\ast$.\nThen there is a $D_{12}$-invariant measure $\\varepsilon$ over $\\mathbb{T}^2$ such that\n$$\\varsigma_{m,n} = \\int_{\\mathbb{T}^2} \\chi_{\\lambda}(\\omega_1,\\omega_2)^m \\chi_{\\mu}(\\omega_1,\\omega_2)^n \\mathrm{d}\\varepsilon(\\omega_1,\\omega_2),$$\nfor all $\\lambda$, $\\mu$.\n\nNote from (\\ref{eqn:moments-nimrep-G2}) that the measure $\\varepsilon$ is a discrete measure which has weight $|\\psi^{(\\nu)}_{\\ast}|^2$ at the points $g(t_{\\nu}) \\in \\mathbb{T}^2$ for $g \\in D_{12}$, $\\nu \\in \\mathrm{Exp}(G)$, and zero everywhere else. Thus the measure $\\varepsilon$ does not depend on the choice of $\\lambda$, $\\mu$, so that the spectral measure over $\\mathbb{T}^2$ is the same for any pair $(G_{\\lambda},G_{\\mu})$, even though the corresponding measures over $\\mathfrak{D}_{\\lambda,\\mu} \\subset \\mathbb{R}^2$, and indeed the subsets $\\mathfrak{D}_{\\lambda,\\mu}$ themselves, are different for each such pair. The same result holds for the spectral measure over $\\mathbb{T}^2$ of a finite subgroup of $G_2$.\n\n\n\n\n\n\n\\section{The finite subgroups of $G_2$} \\label{sect:subgroupsG2}\n\nThe classification of finite subgroups of $G_2$ is due to \\cite{wales:1970, cohen\/wales:1983} (see also \\cite{greiss:1995, he:2003}).\n\nThe reducible (i.e. block-diagonalizable) finite subgroups of $G_2$ are the finite discrete subgroups of $SU(2) \\times SU(2)$ and $SU(3)$ \\cite{wales:1970}. These subgroups are thus well known, and the corresponding spectral measures can be obtained from \\cite{banica\/bisch:2007, evans\/pugh:2009v, evans\/pugh:2010i}.\n\nThe irreducible finite subgroups of $G_2$, of which there are seven up to conjugacy in $G_2$ (or equivalently, up to conjugacy in $GL(V)$, where $V$ is the natural 7-dimensional module for $O(7,\\mathbb{C})$ \\cite[Corollary 1]{greiss:1995}), can be further classified into two types, primitive and imprimitive, where a linear group $\\Gamma \\subset GL(V)$ is imprimitive if there is a non-trivial decomposition $V=\\bigoplus_i V_i$ such that $\\Gamma$ permutes the $V_i$. There are two imprimitive finite subgroups and five primitive ones.\nThese finite subgroups are listed in Table \\ref{Table:subgroupsG2}, where type denotes whether an irreducible subgroup is primitive (P) or imprimitive (I).\n\n\\renewcommand{\\arraystretch}{1}\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\nSubgroup $\\Gamma \\subset G_2$ & Type & $|\\Gamma|$ \\\\\n\\hline\\hline finite subgroups of $SU(2) \\times SU(2)$, $SU(3)$ & - & - \\\\\n\\hline $PSL(2;7) \\cong GL(3;2) \\cong \\Sigma (168) \\subset SU(3)$ & I & 168 \\\\\n\\hline $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ & I & 1344 \\\\\n\\hline $PGL(2;7)$ & P & 336 \\\\\n\\hline $PSL(2;8)$ & P & 504 \\\\\n\\hline $PSL(2;13)$ & P & 1092 \\\\\n\\hline $PU(3;3) \\cong G_2(2)'$ & P & 6048 \\\\\n\\hline $G_2(2)$ & P & 12096 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Finite subgroups of $G_2$.} \\label{Table:subgroupsG2}\n\\end{center}\n\\end{table}\n\nThe McKay graph $\\mathcal{G}^{\\rho}_{\\Gamma}$ is the the fusion graph of the irreducible representation $\\rho$ of $\\Gamma$ acting on the irreducible representations of $\\Gamma$.\nThis graph determines the Bratteli diagram for the tower of relative commutants of the subfactor $P^{\\Gamma} \\subset (M_n \\otimes P)^{\\Gamma}$, where $n$ is the dimension the representation $\\rho$ and $P$ is the type $\\mathrm{II}_1$ factor $\\bigotimes_{n=1}^{\\infty} M_n$ \\cite[$\\S$VI]{wassermann:1988}. This graph is not however the principal graph of this subfactor as it is not bipartite. The prinicpal graph is rather an unfolded version of the McKay graph $\\mathcal{G}^{\\rho}_{\\Gamma}$, with adjacency matrix given by $\\left( \\begin{array}{cc} 0 & \\Delta^{\\rho}_{\\Gamma} \\\\ \\Delta^{\\rho}_{\\Gamma} & 0 \\end{array} \\right)$, where $\\Delta^{\\rho}_{\\Gamma}$ is the adjacency matrix of the (folded) graph $\\mathcal{G}^{\\rho}_{\\Gamma}$.\n\nWe will consider the (joint) spectral measure for the McKay graphs $\\mathcal{G}^j_{\\Gamma} := \\mathcal{G}^{\\varrho_j}_{\\Gamma}$ associated to a finite subgroup $\\Gamma \\subset G_2$, where $\\varrho_j$ are the restrictions of the fundamental representations $\\rho_j$ of $G_2$ to $\\Gamma$, $j=1,2$.\nWe will consider all possible embeddings of the subgroup in $G_2$. Any two such embeddings are conjguate in $G_2$ if and only if they afford the same character on the seven-dimensional representation $\\rho_1$ \\cite[Corollary 1]{greiss:1995}. In some cases there is more than one non-conjugate embedding of the subgroup in $G_2$.\nIn these cases the restricted representation $\\varrho_1$ is not necessarily irreducible.\nIn all cases, even for irreducible $\\varrho_1$, the restriction $\\varrho_2$ of the 14-dimensional representation is not necessarily irreducible.\n\nWe use the following methods to determine embeddings $\\varrho_1$ of $\\Gamma$ in $G_2$.\nFirst, take a seven-dimensional (not necessarily irreducible) representation $\\gamma_1$ of $\\Gamma$.\nThe Kronecker square of $\\rho_1$ decomposes into irreducible representations of $G_2$ as $\\rho_1^2 = \\mathrm{id}_{G_2} + \\rho_1 + \\rho_2 + \\lambda_{(2,0)}$, where $\\lambda_{(2,0)}$ has dimension 21. The Kronecker square of $\\varrho_1$ is obtained by restricting this decomposition to $\\Gamma$, and we see that $\\varrho_1$ appears in the decomposition of $\\varrho_1^2$ into irreducible representations of $\\Gamma$. Thus, if $\\gamma_1$ is not contained in the decomposition of $\\gamma_1^2$ into irreducible representations of $\\Gamma$, we can eliminate $\\gamma_1$ as a possible restriction $\\varrho_1$ of $\\rho_1$. The decomposition of $\\gamma_1^2$ into irreducible representations can be obtained using the character table for $\\Gamma$, by decomposing the character $\\chi_{\\gamma_1^2} = \\chi_{\\gamma_1}^2$ of $\\gamma_1^2$ into the characters of the irreducible representations $\\lambda$ of $\\Gamma$: $\\chi_{\\gamma_1}^2 = \\sum_{\\lambda} a_{\\lambda} \\chi_{\\lambda}$, where $a_{\\lambda} = \\langle \\gamma_1^2, \\lambda \\rangle\/|\\Gamma| = \\sum_{g \\in \\Gamma} \\chi_{\\gamma_1^2}(g)\\chi_{\\lambda}(g)\/|\\Gamma|$.\n\nWe next consider the eigenvalues of the representation matrices of $\\Gamma$. If the elements in a conjugacy class $C_n$ of $\\Gamma$ have order $n$, then $(C_n)^n = Z(\\Gamma)$, where $Z(\\Gamma)$ is the center of $\\Gamma$. If the center is trivial, then the eigenvalues $\\xi$ of the matrices representing these elements must satisfy $\\xi^n = 1$ (this is the case for $PSL$, $PGL$ and $PU(3;3)$). Since $\\chi_{\\lambda}(\\Gamma_j)$ is the sum of the eigenvalues $\\xi$, it is usually possible to write down the complete set of eigenvalues from the information provided by the character table of $\\Gamma$ and the fact that the eigenvalues must be powers of $n^{\\mathrm{th}}$ roots of unity.\nWhere there is some ambiguity, we can pin down the correct choice for the set of eigenvalues from the following considerations. Suppose there is ambiguity regarding the eigenvalues of the conjugacy class $C_{mn}$ whose elements have order $mn$, $m,n \\in \\mathbb{N}$. If there is only one conjugacy class $C_n$ whose elements have order $n$, then for $g \\in C_{mn}$, $g^m \\in C_n$, and since $g, g^m$ commute, their corresponding representation matrices can be simultaneously diagonalised, and thus the eigenvalues of $C_n$ must be $m^{\\mathrm{th}}$ powers of those for $C_{mn}$. Suppose now that there is more than one conjugacy class $C_n^{(j)}$ whose elements have order $n$. Since $g^m$ are all conjugate for conjugate $g$, we see that there exists a $j$ such that $g^m \\in C_n^{(j)}$ for all $g \\in C_{mn}$. It turns out in all the cases considered here that there is only one consistent choice of $j$ such that the eigenvalues of $C_n^{(j)}$ are $m^{\\mathrm{th}}$ powers of those for $C_{mn}$ for all (irreducible) representations.\n\nAs was shown in \\cite[$\\S$4]{evans\/pugh:2009v}, the eigenvalues of the representation matrices of $\\Gamma$ can be written in the form $\\chi_{\\varrho_j}(C) = \\mathrm{Tr}(\\varrho_j(g))$, where $g$ is any element of the conjugacy class $C$ of $\\Gamma$.\nEvery element $g \\in \\Gamma$ is conjugate to an element $d$ in the maximal torus of $G_2$, i.e. $\\varrho_j(h^{-1}gh) = \\varrho_j(d) = (\\varrho_j|_{\\mathbb{T}^2})(t_1,t_2)$ for some $(t_1,t_2) \\in \\mathbb{T}^2$, for $j=1,2$, where $\\varrho_j|_{\\mathbb{T}^2}$ is given by \\cite{evans\/pugh:2012i}\n\\begin{align}\n(\\rho_1|_{\\mathbb{T}^2})&(t_1,t_2) = \\textrm{diag}(D(t_1), D(t_2^{-1}), D(t_1^{-1}t_2), 1), \\label{eqn:restrict_rho1G2_to_T2} \\\\\n(\\rho_2|_{\\mathbb{T}^2})&(t_1,t_2) \\nonumber \\\\\n&= \\textrm{diag}(D(t_1), D(t_2^{-1}), D(t_1^{-1}t_2), D(1), D(t_1t_2), D(t_1^2t_2^{-1}), D(t_1^{-1}t_2^2)), \\label{eqn:restrict_rho2G2_to_T2}\n\\end{align}\nwhere $D(t_i) = \\left( \\begin{array}{cc} \\mathrm{Re}(t_i) & -\\mathrm{Im}(t_i) \\\\ \\mathrm{Im}(t_i) & \\mathrm{Re}(t_i) \\end{array} \\right)$ for $t_i \\in \\mathbb{T}$.\nNow $\\mathrm{Tr}(\\varrho_j(g)) = Tr(\\varrho_j(d)) = \\Phi_j(t_1,t_2)$, thus the eigenvalues of $\\varrho_j(g)$ are all of the form $\\Phi_j(\\omega_1,\\omega_2)$ for $\\omega_1,\\omega_2 \\in \\mathbb{T}$, and hence its spectrum is contained in the interval $I_j$.\nAs shown in \\cite[Sections 3,4]{evans\/pugh:2012i} the spectrum of the fundamental representation $\\rho_j$ of $G_2$, and its restriction to $\\mathbb{T}^2$, is the whole of the interval $I_j = \\chi_{\\rho_j}(\\mathbb{T}^2)$, for $j=1,2$.\nThus the support of the spectral measure $\\mu_{\\Delta_j}$ of $\\Delta_j = \\Delta_{\\mathcal{G}^j_{\\Gamma}}$, the adjacency matrix of $\\mathcal{G}^j_{\\Gamma}$, is contained in $I_j$ when $\\Gamma$ is $G_2$ or one of its finite subgroups.\nThen for $\\Gamma \\subset G_2$, the eigenvalues of every group element in $\\varrho_1$ are necessarily of the form $\\mathcal{E}_{t_1,t_2} := \\{ 1,t_1,t_1^{-1},t_2,t_2^{-1},t_1t_2^{-1},t_1^{-1}t_2 \\}$, where $t_i \\in \\mathbb{T}$. Thus, by \\cite[Proposition]{king\/toumazet\/wybourne:1999}, if the eigenvalues of the group elements in the representation $\\gamma_1$ have this form then $\\gamma_1$ is a restriction $\\varrho_1$ of the seven-dimensional fundamental representation $\\rho_1$ of $G_2$ to $\\Gamma$.\n\nWe now turn to consider the possible restrictions $\\varrho_2$ of the fourteen-dimensional fundamental representation $\\rho_2$ of $G_2$ to $\\Gamma$, for fixed $\\varrho_1$.\nBy dimension considerations one can determine the possible candidates for $\\varrho_2$ from the Kronecker square of $\\varrho_1$.\nLet $\\gamma_2$ be such a candidate, i.e. a fourteen-dimensional representation such that $\\varrho_1^2 = \\mathrm{id}_{\\Gamma} + \\varrho_1 + \\gamma_2 + \\lambda$, where $\\lambda$ is (necessarily) some 27-dimensional representation of $\\Gamma$.\nWe make a choice of pair $(t_1^{C},t_2^{C})$ from the set $X_C$ of eigenvalues of group elements (from the conjugacy class $C$) in $\\varrho_1$ such that $\\mathcal{E}_{t_1^C,t_2^C} = X_C$.\nNote that the choice of $(t_1^C,t_2^C)$ such that $\\mathcal{E}_{t_1^C,t_2^C} = X_C$ is not unique. However, any other pair $(\\tilde{t}_1^C,\\tilde{t}_2^C)$ such that $\\mathcal{E}_{\\tilde{t}_1^C,\\tilde{t}_2^C} = X_C$ will appear in the orbit of $(t_1^C,t_2^C)$ under the action of the Weyl group $D_{12}$ of $G_2$, where the action of $D_{12}$ on $\\mathbb{T}^2$ is given in Section \\ref{sect:measures-different_domains}. Thus $\\Phi_j(\\tilde{t}_1^C,\\tilde{t}_2^C) = \\Phi_j(t_1^C,t_2^C)$ for $j=1,2$.\nThen one checks that $\\Phi_2(t_1^C,t_2^C) = \\chi_{\\gamma_2}(C)$ for each conjugacy class $C$, in which case $\\gamma_2$ is indeed a restriction $\\varrho_2$ of the fourteen-dimensional fundamental representation $\\rho_2$ of $G_2$ to $\\Gamma$.\n\n\\subsection{Spectral measures for finite subgroups of $G_2$} \\label{sect:spec_measure-subgroupsG2}\n\nThere is an $S$-matrix, with rows, columns labelled by the irreducible characters, conjugacy classes respectively of $\\Gamma$, which simultaneously diagonalizes the representations of $\\Gamma$ \\cite{kawai:1989}. The entries of this matrix for the trivial representation 0 are given by $S_{0,C} = \\sqrt{|C|} \\chi_0(C)\/\\sqrt{|\\Gamma|} = \\sqrt{|C|}\/\\sqrt{|\\Gamma|}$, for conjugacy class $C$.\nThen the $m,n^{\\mathrm{th}}$ moment $\\varsigma_{m,n}$ is given over $\\mathfrak{D}$ by (c.f. \\cite[Section 4]{evans\/pugh:2009v} for the case of finite subgroups of $SU(2)$)\n\\begin{equation} \\label{eqn:moments-subgroupG2}\n\\varsigma_{m,n} \\; = \\; \\int_{\\mathfrak{D}} x^m y^n \\mathrm{d}\\nu(x,y) \\; = \\; \\sum_C \\frac{|C|}{|\\Gamma|} \\chi_{\\varrho_1} (C)^m \\chi_{\\varrho_2} (C)^n.\n\\end{equation}\nThere is an analogous statement to (\\ref{eqn:moments-subgroupG2}) for the joint spectral measure $\\nu_{\\lambda,\\mu}$ over $\\mathfrak{D}_{\\lambda,\\mu}$ for any irreducible representations $\\lambda$, $\\mu$ of $\\Gamma$. The weight on the right hand side will again be $|C|\/|\\Gamma|$, since the same $S$-matrix simultaneously diagonalises all the representations of $\\Gamma$. Since this weight does not depend on the representations $\\lambda$, $\\mu$, we see that the $D_{12}$-invariant pullback measure $\\varepsilon$ over $\\mathbb{T}^2$ will be the same for any joint spectral measure $\\nu_{\\lambda,\\mu}$. This is analogous to the situation for nimrep graphs discussed in Section \\ref{sect:spec_measure-nimrepsG2}.\n\nWe wish to compute `inverse' maps $\\widetilde{\\Psi}: \\mathfrak{D} \\rightarrow \\mathbb{T}^2$ such that $\\Psi \\circ \\widetilde{\\Psi} = \\mathrm{id}$.\nThe following equation can easily be checked by substituting in $x=\\Phi_1(\\omega_1,\\omega_2)$, $y=\\Phi_2(\\omega_1,\\omega_2)$:\n$$(\\omega_j+\\omega_j^{-1})^3 + (1-x)(\\omega_j+\\omega_j^{-1})^2 + (y-2)(\\omega_j+\\omega_j^{-1}) +2y-x^2+2x-1 = 0,$$\nwhere $j=1,2$.\nSolving this cubic in $\\omega_j+\\omega_j^{-1} = 2\\cos(\\vartheta_j)$, we obtain for $l \\in \\{ 0,1,2 \\}$\n$$\\vartheta_j(l) = \\cos^{-1}\\left( \\frac{1}{6} \\left( x-1 + 2^{-1\/3} \\epsilon_l P + 2^{1\/3} \\overline{\\epsilon_l} (x^2-2x+7-3y)P^{-1} \\right) \\right)$$\nwhere $2^{1\/3}$ takes a real value, $\\epsilon_l = e^{2 \\pi i l\/3}$ and $P = (2x^3+21x^2-30x+7-45y-9xy + \\sqrt{(4x^3-x^2-2x-10xy-y^2-10y+7)(x^2+2x-7-4y)} \\, )^{1\/3}$. We note that for the roots of a cubic equation it does not matter whether the square root in $P$ is taken to be positive or negative.\nThen we set $\\Psi_{l,l'}(x,y) = (e^{\\vartheta_1(l)i},e^{\\vartheta_2(l')i})$, and we have that $\\Psi(\\Psi_{l,l'}(x,y)) = (x,y)$ for some $l,l' \\in \\{ 0,1,2 \\}$. The particular choice of pair $l,l'$ such that the equality $\\Psi \\circ \\Psi_{l,l'} = \\mathrm{id}$ is satisfied depends on $x,y$, but it is easy to check (eg. using Mathematica) whether a given choice satisfies this equality for any of the examples we consider.\nWe present in Table \\ref{Table:subgroupsG2-orbits(theta1,theta2)} the values of the eigenvalues $(\\chi_{\\varrho_1}(C), \\chi_{\\varrho_2}(C)) = (x,y) \\in \\mathfrak{D}$ which will appear for the finite subgroups of $G_2$, and (the orbits under $D_{12}$ of) the corresponding points $(\\theta_1,\\theta_2) \\in [0,1]^2$ such that $\\Psi(e^{2\\pi i \\theta_1},e^{2\\pi i \\theta_2}) = (x,y)$.\n\n\\renewcommand{\\arraystretch}{1.4}\n\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{|c|c|} \\hline\n$(x,y) \\in \\mathfrak{D}$ & Orbit of $(\\theta_1,\\theta_2) \\in [0,1]^2$ \\\\\n\\hline $(7,14)$ & $(0,0)$ \\\\\n\\hline $(-2,5)$ & $\\left(\\frac{1}{3},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{1}{3}\\right)$ \\\\\n\\hline $(-1,-2)$ & $\\left(0,\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{1}{2}\\right), \\left(\\frac{1}{2},0\\right)$ \\\\\n\\hline $(1,-1)$ & $\\left(0,\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{1}{3}\\right), \\left(\\frac{1}{3},0\\right), \\left(0,\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{2}{3}\\right), \\left(\\frac{2}{3},0\\right)$ \\\\\n\\hline $(3,2)$ & $\\left(0,\\frac{1}{4}\\right), \\left(\\frac{1}{4},\\frac{1}{4}\\right), \\left(\\frac{1}{4},0\\right), \\left(0,\\frac{3}{4}\\right), \\left(\\frac{3}{4},\\frac{3}{4}\\right), \\left(\\frac{3}{4},0\\right)$ \\\\\n\\hline $(-1,2)$ & $\\left(\\frac{1}{4},\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{1}{4}\\right), \\left(\\frac{1}{4},\\frac{3}{4}\\right), \\left(\\frac{3}{4},\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{3}{4}\\right), \\left(\\frac{3}{4},\\frac{1}{4}\\right)$ \\\\\n\\hline $(2,1)$ & $\\left(\\frac{1}{6},\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{1}{6}\\right), \\left(\\frac{1}{6},\\frac{5}{6}\\right), \\left(\\frac{5}{6},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{5}{6}\\right), \\left(\\frac{5}{6},\\frac{1}{6}\\right)$ \\\\\n\\hline $(-1,1)$ & $\\left(\\frac{1}{6},\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{5}{6}\\right), \\left(\\frac{5}{6},\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{1}{6}\\right),$ \\\\\n& $\\left(\\frac{1}{2},\\frac{1}{6}\\right), \\left(\\frac{1}{3},\\frac{1}{2}\\right), \\left(\\frac{5}{6},\\frac{1}{3}\\right), \\left(\\frac{1}{2},\\frac{5}{6}\\right), \\left(\\frac{2}{3},\\frac{1}{2}\\right), \\left(\\frac{1}{6},\\frac{2}{3}\\right)$ \\\\\n\\hline $(0,0)$ & $\\left(\\frac{1}{7},\\frac{3}{7}\\right), \\left(\\frac{3}{7},\\frac{2}{7}\\right), \\left(\\frac{2}{7},\\frac{6}{7}\\right), \\left(\\frac{6}{7},\\frac{4}{7}\\right), \\left(\\frac{4}{7},\\frac{5}{7}\\right), \\left(\\frac{5}{7},\\frac{1}{7}\\right),$ \\\\\n& $\\left(\\frac{3}{7},\\frac{1}{7}\\right), \\left(\\frac{2}{7},\\frac{3}{7}\\right), \\left(\\frac{6}{7},\\frac{2}{7}\\right), \\left(\\frac{4}{7},\\frac{6}{7}\\right), \\left(\\frac{5}{7},\\frac{4}{7}\\right), \\left(\\frac{1}{7},\\frac{5}{7}\\right)$ \\\\\n\\hline $(1,0)$ & $\\left(\\frac{1}{8},\\frac{3}{8}\\right), \\left(\\frac{3}{8},\\frac{1}{4}\\right), \\left(\\frac{1}{4},\\frac{7}{8}\\right), \\left(\\frac{7}{8},\\frac{5}{8}\\right), \\left(\\frac{5}{8},\\frac{3}{4}\\right), \\left(\\frac{3}{4},\\frac{1}{8}\\right),$ \\\\\n& $\\left(\\frac{3}{8},\\frac{1}{8}\\right), \\left(\\frac{1}{4},\\frac{3}{8}\\right), \\left(\\frac{7}{8},\\frac{1}{4}\\right), \\left(\\frac{5}{8},\\frac{7}{8}\\right), \\left(\\frac{3}{4},\\frac{5}{8}\\right), \\left(\\frac{1}{8},\\frac{3}{4}\\right)$ \\\\\n\\hline $(-1,0)$ & $\\left(\\frac{1}{8},\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{3}{8}\\right), \\left(\\frac{3}{8},\\frac{7}{8}\\right), \\left(\\frac{7}{8},\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{5}{8}\\right), \\left(\\frac{5}{8},\\frac{1}{8}\\right),$ \\\\\n& $\\left(\\frac{1}{2},\\frac{1}{8}\\right), \\left(\\frac{3}{8},\\frac{1}{2}\\right), \\left(\\frac{7}{8},\\frac{3}{8}\\right), \\left(\\frac{1}{2},\\frac{7}{8}\\right), \\left(\\frac{5}{8},\\frac{1}{2}\\right), \\left(\\frac{1}{8},\\frac{5}{8}\\right)$ \\\\\n\\hline $(-p,p+q+1)$ & $\\left(\\frac{1}{9},\\frac{4}{9}\\right), \\left(\\frac{4}{9},\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{8}{9}\\right), \\left(\\frac{8}{9},\\frac{5}{9}\\right), \\left(\\frac{5}{9},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{1}{9}\\right),$ \\\\\n& $\\left(\\frac{4}{9},\\frac{1}{9}\\right), \\left(\\frac{1}{3},\\frac{4}{9}\\right), \\left(\\frac{8}{9},\\frac{1}{3}\\right), \\left(\\frac{5}{9},\\frac{8}{9}\\right), \\left(\\frac{2}{3},\\frac{5}{9}\\right), \\left(\\frac{1}{9},\\frac{2}{3}\\right)$ \\\\\n\\hline $(-q,1-p)$ & $\\left(\\frac{1}{9},\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{2}{9}\\right), \\left(\\frac{2}{9},\\frac{8}{9}\\right), \\left(\\frac{8}{9},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{7}{9}\\right), \\left(\\frac{7}{9},\\frac{1}{9}\\right),$ \\\\\n& $\\left(\\frac{1}{3},\\frac{1}{9}\\right), \\left(\\frac{2}{9},\\frac{1}{3}\\right), \\left(\\frac{8}{9},\\frac{2}{9}\\right), \\left(\\frac{2}{3},\\frac{8}{9}\\right), \\left(\\frac{7}{9},\\frac{2}{3}\\right), \\left(\\frac{1}{9},\\frac{7}{9}\\right)$ \\\\\n\\hline $(p+q,1-q)$ & $\\left(\\frac{2}{9},\\frac{5}{9}\\right), \\left(\\frac{5}{9},\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{7}{9}\\right), \\left(\\frac{7}{9},\\frac{4}{9}\\right), \\left(\\frac{4}{9},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{2}{9}\\right),$ \\\\\n& $\\left(\\frac{5}{9},\\frac{2}{9}\\right), \\left(\\frac{1}{3},\\frac{5}{9}\\right), \\left(\\frac{7}{9},\\frac{1}{3}\\right), \\left(\\frac{5}{9},\\frac{7}{9}\\right), \\left(\\frac{2}{3},\\frac{4}{9}\\right), \\left(\\frac{2}{9},\\frac{2}{3}\\right)$ \\\\\n\\hline $(0,-1)$ & $\\left(\\frac{1}{12},\\frac{5}{12}\\right), \\left(\\frac{5}{12},\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{11}{12}\\right), \\left(\\frac{11}{12},\\frac{7}{12}\\right), \\left(\\frac{7}{12},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{1}{12}\\right),$ \\\\\n& $\\left(\\frac{5}{12},\\frac{1}{12}\\right), \\left(\\frac{1}{3},\\frac{5}{12}\\right), \\left(\\frac{11}{12},\\frac{1}{3}\\right), \\left(\\frac{7}{12},\\frac{11}{12}\\right), \\left(\\frac{2}{3},\\frac{7}{12}\\right), \\left(\\frac{1}{12},\\frac{2}{3}\\right)$ \\\\\n\\hline $\\left(\\frac{1+\\sqrt{13}}{2},1\\right)$ & $\\left(\\frac{1}{13},\\frac{4}{13}\\right), \\left(\\frac{4}{13},\\frac{3}{13}\\right), \\left(\\frac{3}{13},\\frac{12}{13}\\right), \\left(\\frac{12}{13},\\frac{9}{13}\\right), \\left(\\frac{9}{13},\\frac{10}{13}\\right), \\left(\\frac{10}{13},\\frac{1}{13}\\right),$ \\\\\n& $\\left(\\frac{4}{13},\\frac{1}{13}\\right), \\left(\\frac{3}{13},\\frac{4}{13}\\right), \\left(\\frac{12}{13},\\frac{3}{13}\\right), \\left(\\frac{9}{13},\\frac{12}{13}\\right), \\left(\\frac{10}{13},\\frac{9}{13}\\right), \\left(\\frac{1}{13},\\frac{10}{13}\\right)$ \\\\\n\\hline $\\left(\\frac{1-\\sqrt{13}}{2},1\\right)$ & $\\left(\\frac{2}{13},\\frac{7}{13}\\right), \\left(\\frac{7}{13},\\frac{5}{13}\\right), \\left(\\frac{5}{13},\\frac{11}{13}\\right), \\left(\\frac{11}{13},\\frac{6}{13}\\right), \\left(\\frac{6}{13},\\frac{8}{13}\\right), \\left(\\frac{8}{13},\\frac{2}{13}\\right),$ \\\\\n& $\\left(\\frac{7}{13},\\frac{2}{13}\\right), \\left(\\frac{5}{13},\\frac{7}{13}\\right), \\left(\\frac{11}{13},\\frac{5}{13}\\right), \\left(\\frac{6}{13},\\frac{11}{13}\\right), \\left(\\frac{8}{13},\\frac{6}{13}\\right), \\left(\\frac{2}{13},\\frac{8}{13}\\right)$ \\\\\n\\hline\n\\end{tabular}\n\\caption{$(x,y) \\in \\mathfrak{D}$ and (the orbits of) the corresponding points $(\\theta_1,\\theta_2) \\in [0,1]^2$. Here $p=2\\cos(4\\pi\/9)$, $q=2\\cos(8\\pi\/9)$} \\label{Table:subgroupsG2-orbits(theta1,theta2)}\n\\end{center}\n\\end{table}\n\n\\renewcommand{\\arraystretch}{1}\n\n\n\\section{Group $PSL(2;7) \\cong GL(3;2) \\cong \\Sigma (168)$} \\label{sect:II1}\n\nThe subgroup $PSL(2;7)$ of $G_2$ is an irreducible imprimitive group of order 168 which is isomorphic to the group $GL(3;2)$, and also to the subgroup $\\Sigma(168)$ of $SU(3)$ which was considered in \\cite{evans\/pugh:2010i}.\n\nThe group $PSL(2;7)$ has irreducible real representations $\\Sigma_d$ of dimensions $d = 1,6,7,8$, and two complex conjugate irreducible representations $\\Sigma_3, \\Sigma_3^{\\ast}$ of dimension 3. Its character table is given in Table \\ref{table:Character_table-II1} \\cite{littlewood:1934}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_2$ & $(C_7,C_7^2,C_7^4)$ & $(C_7^3,C_7^5,C_7^6)$ & $(C_4,C_4^3)$ & $(C_3,C_3^2)$ \\\\\n\\hline $|C|$ & 1 & 21 & 24 & 24 & 42 & 56 \\\\\n\\hline \\hline $\\Sigma_1$ & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_3$ & 3 & -1 & $w$ & $\\overline{w}$ & 1 & 0 \\\\\n\\hline $\\Sigma_3^{\\ast}$ & 3 & -1 & $\\overline{w}$ & $w$ & 1 & 0 \\\\\n\\hline $\\Sigma_6$ & 6 & 2 & -1 & -1 & 0 & 0  \\\\\n\\hline $\\Sigma_7$ & 7 & -1 & 0 & 0 & -1 & 1  \\\\\n\\hline $\\Sigma_8$ & 8 & 0 & 1 & 1 & 0 & -1  \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for group $PSL(2;7)$, where $w = \\eta + \\eta^2 + \\eta^4 = (-1+i\\sqrt{7})\/2$, $\\eta = e^{2\\pi i\/7}$.} \\label{table:Character_table-II1}\n\\end{center}\n\\end{table}\n\nThere are two non-conjugate embeddings of $PSL(2;7)$ in $G_2$ \\cite{king\/toumazet\/wybourne:1999}, given by $\\varrho_1^{(1)} = \\Sigma_7$ and $\\varrho_1^{(2)} = \\Sigma_1 + \\Sigma_3 + \\Sigma_3^{\\ast}$.\nThe McKay graph $\\mathcal{G}^{\\varrho_1^{(1)}}_{PSL(2;7)}$ for $\\varrho_1^{(1)}$ is given in \\cite[Figure 1]{he:2003}. We reproduce it in Figure \\ref{Fig-McKay_Graph-II1-rho1} for completeness, along with the McKay graph $\\mathcal{G}^{\\varrho_1^{(2)}}_{PSL(2;7)}$ for $\\varrho_1^{(2)}$. We use the notation $n$, $n^{\\ast}$ to label the vertices corresponding to the irreducible representations $\\Sigma_n$, $\\Sigma_n^{\\ast}$ respectively.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=110mm]{Fig-McKay_Graph-II1-rho1}\\\\\n \\caption{The McKay graphs $\\mathcal{G}^{\\varrho_1^{(i)}}_{PSL(2;7)}$, $i=1,2$.} \\label{Fig-McKay_Graph-II1-rho1}\n\\end{center}\n\\end{figure}\n\nThe eigenvalues of the representation matrices are given in \\cite[Tables 4a,b]{king\/toumazet\/wybourne:1999}.\nThe decomposition of the Kronecker square of $\\varrho_1^{(i)}$ into irreducibles is given by\n$$(\\varrho_1^{(1)})^2 = \\mathrm{id} + \\varrho_1^{(1)} + \\Sigma_3 + \\Sigma_3^{\\ast} + 2\\Sigma_6 + \\Sigma_7 + 2 \\Sigma_8, \\qquad\n(\\varrho_1^{(2)})^2 = \\mathrm{id} + \\varrho_1^{(2)} + \\Sigma_1 + 2\\Sigma_3 + 2\\Sigma_3^{\\ast} + 2\\Sigma_6 + 2 \\Sigma_8,$$\nwhere $\\mathrm{id} = \\Sigma_1$.\nFrom dimension considerations there are thus two candidates for the fourteen-dimensional representation $\\varrho_2^{(i)}$, which are given by $\\Sigma_3 + \\Sigma_3^{\\ast} + \\Sigma_8$ and $\\Sigma_6 + \\Sigma_8$ for both $i=1,2$.\nHowever, as discussed in Section \\ref{sect:subgroupsG2}, since $\\chi_{\\varrho_2}(C) = \\Phi_2(t_1^C,t_2^C)$, where $(t_1^{C},t_2^{C})$ is a pair from the set of eigenvalues of group elements from the conjugacy class $C$, from knowledge of the eigenvalues from \\cite{king\/toumazet\/wybourne:1999} we see that the decomposition of the fundamental fourteen-dimensional representation into irreducible representations of $PSL(2;7)$ is given by $\\varrho_2 := \\varrho_2^{(i)} = \\Sigma_3 + \\Sigma_3^{\\ast} + \\Sigma_8$ for both $i=1,2$.\nThe values of $x^{(i)} = \\chi_{\\varrho_1^{(i)}}(C) \\in [-2,7]$, $y = \\chi_{\\rho_2}(C) \\in [-2,14]$ for $PSL(2;7)$ are given in Table \\ref{table:(x,y)-II1}, along with the values of $J^2\/64\\pi^4$ for the corresponding pairs $(\\theta_1,\\theta_2) \\in [0,1]^2$ obtained from Table \\ref{Table:subgroupsG2-orbits(theta1,theta2)}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_2$ & $(C_7,C_7^2,C_7^4)$ & $(C_7^3,C_7^5,C_7^6)$ & $(C_4,C_4^3)$ & $(C_3,C_3^2)$ \\\\\n\\hline $\\chi_{\\varrho_1^{(1)}}(C) \\in [-2,7]$ & 7 & -1 & 0 & 0 & -1 & 1  \\\\\n\\hline $\\chi_{\\varrho_1^{(2)}}(C) \\in [-2,7]$ & 7 & -1 & 0 & 0 & 3 & 1  \\\\\n\\hline $\\chi_{\\varrho_2}(C) \\in [-2,14]$ & 14 & -2 & 0 & 0 & 2 & -1 \\\\\n\\hline $J^2(\\theta_1,\\theta_2)\/64\\pi^4$ & 0 & 0 & 49\/4 & 49\/4 & 0 & 0 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{$\\chi_{\\varrho_j}(C)$ for group $PSL(2;7)$, $j=1,2$.} \\label{table:(x,y)-II1}\n\\end{center}\n\\end{table}\n\nLet $\\Omega(\\theta_1,\\theta_2) := \\Phi_1(e^{2\\pi i \\theta_1},e^{2\\pi i \\theta_2})^m \\Phi_2(e^{2\\pi i \\theta_1},e^{2\\pi i \\theta_2})^n \\in \\mathfrak{D}$ and $\\Omega^W(\\theta_1,\\theta_2)$ its orbit under $W=D_{12}$, $\\Omega^W(\\theta_1,\\theta_2) := \\sum_{g \\in D_{12}} \\Omega(g(\\theta_1,\\theta_2))\/12$.\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:Character_table-II1}, \\ref{table:(x,y)-II1} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n$$\\varsigma_{m,n} = \\frac{1}{168} \\Omega^W(0,0) + \\frac{21}{168} \\Omega^W(0,1\/2) + \\frac{56}{168} \\Omega^W(0,1\/3) + \\frac{42}{168} \\Omega' + \\frac{24+24}{168} \\Omega^W(1\/7,3\/7),$$\nwhere $\\Omega'$ is $\\Omega^W(1\/4,1\/2)$ for $\\varrho_1^{(1)}$, and $\\Omega^W(0,1\/4)$ for $\\varrho_1^{(2)}$.\nIt is easy to see that $\\Omega^W(0,0) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\mathrm{d}_1 \\times \\mathrm{d}_1$ and $3\\Omega^W(0,1\/2) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (4 \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 - \\mathrm{d}_1 \\times \\mathrm{d}_1)$.\nNow $12\\Omega^W(1\/7,3\/7) = 4 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}_7 \\times \\mathrm{d}_7$, as illustrated in Figure \\ref{Fig-OmegaWII1}$(a)$ since the Jacobian $J=0$ along the boundaries of the orbit of the fundamental domain, whilst $J^2(g(1\/7,3\/7)\/64\\pi^4) = 49\/4$ for all $g \\in D_{12}$.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=135mm]{Fig-OmegaWII1}\\\\\n \\caption{The orbits of $(a)$ $(1\/7,3\/7)$, \\mbox{$(b)$ $(0,1\/4) \\, \\bullet$ and $(1\/4,1\/2) \\, \\ast$,} $(c)$ $(0,1\/3)$.} \\label{Fig-OmegaWII1}\n\\end{center}\n\\end{figure}\n\nThe orbits of $(0,1\/4)$, $(1\/4,1\/2)$ are illustrated in Figure \\ref{Fig-OmegaWII1}$(b)$, represented by $\\bullet$, $\\ast$ respectively. Both orbits lie on the boundary of the fundamental domains of $\\mathbb{T}^2\/D_{12}$, however, only the orbit of $(1\/4,1\/2)$ lies on the boundary of the fundamental domains of $\\mathbb{T}^2\/S_3$, illustrated in Figure \\ref{fig:fund_domain-A2inT2}. Then $6K(0,1\/4)\\,\\Omega^W(0,1\/4) = 16 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) K(\\theta_1,\\theta_2) \\, \\mathrm{d}_4 \\times \\mathrm{d}_4$ where $K$ is an $S_3$-invariant function on $\\mathbb{T}^2$ which is zero on the boundaries of the fundamental domains of $\\mathbb{T}^2\/S_3$. Such an $S_3$-invariant function is given by the square $\\widetilde{J}^2$ of the Jacobian which appeared for the $A_2$ spectral measures in \\cite{evans\/pugh:2009v, evans\/pugh:2010i}, which is given by $\\widetilde{J}(\\theta_1,\\theta_2) = 4\\pi^2(\\sin(2\\pi(\\theta_1+\\theta_2))-\\sin(2\\pi(2\\theta_1-\\theta_2))-\\sin(2\\pi(2\\theta_2-\\theta_1)))$. Now $|\\widetilde{J}(0,1\/4)| = 8\\pi^2$, thus we take $K = 16\\widetilde{J}^2\/64\\pi^4 = \\widetilde{J}^2\/4\\pi^4$, and we have $K(0,1\/4) = 16$. We can thus also obtain an expression for $\\Omega^W(1\/4,1\/2)$, where from Figure \\ref{Fig-OmegaWII1}$(b)$ we see that $6\\Omega^W(1\/4,1\/2) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\left( (16 - K(\\theta_1,\\theta_2)) \\, \\mathrm{d}_4 \\times \\mathrm{d}_4 - 4 \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 \\right)$.\n\nFinally, the orbit of $(\\theta_1,\\theta_2)=(0,1\/3)$ is illustrated in Figure \\ref{Fig-OmegaWII1}$(c)$, and thus $6\\Omega^W(0,1\/3) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\, (9 \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 - 3 \\, \\mathrm{d}^{(1)})$, and we obtain\n\n\\begin{Thm} \\label{thm:measureII1}\nThe joint spectral measure (over $\\mathbb{T}^2$) for the non-conjugate embeddings of the projective special linear group $PSL(2;7)$ over the finite field $\\mathbb{F}_7$ into the fundamental representations of $G_2$ is\n\\begin{equation} \\label{eqn:measureII1}\n\\mathrm{d}\\varepsilon = \\frac{1}{672\\pi^4} J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7 + \\frac{1}{24} K' \\, \\mathrm{d}_4 \\times \\mathrm{d}_4 + \\frac{1}{2} \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 - \\frac{1}{28} \\, \\mathrm{d}_1 \\times \\mathrm{d}_1 - \\frac{1}{6} \\, \\mathrm{d}^{(1)},\n\\end{equation}\nwhere $K' = 16-K$ for the embedding of $PSL(2;7)$ in $G_2$ given by $\\varrho_1^{(1)} = \\Sigma_7$ and $K' = K$ for the embedding given by $\\varrho_1^{(2)} = \\Sigma_1 + \\Sigma_3 + \\Sigma_3^{\\ast}$, where $K(\\theta_1,\\theta_2) = (\\sin(2\\pi(\\theta_1+\\theta_2))-\\sin(2\\pi(2\\theta_1-\\theta_2))-\\sin(2\\pi(2\\theta_2-\\theta_1)))^2$, $\\mathrm{d}_m$ is the uniform measure over $m^{\\mathrm{th}}$ roots of unity and $\\mathrm{d}^{(k+4)}$ is the uniform measure on the points in $C_k^W$.\n\\end{Thm}\n\n\\begin{Rem}\nNote that measure in Theorem \\ref{thm:measureII1} for the second embedding $\\varrho_1^{(2)}$ of $PSL(2;7)$ in $G_2$ is precisely that for $\\Sigma (168) \\subset SU(3)$ given in \\cite[Theorem 16]{evans\/pugh:2010i}. However, (\\ref{eqn:measureII1}) has a neater expression than that given in \\cite{evans\/pugh:2010i}, because here we were able to use the Jacobian $J$ for $G_2$ which is also 0 along the diagonal, whereas the Jacobian for $SU(3)$ (essentially $K$ in Theorem \\ref{thm:measureII1}) is non-zero along the diagonal. \\end{Rem}\n\n\n\\section{Group $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$} \\label{sect:II2}\n\nThe subgroup $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ of $G_2$ is an irreducible imprimitive group of order 1344.\nIt has eleven irreducible representations (nine real and two complex conjugate representations) and its character table is given in Table \\ref{table:Character_table-II2} (see \\cite{littlewood:1934, he:2003}),\nwhere elements in $C_4$, $C_4^{\\prime}$, $C_4^{\\prime}$, $C_7$, $C_7^{\\prime}$, $C_6$, $C_3$ are of cycle type $(C_4,C_4^3)$, $(C_4^{\\prime},C_4^{\\prime3})$, $(C_4^{\\prime\\prime},C_4^{\\prime\\prime3})$, $(C_7,C_7^2,C_7^4)$, $(C_7^3,C_7^5,C_7^6)$, $(C_6,C_6^5)$, $(C_3,C_3^2)$ respectively.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_2$ & $C_2^{\\prime}$ & $C_2^{\\prime\\prime}$ & $C_4$ & $C_4^{\\prime}$ & $C_4^{\\prime\\prime}$ & $C_7$ & $C_7^{\\prime}$ & $C_6$ & $C_3$ \\\\\n\\hline $|C|$ & 1 & 7 & 42 & 42 & 84 & 168 & 168 & 192 & 192 & 224 & 224 \\\\\n\\hline \\hline $\\Sigma_1$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_3$ & 3 & 3 & -1 & -1 & -1 & 1 & 1 & $w$ & $\\overline{w}$ & 0 & 0 \\\\\n\\hline $\\Sigma_3^{\\ast}$ & 3 & 3 & -1 & -1 & -1 & 1 & 1 & $\\overline{w}$ & $w$ & 0 & 0 \\\\\n\\hline $\\Sigma_6$ & 6 & 6 & 2 & 2 & 2 & 0 & 0 & -1 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_7^{(1)}$ & 7 & -1 & -1 & 3 & -1 & -1 & 1 & 0 & 0 & -1 & 1 \\\\\n\\hline $\\Sigma_7^{(1)\\prime}$ & 7 & -1 & 3 & -1 & -1 & 1 & -1 & 0 & 0 & -1 & 1 \\\\\n\\hline $\\Sigma_7^{(2)}$ & 7 & 7 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 1 & 1 \\\\\n\\hline $\\Sigma_8$ & 8 & 8 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1 \\\\\n\\hline $\\Sigma_{14}$ & 14 & -2 & 2 & 2 & -2 & 0 & 0 & 0 & 0 & 1 & -1 \\\\\n\\hline $\\Sigma_{21}$ & 21 & -3 & -3 & 1 & 1 & 1 & -1 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{21}'$ & 21 & -3 & 1 & -3 & 1 & -1 & 1 & 0 & 0 & 0 & 0 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$, where $w = \\eta + \\eta^2 + \\eta^4 = (-1+i\\sqrt{7})\/2$, $\\eta = e^{2\\pi i\/7}$.} \\label{table:Character_table-II2}\n\\end{center}\n\\end{table}\n\nThere are five non-conjugate seven-dimensional representations, $\\gamma_1^{(1)} = \\Sigma_1 + \\Sigma_3 + \\Sigma_3^{\\ast}$, $\\gamma_1^{(2)} = \\Sigma_1 + \\Sigma_6$, $\\gamma_1^{(3)} = \\Sigma_7^{(1)}$, $\\gamma_1^{(4)} = \\Sigma_7^{(1)\\prime}$ and $\\gamma_1^{(5)} = \\Sigma_7^{(2)}$.\nThese all satisfy the condition that $\\gamma_1^{(i)}$ appears in the decomposition of $(\\gamma_1^{(i)})^2$.\nWe thus consider the eigenvalues of the representation matrices to determine which of the $\\gamma_1^{(i)}$ are embeddings of $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ in $G_2$.\nThese eigenvalues are given in Table \\ref{table:evalues-II2} for representations of dimension less than or equal to 7. As described in Section \\ref{sect:subgroupsG2}, these eigenvalues can be determined from the character table of $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$. The additional information that is needed is to note that the eigenvalues for group elements in $(C_4,C_4^3)$ square to those for elements in $C_2$, those for $(C_4^{\\prime},C_4^{\\prime3})$ square to those for $C_2^{\\prime}$, those for $(C_4^{\\prime\\prime},C_4^{\\prime\\prime3})$ square to those for $C_2^{\\prime\\prime}$, whilst those for $(C_6,C_6^5)$ square to those for $(C_3,C_3^2)$ and also cube to those for $C_2$.\nThese observations follow from \\cite{littlewood:1934} and the fact that, for example, it is impossible to choose eigenvalues for group elements in $(C_6,C_6^5)$ which cube to those for elements in $C_2^{\\prime}$ or $C_2^{\\prime\\prime}$ for all irreducible representations.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|} \\hline\n & $\\Sigma_1$ & $\\Sigma_3$ & $\\Sigma_3^{\\ast}$ & $\\Sigma_6$ \\\\\n\\hline \\hline $C_1$ & 1 & $(1,1,1)$ & $(1,1,1)$ & $(1,1,1,1,1,1)$ \\\\\n\\hline $C_2$ & 1 & $(1,1,1)$ & $(1,1,1)$ & $(1,1,1,1,1,1)$ \\\\\n\\hline $C_2^{\\prime}$ & 1 & $(1,-1,-1)$ & $(1,-1,-1)$ & $(1,1,1,1,-1,-1)$ \\\\\n\\hline $C_2^{\\prime\\prime}$ & 1 & $(1,-1,-1)$ & $(1,-1,-1)$ & $(1,1,1,1,-1,-1)$ \\\\\n\\hline $C_4$ & 1 & $(1,-1,-1)$ & $(1,-1,-1)$ & $(1,1,1,1,-1,-1)$ \\\\\n\\hline $C_4^{\\prime}$ & 1 & $(1,i,-i)$ & $(1,i,-i)$ & $(1,1,-1,-1,i,-i)$ \\\\\n\\hline $C_4^{\\prime\\prime}$ & 1 & $(1,i,-i)$ & $(1,i,-i)$ & $(1,1,-1,-1,i,-i)$ \\\\\n\\hline $C_7$ & 1 & $(\\eta,\\eta^2,\\eta^4)$ & $(\\eta^3,\\eta^5,\\eta^6)$ & $(\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ \\\\\n\\hline $C_7^{\\prime}$ & 1 & $(\\eta^3,\\eta^5,\\eta^6)$ & $(\\eta,\\eta^2,\\eta^4)$ & $(\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ \\\\\n\\hline $C_6$ & 1 & $(1,\\mu^2,\\mu^4)$ & $(1,\\mu^2,\\mu^4)$ & $(1,1,\\mu^2,\\mu^2,\\mu^4,\\mu^4)$ \\\\\n\\hline $C_3$ & 1 & $(1,\\omega,\\omega^2)$ & $(1,\\omega,\\omega^2)$ & $(1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ \\\\\n\\hline\n\\end{tabular} \\\\\n$\\;$ \\\\\n\\begin{tabular}{|c||c|c|c|} \\hline\n & $\\Sigma_7^{(1)}$ & $\\Sigma_7^{(1)\\prime}$ & $\\Sigma_7^{(2)}$ \\\\\n\\hline \\hline $C_1$ & $(1,1,1,1,1,1,1)$ & $(1,1,1,1,1,1,1)$ & $(1,1,1,1,1,1,1)$ \\\\\n\\hline $C_2$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,1,1,1,1,1,1)$ \\\\\n\\hline $C_2^{\\prime}$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,1,1,1,1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1)$ \\\\\n\\hline $C_2^{\\prime\\prime}$ & $(1,1,1,1,1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1)$ \\\\\n\\hline $C_4$ & $(1,-1,-1,i,i,-i,-i)$ & $(1,-1,-1,i,i,-i,-i)$ & $(1,1,1,-1,-1,i,-i)$ \\\\\n\\hline $C_4^{\\prime}$ & $(1,-1,-1,i,i,-i,-i)$ & $(1,1,1,-1,-1,i,-i)$ & $(1,-1,-1,i,i,-i,-i)$ \\\\\n\\hline $C_4^{\\prime\\prime}$ & $(1,1,1,-1,-1,i,-i)$ & $(1,-1,-1,i,i,-i,-i)$ & $(1,-1,-1,i,i,-i,-i)$ \\\\\n\\hline $C_7$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ \\\\\n\\hline $C_7^{\\prime}$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ \\\\\n\\hline $C_6$ & $(1,-1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ & $(1,-1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ & $(1,1,1,\\mu^2,\\mu^2,\\mu^4,\\mu^4)$ \\\\\n\\hline $C_3$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Eigenvalues of group elements in each conjugacy class of $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ for irreducible representations of dimension $\\leq 7$, where $\\omega = e^{2\\pi i\/3}$, $\\mu = e^{2\\pi i\/6}$ and $\\eta = e^{2\\pi i\/7}$.} \\label{table:evalues-II2}\n\\end{center}\n\\end{table}\n\nFrom considering the set of eigenvalues $X_C$ for group elements in $C$ in the representation $\\gamma_1^{(i)}$, we see that there is no choice of $(t_1^C,t_2^C) \\in X_C$ such that $\\mathcal{E}_{t_1^C,t_2^C} = X_C$ for $i=2,3$ when $C = C_2^{\\prime}$ and for $i=4$ when $C = C_2^{\\prime\\prime}$. However, such a choice does exist for $i=1,5$ for all conjugacy classes $C$, thus we set $\\varrho_1^{(1)} = \\gamma_1^{(5)}$, $\\varrho_1^{(2)} = \\gamma_1^{(1)}$. We present one such choice of eigenvalues $(t_1^C,t_2^C)$ in Table \\ref{table:t1,t2-II2}.\nThe McKay graphs $\\mathcal{G}^{\\varrho_1^{(1)}}_{PSL(2;7) \\rtimes \\mathbb{Z}_2^3}$, $\\mathcal{G}^{\\varrho_1^{(2)}}_{PSL(2;7) \\rtimes \\mathbb{Z}_2^3}$ for $\\varrho_1^{(2)}$ for $\\varrho_1^{(1)}$, $\\varrho_1^{(2)}$ are given in Figure \\ref{Fig-McKay_Graph-II2-rho1}. We use the notation $n$, $n^{\\ast}$, $n^{(i)\\prime}$ to label the vertices corresponding to the irreducible representations $\\Sigma_n$, $\\Sigma_n^{\\ast}$, $\\Sigma_n^{(i)\\prime}$ respectively. Since both McKay graphs are not connected, we see that $\\varrho_1^{(i)}$ is not a faithful representation, $i=1,2$.\nNote that the McKay graph for $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ given in \\cite[Figure 1]{he:2003} is not the McKay graph for a restriction of the fundamental seven-dimensional representation, as claimed there, but rather for the irreducible representation $\\Sigma_7^{(1)}$ (or equivalently the representation $\\Sigma_7^{(1)\\prime}$).\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=115mm]{Fig-McKay_Graph-II2-rho1}\\\\\n \\caption{The McKay graphs $\\mathcal{G}^{\\varrho_1^{(i)}}_{PSL(2;7) \\rtimes \\mathbb{Z}_2^3}$, $i=1,2$.} \\label{Fig-McKay_Graph-II2-rho1}\n\\end{center}\n\\end{figure}\n\nThe decomposition of the Kronecker square of $\\varrho_1^{(i)}$ into irreducibles is given by\n$$(\\varrho_1^{(1)})^2 = \\mathrm{id} + \\varrho_1^{(1)} + \\Sigma_3 + \\Sigma_3^{\\ast} + 2\\Sigma_6 + \\Sigma_7^{(2)} + 2 \\Sigma_8, \\qquad\n(\\varrho_1^{(2)})^2 = \\mathrm{id} + \\varrho_1^{(2)} + \\Sigma_1 + 2\\Sigma_3 + 2\\Sigma_3^{\\ast} + 2\\Sigma_6 + 2 \\Sigma_8,$$\nwhere $\\mathrm{id} = \\Sigma_1$.\nFrom dimension considerations there are thus two candidates for the fourteen-dimensional representation $\\varrho_2^{(i)}$, which are given by $\\gamma_2^{(1)} = \\Sigma_3 + \\Sigma_3^{\\ast} + \\Sigma_8$ and $\\gamma_2^{(2)} = \\Sigma_6 + \\Sigma_8$ for both $i=1,2$.\nHowever, since $\\chi_{\\varrho_2^{(i)}}(C) = \\Phi_2(t_1^C,t_2^C)$, we see from Table \\ref{table:t1,t2-II2} that the decomposition of the fundamental fourteen-dimensional representation into irreducible representations of $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ is given by $\\varrho_2 = \\gamma_2^{(1)} = \\Sigma_3 + \\Sigma_3^{\\ast} + \\Sigma_8$ for both $i=1,2$.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|} \\hline\n& \\multicolumn{2}{|c|}{$\\varrho_1^{(1)}$} & \\multicolumn{2}{|c|}{$\\varrho_1^{(2)}$} & & \\\\\n$C$ & $(t_1^C,t_2^C)$ & $\\chi_{\\varrho_2^{(1)}}(C)$ & $(t_1^C,t_2^C)$ & $\\chi_{\\varrho_2^{(2)}}(C)$ & $\\chi_{\\gamma_2^{(1)}}(C)$ & $\\chi_{\\gamma_2^{(2)}}(C)$ \\\\\n\\hline \\hline $C_1$ & $(1,1)$ & 14 & $(1,1)$ & 14 & 14 & 14 \\\\\n\\hline $C_2$ & $(1,1)$ & 14 & $(1,1)$ & 14 & 14 & 14 \\\\\n\\hline $C_2^{\\prime}$ & $(1,-1)$ & -2 & $(1,-1)$ & -2 & -2 & 2 \\\\\n\\hline $C_2^{\\prime\\prime}$ & $(1,-1)$ & -2 & $(1,-1)$ & -2 & -2 & 2 \\\\\n\\hline $C_4$ & $(1,-1)$ & -2 & $(1,-1)$ & -2 & -2 & 2 \\\\\n\\hline $C_4^{\\prime}$ & $(-1,i)$ & 2 & $(1,i)$ & 2 & 2 & 0 \\\\\n\\hline $C_4^{\\prime\\prime}$ & $(-1,i)$ & 2 & $(1,i)$ & 2 & 2 & 0 \\\\\n\\hline $C_7$ & $(\\eta,\\eta^5)$ & 0 & $(\\eta,\\eta^5)$ & 0 & 0 & 0 \\\\\n\\hline $C_7^{\\prime}$ & $(\\eta^2,\\eta^3)$ & 0 & $(\\eta^2,\\eta^3)$ & 0 & 0 & 0 \\\\\n\\hline $C_6$ & $(1,\\mu^2)$ & -1 & $(1,\\mu^2)$ & -1 & -1 & -1 \\\\\n\\hline $C_3$ & $(1,\\omega)$ & -1 & $(1,\\omega)$ & -1 & -1 & -1 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Choice of eigenvalues $(t_1^C,t_2^C)$ for $\\varrho_1^{(i)}$, $i=1,2$, and corresponding values of $\\chi_{\\varrho_2^{(i)}}(C)$.} \\label{table:t1,t2-II2}\n\\end{center}\n\\end{table}\n\nThe values of $x^{(i)} = \\chi_{\\varrho_1^{(i)}}(C) \\in [-2,7]$, $y = \\chi_{\\rho_2}(C) \\in [-2,14]$ for $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ are given in Table \\ref{table:(x,y)-II1}, along with the values of $J^2\/64\\pi^4$ for the corresponding pairs $(\\theta_1,\\theta_2) \\in [0,1]^2$ obtained from Table \\ref{Table:subgroupsG2-orbits(theta1,theta2)}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_2$ & $C_2^{\\prime}$ & $C_2^{\\prime\\prime}$ & $C_4$ & $C_4^{\\prime}$ & $C_4^{\\prime\\prime}$ & $C_7$ & $C_7^{\\prime}$ & $C_6$ & $C_3$ \\\\\n\\hline $\\chi_{\\varrho_1^{(1)}}(\\Gamma_j) \\in [-2,7]$ & 7 & 7 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 1 & 1 \\\\\n\\hline $\\chi_{\\varrho_1^{(2)}}(\\Gamma_j) \\in [-2,7]$ & 7 & 7 & -1 & -1 & -1 & 3 & 3 & 0 & 0 & 1 & 1 \\\\\n\\hline $\\chi_{\\varrho_2}(\\Gamma_j) \\in [-2,14]$ & 14 & 14 & -2 & -2 & -2 & 2 & 2 & 0 & 0 & -1 & -1 \\\\\n\\hline $J^2(\\theta_1,\\theta_2)\/64\\pi^4$ & 0 & 0 & 0 & 0 & 0 & 8 & 8 & 49\/4 & 49\/4 & 9 & 0 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{$\\chi_{\\varrho_j}(C)$ for group $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$, $j=1,2$.} \\label{table:(x,y)-II2}\n\\end{center}\n\\end{table}\n\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:Character_table-II2}, \\ref{table:(x,y)-II2} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n\\begin{align*}\n\\varsigma_{m,n} & = \\frac{1+7}{1344} \\Omega^W(0,0) + \\frac{42+42+84}{1344} \\Omega^W(0,1\/2) + \\frac{224+224}{1344} \\Omega^W(0,1\/3) \\\\\n& \\quad + \\frac{168+168}{1344} \\Omega' + \\frac{192+192}{1344} \\Omega^W(1\/7,3\/7),\n\\end{align*}\nwhere $\\Omega^W(\\theta_1,\\theta_2)$ is as in Section \\ref{sect:II1}, and $\\Omega'$ is $\\Omega^W(1\/4,1\/2)$ for $\\varrho_1^{(1)}$, and $\\Omega^W(0,1\/4)$ for $\\varrho_1^{(2)}$. Thus the joint moments are precisely those for the two embeddings of $PSL(2;7)$ in $G_2$, and we obtain:\n\n\\begin{Thm}\nThe joint spectral measure (over $\\mathbb{T}^2$) for the non-conjugate embeddings of $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ into the fundamental representations of $G_2$ is given by (\\ref{eqn:measureII1}),\nwhere $K' = 16-K$ for the embedding of $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ in $G_2$ given by $\\varrho_1^{(1)} = \\Sigma_7^{(2)}$ and $K' = K$ for the embedding given by $\\varrho_1^{(2)} = \\Sigma_1 + \\Sigma_3 + \\Sigma_3^{\\ast}$, and where $K$ is again given by $-4K(\\omega_1,\\omega_2) = (\\omega_1\\omega_2 - \\omega_1^{-1}\\omega_2^{-1} - \\omega_1^2\\omega_2^{-1} + \\omega_1^{-2}\\omega_2 + \\omega_1\\omega_2^{-2} - \\omega_1^{-1}\\omega_2^2)^2$ for $\\omega_1,\\omega_2\\in\\mathbb{T}$.\n\\end{Thm}\n\n\n\n\n\\section{Group $PGL(2;7)$}\n\nThe subgroup $PGL(2;7)$ of $G_2$ is an irreducible primitive group of order 336.\nIt has nine irreducible representations, all real, and its character table is given in Table \\ref{table:Character_table-IP3} \\cite{collins:1990}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_2$ & $C_2^{\\prime}$ & $C_8$ & $C_8^{\\prime}$ & $C_4$ & $C_7$ & $C_6$ & $C_3$ \\\\\n\\hline $|C|$ & 1 & 21 & 28 & 42 & 42 & 42 & 48 & 56 & 56 \\\\\n\\hline \\hline $\\Sigma_1$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_1^{\\prime}$ & 1 & 1 & -1 & -1 & -1 & 1 & 1 & -1 & 1 \\\\\n\\hline $\\Sigma_6^{(1)}$ & 6 & -2 & 0 & 0 & 0 & 2 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_6^{(2)}$ & 6 & 2 & 0 & $\\sqrt{2}$ & $-\\sqrt{2}$ & 0 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_6^{(2)\\prime}$ & 6 & 2 & 0 & $-\\sqrt{2}$ & $\\sqrt{2}$ & 0 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_7$ & 7 & -1 & 1 & -1 & -1 & -1 & 0 & 1 & 1 \\\\\n\\hline $\\Sigma_7^{\\prime}$ & 7 & -1 & -1 & 1 & 1 & -1 & 0 & -1 & 1 \\\\\n\\hline $\\Sigma_8$ & 8 & 0 & 2 & 0 & 0 & 0 & 1 & -1 & -1 \\\\\n\\hline $\\Sigma_8^{\\prime}$ & 8 & 0 & -2 & 0 & 0 & 0 & 1 & 1 & -1 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for $PGL(2;7)$.} \\label{table:Character_table-IP3}\n\\end{center}\n\\end{table}\n\nThere are eight non-conjugate seven-dimensional representations, $\\gamma_1^{(1)} = \\Sigma_1 + \\Sigma_6^{(1)}$, $\\gamma_1^{(2)} = \\Sigma_1 + \\Sigma_6^{(2)}$, $\\gamma_1^{(3)} = \\Sigma_1 + \\Sigma_6^{(2)\\prime}$, $\\gamma_1^{(4)} = \\Sigma_1^{\\prime} + \\Sigma_6^{(1)}$, $\\gamma_1^{(5)} = \\Sigma_1^{\\prime} + \\Sigma_6^{(2)}$, $\\gamma_1^{(6)} = \\Sigma_1^{\\prime} + \\Sigma_6^{(2)\\prime}$, $\\gamma_1^{(7)} = \\Sigma_7$ and $\\gamma_1^{(8)} = \\Sigma_7^{\\prime}$.\nThe decomposition of the Kronecker squares of the $\\gamma_1^{(i)}$ are given by:\n\\begin{align*}\n(\\gamma_1^{(1)})^2 & = \\mathrm{id} + \\gamma_1^{(1)} + \\Sigma_1^{\\prime} + 2\\Sigma_6^{(1)} + \\Sigma_6^{(2)} + \\Sigma_6^{(2)\\prime} + \\Sigma_8 + \\Sigma_8^{\\prime}, \\\\\n(\\gamma_1^{(2)})^2 & = \\mathrm{id} + \\gamma_1^{(2)} + 2\\Sigma_6^{(2)} + \\Sigma_6^{(2)\\prime} + \\Sigma_7^{\\prime} + \\Sigma_8 + \\Sigma_8^{\\prime}, \\\\\n(\\gamma_1^{(4)})^2 & = \\mathrm{id} + \\gamma_1^{(4)} + \\Sigma_1 + 2\\Sigma_6^{(1)} + \\Sigma_6^{(2)} + \\Sigma_6^{(2)\\prime} + \\Sigma_8 + \\Sigma_8^{\\prime}, \\\\\n(\\gamma_1^{(5)})^2 & = \\mathrm{id} + \\Sigma_1 + 3\\Sigma_6^{(2)} + \\Sigma_6^{(2)\\prime} + \\Sigma_7^{\\prime} + \\Sigma_8 + \\Sigma_8^{\\prime}, \\\\\n(\\gamma_1^{(7)})^2 & = \\mathrm{id} + \\gamma_1^{(7)} + \\Sigma_1 + \\Sigma_6^{(1)} + \\Sigma_6^{(2)} + \\Sigma_6^{(2)\\prime} + \\Sigma_7^{\\prime} + \\Sigma_8 + \\Sigma_8^{\\prime}, \\\\\n(\\gamma_1^{(8)})^2 & = \\mathrm{id} + \\gamma_1^{(8)} + \\Sigma_1 + \\Sigma_6^{(1)} + \\Sigma_6^{(2)} + \\Sigma_6^{(2)\\prime} + \\Sigma_7 + \\Sigma_8 + \\Sigma_8^{\\prime},\n\\end{align*}\nwhere $\\mathrm{id} = \\Sigma_1$.\nNote, we have omitted the decompositions for $\\gamma_1^{(3)}$, $\\gamma_1^{(6)}$, since from the character table we see that these representations are essentially the same as $\\gamma_1^{(2)}$, $\\gamma_1^{(5)}$ respectively.\nThen we see that $\\gamma_1^{(i)}$ does not appear in the decomposition of $(\\gamma_1^{(i)})^2$ for $i=5,6$, therefore they are not embeddings of $PGL(2;7)$ in $G_2$.\nFrom dimension considerations, we see that candidates $\\gamma_2^{(j)}$ for $\\varrho_2$ are $\\gamma_2^{(1)}=\\Sigma_6^{(1)}+\\Sigma_8$, $\\gamma_2^{(2)}=\\Sigma_6^{(1)}+\\Sigma_8^{\\prime}$, $\\gamma_2^{(3)}=\\Sigma_6^{(2)}+\\Sigma_8$, $\\gamma_2^{(4)}=\\Sigma_6^{(2)}+\\Sigma_8^{\\prime}$, $\\gamma_2^{(5)}=\\Sigma_6^{(2)\\prime}+\\Sigma_8$ and $\\gamma_2^{(6)}=\\Sigma_6^{(2)\\prime}+\\Sigma_8^{\\prime}$, where $1 \\leq j \\leq 6$ for $\\gamma_1^{(i)}$ when $i=1,4,7,8$, and $3 \\leq j \\leq 6$ when $i=2$.\nThen with $(x_i^C,y_j^C) = (\\chi_{\\gamma_1^{(i)}}(C),\\chi_{\\gamma_2^{(j)}}(C))$, we see that $(x_i^C,y_j^C) \\not \\in \\mathfrak{D}$ for any candidate $\\gamma_2^{(j)}$ in the cases $i=1,2,3,7$ when $C=C_2^{\\prime}$. Thus $\\gamma_1^{(i)}$ cannot define an embedding of $PGL(2;7)$ in $G_2$ for $i=1,2,3,7$.\nWe also see that $(x_8^C,y_j^C) \\not \\in \\mathfrak{D}$ for $j=3,4,5,6$.\n\nThus we have candidates $(\\gamma_1^{(i)},\\gamma_2^{(j)})$ for $(\\varrho_1,\\varrho_2)$ when $i=4,8$, where $1 \\leq j \\leq 6$ for $i=4$ and $j \\in \\{ 1,2 \\}$ for $i=8$.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|} \\hline\n & $\\Sigma_1$ & $\\Sigma_1^{\\prime}$ & $\\Sigma_6^{(1)}$ & $\\Sigma_6^{(2)}$ & $\\Sigma_7$ \\\\\n\\hline \\hline $C_1$ & 1 & 1 & $(1,1,1,1,1,1)$ & $(1,1,1,1,1,1)$ & $(1,1,1,1,1,1,1)$ \\\\\n\\hline $C_2$ & 1 & 1 & $(1,1,-1,-1,-1,-1)$ & $(1,1,1,1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1)$ \\\\\n\\hline $C_2^{\\prime}$ & 1 & -1 & $(1,1,1,-1,-1,-1)$ & $(1,1,1,-1,-1,-1)$ & $(1,1,1,1,-1,-1,-1)$ \\\\\n\\hline $C_8$ & 1 & -1 & $(1,-1,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,i,-i,\\nu,\\nu^7)$ & $(-1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ \\\\\n\\hline $C_8^{\\prime}$ & 1 & -1 & $(1,-1,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,i,-i,\\nu^3,\\nu^5)$ & $(-1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ \\\\\n\\hline $C_4$ & 1 & 1 & $(1,1,i,i,-i,-i)$ & $(1,1,-1,-1,i,-i)$ & $(1,-1,-1,i,i,-i,-i)$ \\\\\n\\hline $C_7$ & 1 & 1 & $(\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ \\\\\n\\hline $C_6$ & 1 & -1 & $(1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ & $(1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ & $(1,1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ \\\\\n\\hline $C_3$ & 1 & 1 & $(1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ \\\\\n\\hline\n\\end{tabular} \\\\\n\\begin{tabular}{|c||c|c|c|} \\hline\n & $\\Sigma_7^{\\prime}$ & $\\Sigma_8$ & $\\Sigma_8^{\\prime}$ \\\\\n\\hline \\hline $C_1$ & $(1,1,1,1,1,1,1)$ & $(1,1,1,1,1,1,1,1)$ & $(1,1,1,1,1,1,1,1)$ \\\\\n\\hline $C_2$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,1,1,1,-1,-1,-1,-1)$ & $(1,1,1,1,-1,-1,-1,-1)$ \\\\\n\\hline $C_2^{\\prime}$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,1,1,1,1,-1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1,-1)$ \\\\\n\\hline $C_8$ & $(1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ \\\\\n\\hline $C_8^{\\prime}$ & $(1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ \\\\\n\\hline $C_4$ & $(1,-1,-1,i,i,-i,-i)$ & $(1,1,-1,-1,i,i,-i,-i)$ & $(1,1,-1,-1,i,i,-i,-i)$ \\\\\n\\hline $C_7$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ \\\\\n\\hline $C_6$ & $(1,-1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ & $(1,-1,\\mu,\\mu^2,\\mu^2,\\mu^4,\\mu^4,\\mu^5)$ & $(1,-1,\\mu,\\mu,\\mu^2,\\mu^4,\\mu^5,\\mu^5)$ \\\\\n\\hline $C_3$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,1,\\omega,\\omega,\\omega,\\omega^2,\\omega^2,\\omega^2)$ & $(1,1,\\omega,\\omega,\\omega,\\omega^2,\\omega^2,\\omega^2)$ \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Eigenvalues of group elements in each conjugacy class of $PGL(2;7)$, where $\\omega = e^{2\\pi i\/3}$, $\\mu = e^{2\\pi i\/6}$ and $\\eta = e^{2\\pi i\/7}$.} \\label{table:evalues-IP3}\n\\end{center}\n\\end{table}\n\nWe now consider the eigenvalues of the representation matrices to determine which of the remaining $\\gamma_1^{(i)}$ are embeddings of $PSL(2;7)$ in $G_2$.\nThese eigenvalues are given in Table \\ref{table:evalues-IP3}. As described in Section \\ref{sect:subgroupsG2}, these eigenvalues can be determined from the character table of $PGL(2;7)$. The additional information that is needed is to note that the eigenvalues for group elements in $C_4$ square to those for elements in $C_2$, those for $C_8$, $C_8^{\\prime}$ square to those for $C_4$, those for $C_6$ square to those for $C_3)$ and also cube to those for $C_2^{\\prime}$.\nThese observations follow from the fact that, for example, it is impossible to choose eigenvalues for group elements in $C_4$ which square to those for elements in $C_2^{\\prime}$ for all irreducible representations.\nNote also that we have omitted the eigenvalues for matrices in the irreducible representation $\\Sigma_6^{(2)\\prime}$. These eigenvalues are identical to those for $\\Sigma_6^{(2)}$, except for elements in the conjugacy classes $C_8$, $C_8^{\\prime}$, where for $\\Sigma_6^{(2)\\prime}$ the eigenvalues for elements in $C_8$, $C_8^{\\prime}$ respectively are given by those for elements in $C_8^{\\prime}$, $C_8$ respectively in the representation $\\Sigma_6^{(2)}$.\n\nFrom considering the set of eigenvalues $X_C$ for group elements in $C$ in the representation $\\gamma_1^{(i)}$, $i=4,8$, we see that there is a choice of $(t_1^C,t_2^C) \\in X_C$ such that $\\mathcal{E}_{t_1^C,t_2^C} = X_C$, for all conjugacy classes $C$. We present one such choice of eigenvalues $(t_1^C,t_2^C)$ in Table \\ref{table:t1,t2-IP3}. Thus we set $\\varrho_1^{(1)} = \\Sigma_7^{\\prime}$, $\\varrho_1^{(2)} = \\Sigma_1^{\\prime} + \\Sigma_6^{(1)}$.\nThe McKay graphs $\\mathcal{G}^{\\varrho_1^{(1)}}_{PGL(2;7)}$, $\\mathcal{G}^{\\varrho_1^{(2)}}_{PGL(2;7)}$ for $\\varrho_1^{(1)}$, $\\varrho_1^{(2)}$ are given in Figure \\ref{Fig-McKay_Graph-IP3-rho1}, where we use the same notation as previously.\nNote that the graph given in \\cite[Figure 2]{he:2003} is not the McKay graph for the restriction $\\varrho_1$ of the fundamental seven-dimensional representation of $G_2$ as claimed in \\cite{he:2003}, but is rather the McKay graph for the seven-dimensional representation $\\Sigma_7$, which as shown above does not define an embedding of $PGL(2;7)$ in $G_2$.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=110mm]{Fig-McKay_Graph-IP3-rho1}\\\\\n \\caption{The McKay graphs $\\mathcal{G}^{\\varrho_1^{(i)}}_{PGL(2;7)}$, $i=1,2$.} \\label{Fig-McKay_Graph-IP3-rho1}\n\\end{center}\n\\end{figure}\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|} \\hline\n& \\multicolumn{2}{|c|}{$\\varrho_1^{(1)}$} & \\multicolumn{2}{|c|}{$\\varrho_1^{(2)}$} & & & & \\\\\n$C$ & $(t_1^C,t_2^C)$ & $\\chi_{\\varrho_2^{(1)}}(C)$ & $(t_1^C,t_2^C)$ & $\\chi_{\\varrho_2^{(2)}}(C)$ & $\\chi_{\\gamma_2^{(1)}}(C)$ & $\\chi_{\\gamma_2^{(2)}}(C)$ & $\\chi_{\\gamma_2^{(3)}}(C)$ & $\\chi_{\\gamma_2^{(4)}}(C)$ \\\\\n\\hline \\hline $C_1$ & $(1,1)$ & 14 & $(1,1)$ & 14 & 14 & 14 & 14 & 14 \\\\\n\\hline $C_2$ & $(1,-1)$ & -2 & $(1,-1)$ & -2 & -2 & -2 & 2 & 2 \\\\\n\\hline $C_2^{\\prime}$ & $(1,-1)$ & -2 & $(1,-1)$ & -2 & 2 & -2 & 2 & -2 \\\\\n\\hline $C_8$ & $(i,\\nu^7)$ & 0 & $(-1,\\nu)$ & 0 & 0 & 0 & $\\sqrt{2}$ & $\\sqrt{2}$ \\\\\n\\hline $C_8^{\\prime}$ & $(i,\\nu^5)$ & 0 & $(-1,\\nu)$ & 0 & 0 & 0 & $-\\sqrt{2}$ & $-\\sqrt{2}$ \\\\\n\\hline $C_4$ & $(-1,i)$ & 2 & $(1,i)$ & 2 & 2 & 2 & 0 & 0 \\\\\n\\hline $C_7$ & $(\\eta,\\eta^5)$ & 0 & $(\\eta,\\eta^5)$ & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline $C_6$ & $(-1,\\mu)$ & 1 & $(-1,\\mu)$ & 1 & -1 & 1 & -1 & 1 \\\\\n\\hline $C_3$ & $(1,\\omega)$ & -1 & $(1,\\omega)$ & -1 & -1 & -1 & -1 & -1 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Choice of eigenvalues $(t_1^C,t_2^C)$ for $\\varrho_1^{(i)}$, $i=1,2$, and corresponding values of $\\chi_{\\varrho_2^{(i)}}(C)$.} \\label{table:t1,t2-IP3}\n\\end{center}\n\\end{table}\n\nSince $\\chi_{\\varrho_2^{(i)}}(C) = \\Phi_2(t_1^C,t_2^C)$, we see from Table \\ref{table:t1,t2-II2} that the decomposition of the fundamental fourteen-dimensional representation into irreducible representations of $PGL(2;7)$ is given by $\\varrho_2 = \\gamma_2^{(2)} = \\Sigma_6^{(1)} + \\Sigma_8^{\\prime}$ for both $i=1,2$.\nThe values of $x^{(i)} = \\chi_{\\varrho_1^{(i)}}(C) \\in [-2,7]$, $y = \\chi_{\\varrho_2}(C) \\in [-2,14]$ for $PGL(2;7)$ are given in Table \\ref{table:(x,y)-IP3}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_2$ & $C_2^{\\prime}$ & $C_8$ & $C_8^{\\prime}$ & $C_4$ & $C_7$ & $C_6$ & $C_3$ \\\\\n\\hline $\\chi_{\\varrho_1^{(1)}}(C) \\in [-2,7]$ & 7 & -1 & -1 & 1 & 1 & -1 & 0 & -1 & 1 \\\\\n\\hline $\\chi_{\\varrho_1^{(2)}}(C) \\in [-2,7]$ & 7 & -1 & -1 & -1 & -1 & 3 & 0 & -1 & 1 \\\\\n\\hline $\\chi_{\\varrho_2}(C) \\in [-2,14]$ & 14 & -2 & -2 & 0 & 0 & 2 & 0 & 1 & -1 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{$\\chi_{\\varrho_j}(C)$ for group $PGL(2;7)$, $j=1,2$.} \\label{table:(x,y)-IP3}\n\\end{center}\n\\end{table}\n\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:Character_table-IP3}, \\ref{table:(x,y)-IP3} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n\\begin{align*}\n\\varsigma_{m,n} & = \\frac{1}{336} \\Omega^W(0,0) + \\frac{21+28}{336} \\Omega^W(0,1\/2) + \\frac{56}{336} \\Omega^W(0,1\/3) \\\\\n& \\quad + \\frac{56}{336} \\Omega^W(1\/6,1\/2) + \\frac{48}{336} \\Omega^W(1\/7,3\/7) + \\frac{42}{336} \\Omega' + \\frac{42+42}{336} \\Omega'',\n\\end{align*}\nwhere $\\Omega^W(\\theta_1,\\theta_2)$ is as in Section \\ref{sect:II1} and $(\\Omega',\\Omega'')$ is $(\\Omega^W(1\/4,1\/2),\\Omega^W(1\/8,3\/8))$ for $\\varrho_1^{(1)}$, and $(\\Omega^W(0,1\/4),\\Omega^W(1\/8,1\/2))$ for $\\varrho_1^{(2)}$.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=135mm]{Fig-OmegaWIP3}\\\\\n \\caption{The orbits of $(a)$ $(1\/6,1\/2)$, $(b)$ $(1\/8,3\/8) \\, \\bullet$ and $(1\/8,1\/2) \\, \\ast$.} \\label{Fig-OmegaWIP3}\n\\end{center}\n\\end{figure}\n\nNow $12\\Omega^W(1\/6,1\/2) = 4 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}_6 \\times \\mathrm{d}_6$, as illustrated in Figure \\ref{Fig-OmegaWIP3}$(a)$ since the Jacobian $J=0$ along the boundaries of the orbit of the fundamental domain, whilst $J^2(g(1\/6,1\/2)\/64\\pi^4) = 9$ for all $g \\in D_{12}$.\n\nThe orbits of $(1\/8,3\/8)$, $(1\/8,1\/2)$ are illustrated in Figure \\ref{Fig-OmegaWIP3}$(b)$, represented by $\\bullet$, $\\ast$ respectively.\nNeither orbit gives a linear combination of the measures in Definition \\ref{def:4measures}, thus we have $12\\Omega^W(1\/8,3\/8) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\left( \\sum_{g \\in D_{12}} \\delta_{g(e^{\\pi i\/4},e^{3\\pi i\/4})} \\right)$, $12\\Omega^W(1\/8,1\/2) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\left( \\sum_{g \\in D_{12}} \\delta_{g(e^{\\pi i\/4},-1)} \\right)$.\n\nThe measures $J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7$, $K' \\, \\mathrm{d}_4 \\times \\mathrm{d}_4$, $\\mathrm{d}_3 \\times \\mathrm{d}_3$, $\\mathrm{d}_2 \\times \\mathrm{d}_2$ and $\\mathrm{d}_1 \\times \\mathrm{d}_1$ supported by the other points have all appeared in the previous sections, so we obtain:\n\n\\begin{Thm}\nThe joint spectral measure (over $\\mathbb{T}^2$) for the non-conjugate embeddings of $PGL(2;7)$ into the fundamental representations of $G_2$ is\n\\begin{equation}\n\\begin{split}\n\\mathrm{d}\\varepsilon & = \\frac{1}{1344\\pi^4} J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7 + \\frac{1}{1152\\pi^4} J^2 \\, \\mathrm{d}_6 \\times \\mathrm{d}_6 + \\frac{1}{24} K' \\, \\mathrm{d}_4 \\times \\mathrm{d}_4 + \\frac{1}{4} \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 \\\\\n& \\quad + \\frac{1}{9} \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 - \\frac{23}{504} \\, \\mathrm{d}_1 \\times \\mathrm{d}_1 + \\frac{1}{48} \\sum_{g \\in D_{12}} \\delta_{g(e^{\\pi i\/4},t)},\n\\end{split}\n\\end{equation}\nwhere $K' = 16-K$ for the embedding of $PGL(2;7)$ in $G_2$ given by $\\varrho_1^{(1)} = \\Sigma_7^{\\prime}$ and $K' = K$ for the embedding given by $\\varrho_1^{(2)} = \\Sigma_1^{\\prime} + \\Sigma_6$, with $K$ as in Theorem \\ref{thm:measureII1}, whilst $t = e^{3\\pi i\/4}$ for the embedding of $PGL(2;7)$ in $G_2$ given by $\\varrho_1^{(1)}$ and $t=-1$ for the embedding given by $\\varrho_1^{(2)}$,\nand where $\\mathrm{d}_m$ is the uniform measure over $m^{\\mathrm{th}}$ roots of unity and $\\delta_x$ is the Dirac measure at the point $x$.\n\\end{Thm}\n\n\n\n\n\\section{Group $PSL(2;8)$}\n\nThe subgroup $PSL(2;8)$ of $G_2$ is an irreducible primitive group of order 504.\nIt has nine irreducible representations, all real, five of which have dimension less than or equal to 7. The character table for $PSL(2;8)$ is given in\nTable \\ref{table:Character_table-IP2} \\cite{james\/liebeck:2001} (the orders of the elements in each conjugacy class can be obtained from \\cite{lopez_pena\/majid\/rietsch:2010}).\nThe spectral measure for the first three seven-dimensional representations are equal, since the conjugacy classes $C_{9}$, $C_{9}^{\\prime}$, $C_{9}^{\\prime\\prime}$ each have the same order.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_3$ & $C_9$ & $C_9^{\\prime}$ & $C_9^{\\prime\\prime}$ & $C_2$ & $C_7$ & $C_7^{\\prime}$ & $C_7^{\\prime\\prime}$ \\\\\n\\hline $|C|$ & 1 & 56 & 56 & 56 & 56 & 63 & 72 & 72 & 72 \\\\\n\\hline \\hline $\\Sigma_1$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_7^{(1)}$ & 7 & 1 & $-p$ & $-q$ & $p+q$ & -1 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_7^{(1)\\prime}$ & 7 & 1 & $p+q$ & $-p$ & $-q$ & -1 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_7^{(1)\\prime\\prime}$ & 7 & 1 & $-q$ & $p+q$ & $-p$ & -1 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_7^{(2)}$ & 7 & -2 & 1 & 1 & 1 & -1 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_8$ & 8 & -1 & -1 & -1 & -1 & 0 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_9$ & 9 & 0 & 0 & 0 & 0 & 1 & $2\\cos(2\\pi\/7)$ & $2\\cos(4\\pi\/7)$ & $2\\cos(6\\pi\/7)$ \\\\\n\\hline $\\Sigma_9^{\\prime}$ & 9 & 0 & 0 & 0 & 0 & 1 & $2\\cos(4\\pi\/7)$ & $2\\cos(6\\pi\/7)$ & $2\\cos(2\\pi\/7)$ \\\\\n\\hline $\\Sigma_9^{\\prime\\prime}$ & 9 & 0 & 0 & 0 & 0 & 1 & $2\\cos(6\\pi\/7)$ & $2\\cos(2\\pi\/7)$ & $2\\cos(4\\pi\/7)$ \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for $PSL(2;8)$, where $p=2\\cos(4\\pi\/9)$, $q=2\\cos(8\\pi\/9)$.} \\label{table:Character_table-IP2}\n\\end{center}\n\\end{table}\n\nWe thus have four candidates for the restriction $\\varrho_1$ of the fundamental representation $\\rho_1$ of $G_2$ to $PSL(2;8)$. The Kronecker squares of $\\Sigma_7^{(1)}$, $\\Sigma_7^{(2)}$ decompose into irreducibles as\n$$(\\Sigma_7^{(1)})^2 = \\mathrm{id} + \\Sigma_7^{(1)} + \\Sigma_7^{(1)\\prime} + \\Sigma_7^{(2)} + \\Sigma_9 + \\Sigma_9^{\\prime} + \\Sigma_9^{\\prime\\prime}, \\qquad\n(\\Sigma_7^{(2)})^2 = \\mathrm{id} + \\Sigma_7^{(1)} + \\Sigma_7^{(1)\\prime} + \\Sigma_7^{(1)\\prime\\prime} + \\Sigma_9 + \\Sigma_9^{\\prime} + \\Sigma_9^{\\prime\\prime},$$\nwhere $\\mathrm{id} = \\Sigma_1$. The irreducible representation $\\Sigma_7^{(2)}$ does not appear in the decomposition of its Kronecker square into irreducibles, and therefore does not give an embedding of $PSL(2;8)$ into $G_2$.\nThe other three seven-dimensional representations give non-conjugate embeddings of $PSL(2;8)$ into $G_2$. We will fix $\\varrho_1 = \\Sigma_7$ for the remainder of this section.\n\nThe McKay graph for $\\varrho_1$ is given in Figure \\ref{Fig-McKay_Graph-IP2-rho1}, where we use the same notation as previously.\nNote that the graph given in \\cite[Figure 2]{he:2003} is not the McKay graph for the restriction $\\varrho_1$ of the fundamental seven-dimensional representation of $G_2$ as claimed in \\cite{he:2003}, but is rather the McKay graph for the seven-dimensional representation $\\Sigma_7^{(2)}$.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=55mm]{Fig-McKay_Graph-IP2-rho1}\\\\\n \\caption{The McKay graph $\\mathcal{G}^{\\varrho_1}_{PSL(2;8)}$.} \\label{Fig-McKay_Graph-IP2-rho1}\n\\end{center}\n\\end{figure}\n\nFrom the decomposition of the Kronecker square of $\\varrho_1$ and dimension considerations there is only one possibility for the fourteen-dimensional representation $\\varrho_2$, that is, $\\varrho_2 = \\Sigma_7^{(1)\\prime} + \\Sigma_7^{(2)}$.\nThe values of $x = \\chi_{\\varrho_1}(C) \\in [-2,7]$, $y = \\chi_{\\varrho_2}(C) \\in [-2,14]$ for $PSL(2;8)$ are given in Table \\ref{table:(x,y)-IP2}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_3$ & $C_9$ & $C_9^{\\prime}$ & $C_9^{\\prime\\prime}$ & $C_2$ & $C_7$ & $C_7^{\\prime}$ & $C_7^{\\prime\\prime}$ \\\\\n\\hline $\\chi_{\\varrho_1}(C) \\in [-2,7]$ & 7 & 1 & $-p$ & $-q$ & $p+q$ & -1 & 0 & 0 & 0 \\\\\n\\hline $\\chi_{\\varrho_2}(C) \\in [-2,14]$ & 14 & -1 & $p+q+1$ & $1-p$ & $1-q$ & -2 & 0 & 0 & 0 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{$\\chi_{\\varrho_j}(C)$ for group $PSL(2;8)$, $j=1,2$.} \\label{table:(x,y)-IP2}\n\\end{center}\n\\end{table}\n\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:Character_table-IP2} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n\\begin{align*}\n\\varsigma_{m,n} & = \\frac{1}{504} \\Omega^W(0,0) + \\frac{56}{504} \\Omega^W(0,1\/3) + \\frac{63}{504} \\Omega^W(0,1\/2) + \\frac{72+72+72}{504} \\Omega^W(1\/7,3\/7) \\\\\n& \\quad + \\frac{56}{504} \\Omega^W(1\/9,4\/9) + \\frac{56}{504} \\Omega^W(1\/9,1\/3) + \\frac{56}{504} \\Omega^W(2\/9,5\/9),\n\\end{align*}\nwhere $\\Omega^W(\\theta_1,\\theta_2)$ is as in Section \\ref{sect:II1}.\nNow $J^2(g(1\/9,4\/9)\/64\\pi^4) = 9(3+2\\cos(\\pi\/9)+2\\cos(2\\pi\/9))\/4 =: a_1$, $J^2(g(1\/9,1\/3)\/64\\pi^4) = 9(3-2\\cos(2\\pi\/9)+2\\sin(\\pi\/18))\/4 =: a_2$ and $J^2(g(2\/9,5\/9)\/64\\pi^4) = 9(3-2\\cos(\\pi\/9)-2\\sin(\\pi\/18))\/4 =: a_3$, for all $g \\in D_{12}$.\nThus $a_1 \\Omega^W(1\/9,4\/9) = 18 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}^{((9\/2))}$, $a_2 \\Omega^W(1\/9,1\/3) = 18 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}^{((9\/4))}$ and $a_3 \\Omega^W(2\/9,5\/9) = 18 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}^{((9))}$, as illustrated in Figure \\ref{Fig-OmegaWIP2} since the Jacobian $J=0$ along the boundaries of the orbit of the fundamental domain.\nThe measures $J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7$, $\\mathrm{d}_3 \\times \\mathrm{d}_3$, $\\mathrm{d}_2 \\times \\mathrm{d}_2$, $\\mathrm{d}_1 \\times \\mathrm{d}_1$ and $\\mathrm{d}^{(1)}$ supported by the other points above have all appeared in the previous sections, so we obtain:\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=55mm]{Fig-OmegaWIP2}\\\\\n \\caption{The orbits of $(1\/9,4\/9)\\textcolor{red}{\\scriptscriptstyle{\\bullet}}$, $(1\/9,1\/3)\\textcolor{blue}{\\scriptscriptstyle{\\bullet}}$, $(2\/9,5\/9)\\textcolor{green}{\\scriptscriptstyle{\\bullet}}$.} \\label{Fig-OmegaWIP2}\n\\end{center}\n\\end{figure}\n\n\\begin{Thm}\nThe joint spectral measure (over $\\mathbb{T}^2$) for all embeddings of $PSL(2;8)$ into the fundamental representations of $G_2$ is\n\\begin{equation}\n\\begin{split}\n\\mathrm{d}\\varepsilon & = \\frac{1}{448\\pi^4} J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7 + \\frac{1}{6} \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 + \\frac{1}{6} \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 - \\frac{5}{126} \\, \\mathrm{d}_1 \\times \\mathrm{d}_1 - \\frac{1}{18} \\, \\mathrm{d}^{(1)} \\\\\n& \\quad + \\frac{1}{384\\pi^4} a_3^{-1} J^2 \\, \\mathrm{d}^{((9))} + \\frac{1}{384\\pi^4} a_1^{-1} J^2 \\, \\mathrm{d}^{((9\/2))} + \\frac{1}{384\\pi^4} a_2^{-1} J^2 \\, \\mathrm{d}^{((9\/4))},\n\\end{split}\n\\end{equation}\nwhere $\\mathrm{d}^{((n))}$ is as in Definition \\ref{def:4measures}, $\\mathrm{d}_m$ is the uniform measure over $m^{\\mathrm{th}}$ roots of unity and $\\mathrm{d}^{(k+4)}$ is the uniform measure on the points in $C_k^W$.\n\\end{Thm}\n\n\n\n\\section{Group $PSL(2;13)$}\n\nThe subgroup $PSL(2;13)$ of $G_2$ is an irreducible primitive group of order 1092. It has nine irreducible representations, all real, and its character table is given in Table \\ref{table:Character_table-IP1} \\cite{cohen:1998}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_{13}$ & $C_{13}^{\\prime}$ & $C_2$ $C_7$ & $C_7^{\\prime}$ & $C_7^{\\prime\\prime}$ & $C_6$ & $C_3$ &  \\\\\n\\hline $|C|$ & 1 & 84 & 84 & 91 & 156 & 156 & 156 & 182 & 182 \\\\\n\\hline \\hline $\\Sigma_1$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_7$ & 7 & $p_-$ & $p_+$ & -1 & 0 & 0 & 0 & -1 & 1 \\\\\n\\hline $\\Sigma_7^{\\prime}$ & 7 & $p_+$ & $p_-$ & -1 & 0 & 0 & 0 & -1 & 1 \\\\\n\\hline $\\Sigma_{12}$ & 12 & -1 & -1 & 0 & $r_1$ & $r_2$ & $r_3$ & 0 & 0 \\\\\n\\hline $\\Sigma_{12}^{\\prime}$ & 12 & -1 & -1 & 0 & $r_2$ & $r_3$ & $r_1$ & 0 & 0 \\\\\n\\hline $\\Sigma_{12}^{\\prime\\prime}$ & 12 & -1 & -1 & 0 & $r_3$ & $r_1$ & $r_2$ & 0 & 0 \\\\\n\\hline $\\Sigma_{13}$ & 13 & 0 & 0 & 1 & -1 & -1 & -1 & 1 & 1 \\\\\n\\hline $\\Sigma_{14}$ & 14 & 1 & 1 & -2 & 0 & 0 & 0 & 1 & -1 \\\\\n\\hline $\\Sigma_{14}^{\\prime}$ & 14 & 1 & 1 & 2 & 0 & 0 & 0 & -1 & -1 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for $PSL(2;13)$, where $p_{\\pm} = (1\\pm\\sqrt{13})\/2$, $r_j=-2\\cos(2j\\pi\/7)$.} \\label{table:Character_table-IP1}\n\\end{center}\n\\end{table}\n\nOnly three of the irreducible representation have dimension less than or equal to 7. These are the identity representation, and two seven-dimensional representations $\\Sigma_7$, $\\Sigma_7^{\\prime}$ whose character values only differ on the two conjugacy classes $C_{13}$, $C_{13}^{\\prime}$ whose elements have order 13. Here $\\chi_{\\Sigma_7}(g) = p,q$ for $g \\in C_{13},C_{13}^{\\prime}$ respectively, whilst $\\chi_{\\Sigma_7^{\\prime}}(g) = q,p$ respectively, where $p=(1+\\sqrt{13})\/2=1+\\zeta+\\zeta^3+\\zeta^4+\\zeta^9+\\zeta^{10}+\\zeta^{12}$, $q=(1-\\sqrt{13})\/2=1+\\zeta^2+\\zeta^5+\\zeta^6+\\zeta^7+\\zeta^8+\\zeta^{11}$, for $\\zeta = e^{2\\pi i\/13}$.\nThus the spectral measure for both seven-dimensional representations are equal, since $C_{13}$, $C_{13}^{\\prime}$ both have the same order, and both give embeddings of $PSL(2;13)$ in $G_2$.\nThe McKay graph for the fundamental seven-dimensional representation is given in \\cite[Figure 2]{he:2003}, and is reproduced (twice) in Figure \\ref{Fig-McKay_Graph-IP1-rho1} for completeness, where we use the same notation as previously. The figure on the right hand side illustrates the resemblance of $\\mathcal{G}^{\\varrho_1}_{PSL(2;13)}$ with the McKay graph of $G_2$ itself.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=60mm]{Fig-McKay_Graph-IP1-rho1} \\hspace{15mm} \\includegraphics[width=60mm]{Fig-McKay_Graph-IP1-rho1-2}\\\\\n \\caption{Two presentations of the McKay graph $\\mathcal{G}^{\\varrho_1}_{PSL(2;13)}$.} \\label{Fig-McKay_Graph-IP1-rho1}\n\\end{center}\n\\end{figure}\n\nThe set $X_C$ of eigenvalues of group elements in each conjugacy class for $\\varrho_1 = \\Sigma_7$ are given in Table \\ref{table:evalues-IP1}, along with a choice of $(t_1^C,t_2^C) \\in X_C$ such that $\\mathcal{E}_{t_1^C,t_2^C} = X_C$.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|} \\hline\n$C$ & $X_C$ & $(t_1^C,t_2^C)$ & $\\chi_{\\varrho_1}(C)$ & $\\chi_{\\varrho_2}(C)$ & $\\chi_{\\Sigma_{14}}(C)$ & $\\chi_{\\Sigma_{14}^{\\prime}}(C)$ \\\\\n\\hline \\hline $C_1$ & $(1,1,1,1,1,1,1)$ & $(1,1)$ & 7 & 14 & 14 & 14 \\\\\n\\hline $C_{13}$ & $(1,\\zeta,\\zeta^3,\\zeta^4,\\zeta^9,\\zeta^{10},\\zeta^{12})$ & $(\\zeta,\\zeta^4)$ & $(1+\\sqrt{13})\/2$ & 1 & 1 & 1 \\\\\n\\hline $C_{13}^{\\prime}$ & $(1,\\zeta^2,\\zeta^5,\\zeta^6,\\zeta^7,\\zeta^8,\\zeta^{11})$ & $(\\zeta^2,\\zeta^7)$ & $(1-\\sqrt{13})\/2$ & 1 & 1 & 1 \\\\\n\\hline $C_2$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,-1)$ & -1 & -2 & -2 & 2\\\\\n\\hline $C_7$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(\\eta,\\eta^5)$ & 0 & 0 & 0 & 0 \\\\\n\\hline $C_7^{\\prime}$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(\\eta,\\eta^5)$ & 0 & 0 & 0 & 0 \\\\\n\\hline $C_7^{\\prime\\prime}$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(\\eta,\\eta^5)$ & 0 & 0 & 0 & 0 \\\\\n\\hline $C_6$ & $(1,-1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ & $(-1,\\mu)$ & -1 & 1 & 1 & -1 \\\\\n\\hline $C_3$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,\\omega)$ & 1 & -1 & -1 & -1\\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Choice of eigenvalues $(t_1^C,t_2^C)$ for $\\varrho_1$ and corresponding values of $\\chi_{\\varrho_2}(C)$, where $\\omega = e^{2\\pi i\/3}$, $\\mu = e^{2\\pi i\/6}$, $\\eta = e^{2\\pi i\/7}$ and $\\zeta = e^{2\\pi i\/13}$.} \\label{table:evalues-IP1}\n\\end{center}\n\\end{table}\n\nThe decomposition of the Kronecker square of $\\varrho_1$ into irreducibles is given by\n$$\\varrho_1^2 = \\mathrm{id} + \\varrho_1 + \\Sigma_{13} + \\Sigma_{14} + \\Sigma_{14}^{\\prime},$$\nwhere as before the notation $\\Sigma_n$ denotes an irreducible representation of $PSL(2;13)$ of dimension $n$.\nFrom dimension considerations there are thus two candidates for the fourteen-dimensional representation $\\varrho_2$, which are given by the two fourteen-dimensional irreducible representations.\nHowever, since $\\chi_{\\varrho_2}(C) = \\Phi_2(t_1^C,t_2^C)$, we see from Table \\ref{table:evalues-IP1} that the decomposition of the fundamental fourteen-dimensional representation into irreducible representations of $PSL(2;13)$ is given by $\\varrho_2 = \\Sigma_{14}$, not $\\Sigma_{14}^{\\prime}$.\n\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:evalues-IP1} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n\\begin{align*}\n\\varsigma_{m,n} & = \\frac{1}{1092} \\Omega^W(0,0) + \\frac{91}{1092} \\Omega^W(0,1\/2) + \\frac{182}{1092} \\Omega^W(0,1\/3) + \\frac{182}{1092} \\Omega^W(1\/6,1\/2) \\\\\n& \\quad + \\frac{156+156+156}{1092} \\Omega^W(1\/7,3\/7) + \\frac{84}{1092} \\Omega^W(1\/13,4\/13)  + \\frac{84}{1092} \\Omega^W(2\/13,7\/13),\n\\end{align*}\nwhere $\\Omega^W(\\theta_1,\\theta_2)$ is as in Section \\ref{sect:II1}.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=55mm]{Fig-OmegaWIP1}\\\\\n \\caption{The orbit of $(1\/13,4\/13)$ and $(2\/13,7\/13)$.} \\label{Fig-OmegaWIP1}\n\\end{center}\n\\end{figure}\n\nThe orbit of the points $(1\/13,4\/13)$, $(2\/13,7\/13)$, illustrated in Figure \\ref{Fig-OmegaWIP1}, do not give a linear combination of the measures in Definition \\ref{def:4measures}, thus we have $12\\Omega^W(1\/13,4\/13) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\left( \\sum_{g \\in D_{12}} \\delta_{g(\\zeta,\\zeta^4)} \\right)$ and $12\\Omega^W(2\/13,7\/13) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\left( \\sum_{g \\in D_{12}} \\delta_{g(\\zeta^2,\\zeta^7)} \\right)$.\nThe measures $J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7$, $J^2 \\, \\mathrm{d}_6 \\times \\mathrm{d}_6$, $\\mathrm{d}_3 \\times \\mathrm{d}_3$, $\\mathrm{d}_2 \\times \\mathrm{d}_2$, $\\mathrm{d}_1 \\times \\mathrm{d}_1$ and $\\mathrm{d}^{(1)}$ supported by the other points above have all appeared in the previous sections, so we obtain:\n\n\\begin{Thm}\nThe joint spectral measure (over $\\mathbb{T}^2$) for all embeddings of $PSL(2;13)$ into the fundamental representations of $G_2$ is\n\\begin{equation}\n\\begin{split}\n\\mathrm{d}\\varepsilon & = \\frac{1}{448\\pi^4} J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7 + \\frac{1}{1152\\pi^4} J^2 \\, \\mathrm{d}_6 \\times \\mathrm{d}_6 + \\frac{1}{4} \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 + \\frac{1}{9} \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 \\\\\n& \\quad - \\frac{22}{819} \\, \\mathrm{d}_1 \\times \\mathrm{d}_1 - \\frac{1}{12} \\, \\mathrm{d}^{(1)} + \\frac{7}{192} \\sum_{g \\in D_{12}} (\\delta_{g(\\zeta,\\zeta^4)} + \\delta_{g(\\zeta^2,\\zeta^7)})\n\\end{split}\n\\end{equation}\nwhere $\\mathrm{d}_m$ is the uniform measure over $m^{\\mathrm{th}}$ roots of unity, $\\mathrm{d}^{(k+4)}$ is the uniform measure on the points in $C_k^W$, $\\delta_x$ is the Dirac measure at the point $x$ and $\\zeta = e^{2 \\pi i\/13}$.\n\\end{Thm}\n\n\n\\section{Group $PU(3;3) \\cong G_2(2)'$}\n\nThe subgroup $PU(3;3)$ of $G_2$ is an irreducible primitive group of order 6048.\nIt has fourteen irreducible representations (seven real repreentations, one quaternionic representation $\\Sigma_6$, and three pairs of complex conjugate representations) and its character table is given in Table \\ref{table:Character_table-IP4} \\cite{conway\/curtis\/norton\/parker\/wilson:1985}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|@{\\hspace{1.5mm}}c@{\\hspace{1.5mm}}||@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|} \\hline\n$C$ & $C_1$ & $C_3$ & $C_2$ & $C_4$ & $C_4^{\\prime}$ & $C_4^{\\prime\\prime}$ & $C_{12}$ & $C_{12}^{\\prime}$ & $C_6$ & $C_3^{\\prime}$ & $C_8$ & $C_8^{\\prime}$ & $C_7$ & $C_7^{\\prime}$ \\\\\n\\hline $|C|$ & 1 & 56 & 63 & 63 & 63 & 378 & 504 & 504 & 504 & 672 & 756 & 756 & 864 & 864 \\\\\n\\hline \\hline $\\Sigma_1$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_6$ & 6 & -3 & -2 & -2 & -2 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & -1 & -1 \\\\\n\\hline $\\Sigma_7$ & 7 & -2 & 3 & $-1-2i$ & $-1+2i$ & 1 & $-1-i$ & $-1+i$ & 0 & 1 & $-i$ & $i$ & 0 & 0 \\\\\n\\hline $\\Sigma_7^{\\ast}$ & 7 & -2 & 3 & $-1+2i$ & $-1-2i$ & 1 & $-1+i$ & $-1-i$ & 0 & 1 & $i$ & $-i$ & 0 & 0 \\\\\n\\hline $\\Sigma_7^{\\prime}$ & 7 & -2 & -1 & 3 & 3 & -1 & 0 & 0 & 2 & 1 & -1 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_{14}$ & 14 & 5 & -2 & 2 & 2 & 2 & -1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{21}$ & 21 & 3 & 1 & $-3-2i$ & $-3+2i$ & -1 & $-i$ & $i$ & 1 & 0 & $i$ & $-i$ & 0 & 0 \\\\\n\\hline $\\Sigma_{21}^{\\ast}$ & 21 & 3 & 1 & $-3+2i$ & $-3-2i$ & -1 & $i$ & $-i$ & 1 & 0 & $-i$ & $i$ & 0 & 0 \\\\\n\\hline $\\Sigma_{21}^{\\prime}$ & 21 & 3 & 5 & 1 & 1 & 1 & 1 & 1 & -1 & 0 & -1 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_{27}$ & 27 & 0 & 3 & 3 & 3 & -1 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1 \\\\\n\\hline $\\Sigma_{28}$ & 28 & 1 & -4 & $-4i$ & $4i$ & 0 & $i$ & $-i$ & -1 & 1 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{28}^{\\ast}$ & 28 & 1 & -4 & $4i$ & $-4i$ & 0 & $-i$ & $i$ & -1 & 1 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{32}$ & 32 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & $\\frac{1+i\\sqrt{7}}{2}$ & $\\frac{1-i\\sqrt{7}}{2}$ \\\\\n\\hline $\\Sigma_{32}^{\\ast}$ & 32 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & $\\frac{1-i\\sqrt{7}}{2}$ & $\\frac{1+i\\sqrt{7}}{2}$ \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for $PU(3;3)$.} \\label{table:Character_table-IP4}\n\\end{center}\n\\end{table}\n\nThere are two non-conjugate real seven-dimensional representations, $\\gamma_1^{(1)} = \\Sigma_1 + \\Sigma_6$ and $\\gamma_1^{(2)} = \\Sigma_7^{\\prime}$.\nThese both satisfy the condition that $\\gamma_1^{(i)}$ appears in the decomposition of $(\\gamma_1^{(i)})^2$.\nWe thus consider the eigenvalues of the representation matrices to determine which of the $\\gamma_1^{(i)}$ are embeddings of $PU(3;3)$ in $G_2$.\nThese eigenvalues are given in Table \\ref{table:evalues-IP4} for the representations $\\Sigma_1$, $\\Sigma_6$ and $\\Sigma_7^{\\prime}$. As described in Section \\ref{sect:subgroupsG2}, these eigenvalues can be determined from the character table of $PU(3;3)$. The additional information that is needed is to note that the eigenvalues for group elements in $C_4$, $C_4^{\\prime}$ and $C_4^{\\prime\\prime}$ all square to those for elements in $C_2$, those for $C_8$, $C_8^{\\prime}$ square to those for $C_4$, $C_4^{\\prime}$ respectively, those for $C_6$ square to those for $C_3$ and also cube to those for $C_2$, those for $C_{12}$ square to those for $C_6$ and cube to those for $C_4$ whilst those for $C_{12}^{\\prime}$ also square to those for $C_6$ but cube to those for $C_4^{\\prime}$ (see \\cite{conway\/curtis\/norton\/parker\/wilson:1985}).\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c||c|} \\hline\n & $\\Sigma_1$ & $\\Sigma_6$ & $\\Sigma_7^{\\prime}$ & $(t_1^C,t_2^C)$ \\\\\n\\hline \\hline $C_1$ & 1 & $(1,1,1,1,1,1)$ & $(1,1,1,1,1,1,1)$ & $(1,1)$ \\\\\n\\hline $C_3$ & 1 & $(\\omega,\\omega,\\omega,\\omega^2,\\omega^2,\\omega^2)$ & $(1,\\omega,\\omega,\\omega,\\omega^2,\\omega^2,\\omega^2)$  & $(\\omega,\\omega^2)$ \\\\\n\\hline $C_2$ & 1 & $(1,1,-1,-1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1)$  & $(1,-1)$ \\\\\n\\hline $C_4$ & 1 & $(-1,-1,i,i,-i,-i)$ & $(1,1,1,i,i,-i,-i)$  & $(1,i)$ \\\\\n\\hline $C_4^{\\prime}$ & 1 & $(-1,-1,i,i,-i,-i)$ & $(1,1,1,i,i,-i,-i)$  & $(1,i)$ \\\\\n\\hline $C_4^{\\prime\\prime}$ & 1 & $(1,1,i,i,-i,-i)$ & $(1,-1,-1,i,i,-i,-i)$  & $(-1,i)$ \\\\\n\\hline $C_{12}$ & 1 & $(\\xi,\\xi^2,\\xi^5,\\xi^7,\\xi^{10},\\xi^{11})$ & $(1,\\xi,\\xi^4,\\xi^5,\\xi^7,\\xi^8,\\xi^{11})$  & $(\\xi,\\xi^5)$ \\\\\n\\hline $C_{12}^{\\prime}$ & 1 & $(\\xi,\\xi^2,\\xi^5,\\xi^7,\\xi^{10},\\xi^{11})$ & $(1,\\xi,\\xi^4,\\xi^5,\\xi^7,\\xi^8,\\xi^{11})$  & $(\\xi,\\xi^5)$ \\\\\n\\hline $C_6$ & 1 & $(\\mu,\\mu,\\mu^2,\\mu^4,\\mu^5,\\mu^5)$ & $(1,\\mu,\\mu,\\mu^2,\\mu^4,\\mu^5,\\mu^5)$  & $(\\mu,\\mu^2)$ \\\\\n\\hline $C_3^{\\prime}$ & 1 & $(1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$  & $(1,\\omega)$ \\\\\n\\hline $C_8$ & 1 & $(i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,-1,\\nu,\\nu^3,\\nu^5,\\nu^7)$  & $(-1,\\nu)$ \\\\\n\\hline $C_8^{\\prime}$ & 1 & $(i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,-1,\\nu,\\nu^3,\\nu^5,\\nu^7)$  & $(-1,\\nu)$ \\\\\n\\hline $C_7$ & 1 & $(\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$  & $(\\eta,\\eta^5)$ \\\\\n\\hline $C_7^{\\prime}$ & 1 & $(\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$  & $(\\eta,\\eta^5)$ \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Eigenvalues of group elements in each conjugacy class of $PU(3;3)$ for the irreducible representations $\\Sigma_1$, $\\Sigma_6$ and $\\Sigma_7^{\\prime}$, where $\\omega = e^{2\\pi i\/3}$, $\\mu = e^{2\\pi i\/6}$, $\\eta = e^{2\\pi i\/7}$, $\\nu = e^{2\\pi i\/8}$ and $\\xi = e^{2\\pi i\/12}$} \\label{table:evalues-IP4}\n\\end{center}\n\\end{table}\n\nFrom considering the set of eigenvalues $X_C$ for group elements in $C$ in the representation $\\gamma_1^{(i)}$, we see that there is no choice of $(t_1^C,t_2^C) \\in X_C$ such that $\\mathcal{E}_{t_1^C,t_2^C} = X_C$ for $i=1$ when $C=C_{12},C_{12}^{\\prime}$. However, such a choice does exist for all conjugacy classes $C$ for $i=2$, thus we have $\\varrho_1 = \\gamma_1^{(2)}$. We present one such choice of eigenvalues $(t_1^C,t_2^C)$ in the final column of Table \\ref{table:evalues-IP4}.\nThe McKay graph $\\mathcal{G}^{\\varrho_1}_{PU(3;3)}$ is given in \\cite[Figure 2]{he:2003}, and we reproduce it here in Figure \\ref{Fig-McKay_Graph-IP4-rho1} for completeness.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=70mm]{Fig-McKay_Graph-IP4-rho1}\\\\\n \\caption{The McKay graphs $\\mathcal{G}^{\\varrho_1}_{PU(3;3)}$.} \\label{Fig-McKay_Graph-IP4-rho1}\n\\end{center}\n\\end{figure}\n\nThe decomposition of the Kronecker square of $\\varrho_1$ into irreducibles is given by\n$$\\varrho_1^2 = \\mathrm{id} + \\varrho_1 + \\Sigma_{14} + \\Sigma_{27},$$\nwhere $\\mathrm{id} = \\Sigma_1$.\nThus the fourteen-dimensional representation $\\varrho_2$ must be $\\Sigma_{14}$, and we note that $\\chi_{\\varrho_2}(C) = \\Phi_2(t_1^C,t_2^C)$ as required.\n\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:Character_table-IP4} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n\\begin{align*}\n\\varsigma_{m,n} & = \\frac{1}{6048} \\Omega^W(0,0) + \\frac{56}{6048} \\Omega^W(1\/3,2\/3) + \\frac{63}{6048} \\Omega^W(0,1\/2) + \\frac{672}{6048} \\Omega^W(0,1\/3) \\\\\n& \\quad + \\frac{63+63}{6048} \\Omega^W(0,1\/4) + \\frac{378}{6048} \\Omega^W(1\/4,1\/2) + \\frac{504}{6048} \\Omega^W(1\/6,1\/3) \\\\\n& \\quad + \\frac{864+864}{6048} \\Omega^W(1\/7,3\/7) + \\frac{756+756}{6048} \\Omega^W(1\/8,1\/2) + \\frac{504+504}{6048} \\Omega^W(1\/12,5\/12),\n\\end{align*}\nwhere $\\Omega^W(\\theta_1,\\theta_2)$ is as in Section \\ref{sect:II1}.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=135mm]{Fig-OmegaWIP4}\\\\\n \\caption{The orbits of $(a)$ $(1\/6,1\/3)$, $(b)$ $(1\/12,5\/12)$.} \\label{Fig-OmegaWIP4}\n\\end{center}\n\\end{figure}\n\nNow $2\\Omega^W(1\/3,2\/3) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\, (3 \\mathrm{d}^{(1)} - \\mathrm{d}_1 \\times \\mathrm{d}_1)$.\nThe points $\\circ$ in Figure \\ref{Fig-OmegaWIP4}$(a)$ give the measure $3 \\, \\mathrm{d}^{(1)}$ whilst the points $\\diamond$ in Figure \\ref{Fig-OmegaWIP4}$(a)$ give the measure $4 \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 - \\mathrm{d}_1 \\times \\mathrm{d}_1$, thus we see that $6\\Omega^W(1\/6,1\/3) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\left( 12 \\, \\mathrm{d}^{(2)} - 3 \\, \\mathrm{d}^{(1)} - 4 \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 + \\mathrm{d}_1 \\times \\mathrm{d}_1 \\right)$.\n\nFinally, $12\\Omega^W(1\/12,5\/12) = 18 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}^{(4)}$, as illustrated in Figure \\ref{Fig-OmegaWIP4}$(b)$ since the Jacobian $J=0$ along the boundaries of the orbit of the fundamental domain, whilst $J^2(g(1\/12,5\/12)\/64\\pi^4) = 12$ for all $g \\in D_{12}$.\n\nThe measures $J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7$, $(24-K) \\, \\mathrm{d}_4 \\times \\mathrm{d}_4$, $\\mathrm{d}_3 \\times \\mathrm{d}_3$ and $\\sum_{g \\in D_{12}} \\delta_{g(e^{\\pi i\/4},-1)}$ supported by the other points have all appeared in the previous sections, so we obtain:\n\n\\begin{Thm}\nThe joint spectral measure (over $\\mathbb{T}^2$) for all embeddings of $PU(3;3)$ into the fundamental representations of $G_2$ is\n\\begin{equation}\n\\begin{split}\n\\mathrm{d}\\varepsilon & = \\frac{1}{672\\pi^4} J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7 + \\frac{1}{144} (24-K) \\, \\mathrm{d}_4 \\times \\mathrm{d}_4 + \\frac{1}{6} \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 - \\frac{1}{12} \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 \\\\\n& \\quad + \\frac{1}{168} \\, \\mathrm{d}_1 \\times \\mathrm{d}_1 + \\frac{1}{1152\\pi^4} J^2 \\, \\mathrm{d}^{(4)} + \\frac{1}{6} \\, \\mathrm{d}^{(2)} - \\frac{1}{12} \\, \\mathrm{d}^{(1)} + \\frac{1}{48} \\sum_{g \\in D_{12}} \\delta_{g(e^{\\pi i\/4},-1)},\n\\end{split}\n\\end{equation}\nwhere $K(\\theta_1,\\theta_2) = (\\sin(2\\pi(\\theta_1+\\theta_2))-\\sin(2\\pi(2\\theta_1-\\theta_2))-\\sin(2\\pi(2\\theta_2-\\theta_1)))^2$, $\\mathrm{d}_m$ is the uniform measure over $m^{\\mathrm{th}}$ roots of unity, $\\mathrm{d}^{(k+4)}$ is the uniform measure on the points in $C_k^W$, and $\\delta_x$ is the Dirac measure at the point $x$.\n\\end{Thm}\n\n\n\\section{Group $G_2(2)$} \\label{sect:IP5}\n\nThe subgroup $G_2(2) = G_2(\\mathbb{F}_2)$ of $G_2 = G_2(\\mathbb{C})$ is an irreducible primitive group of order 12096. It is the group $G_2$ defined over the Galois field $\\mathbb{F}_2$.\nIt has sixteen irreducible representations (fourteen real and two complex conjugate representations), and its character table is given in\n\\cite{he:2003} (the orders of the elements in each conjugacy class can be obtained from \\cite{koca\/koc:1994}).\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|@{\\hspace{1.5mm}}c@{\\hspace{1.5mm}}||@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|} \\hline\n$C$ & $C_1$ & $C_3$ & $C_2$ & $C_4$ & $C_2^{\\prime}$ & $C_4^{\\prime}$ & $C_4^{\\prime\\prime}$ & $C_6$ & $C_3^{\\prime}$ & $C_{12}$ & $C_{12}^{\\prime}$ & $C_{12}^{\\prime\\prime}$ & $C_8$ & $C_8^{\\prime}$ & $C_7$ & $C_6^{\\prime}$ \\\\\n\\hline $|C|$                & 1  & 56 & 63 & 126 & 252 & 252 & 378 & 504 & 672 & 1008 & 1008 & 1008 & 1512 & 1512 & 1728 & 2016 \\\\\n\\hline $\\Sigma_1$           & 1  & 1  & 1  & 1  & 1  & 1  & 1  & 1  & 1  & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_1'$          & 1  & 1  & 1  & 1  & -1 & -1 & 1  & 1  & 1  & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\\\\n\\hline $\\Sigma_6$           & 6  & -3 & -2 & -2 & 0  & 0  & 2  & 1  & 0  & $i\\sqrt{3}$ & $-i\\sqrt{3}$ & 1 & 0 & 0 & -1 & 0 \\\\\n\\hline $\\Sigma_6^{\\ast}$    & 6  & -3 & -2 & -2 & 0  & 0  & 2  & 1  & 0  & $-i\\sqrt{3}$ & $i\\sqrt{3}$ & 1 & 0 & 0 & -1 & 0 \\\\\n\\hline $\\Sigma_7$           & 1  & -2 & -1 & 3  & -1 & 3  & -1 & 2  & 1  & 0 & 0 & 0 & 1 & -1 & 0 & -1 \\\\\n\\hline $\\Sigma_7'$          & 1  & -2 & -1 & 3  & 1  & -3 & -1 & 2  & 1  & 0 & 0 & 0 & -1 & -1 & 0 & 1 \\\\\n\\hline $\\Sigma_{14}$        & 14 & 5  & -2 & 2  & -2 & 2  & 2  & 1  & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 1 \\\\\n\\hline $\\Sigma_{14}'$       & 14 & -4 & 6  & -2 & 0  & 0  & 2  & 0  & 2  & 0 & 0 & -2 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{14}''$      & 14 & 5  & -2 & 2  & 2  & -2 & 2  & 1  & -1 & 1 & 1 & -1 & 0 & 0 & 0 & -1 \\\\\n\\hline $\\Sigma_{21}$        & 21 & 3  & 5  & 1  & 3  & -1 & 1  & -1 & 0  & -1 & -1 & 1 & 1 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_{21}'$       & 21 & 3  & 5  & 1  & -3 & 1  & 1  & -1 & 0  & 1 & 1 & 1 & -1 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_{27}$        & 27 & 0  & 3  & 3  & 3  & 3  & -1 & 0  & 0  & 0 & 0 & 0 & -1 & 1 & -1 & 0 \\\\\n\\hline $\\Sigma_{27}'$       & 27 & 0  & 3  & 3  & -3 & -3 & -1 & 0  & 0  & 0 & 0 & 0 & 1 & 1 & -1 & 0 \\\\\n\\hline $\\Sigma_{42}$        & 42 & 6  & 2  & -6 & 0  & 0  & -2 & 2  & 0  & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{56}$        & 56 & 2  & -8 & 0  & 0  & 0  & 0  & -2 & 2  & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{64}$        & 64 & -8 & 0  & 0  & 0  & 0  & 0  & 0  & -2 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for $G_2(2)$.} \\label{table:Character_table-IP5}\n\\end{center}\n\\end{table}\n\nThe group $G_2(2)$ has only two seven-dimensional real representations, both of which are irreducible. Of these two, only $\\Sigma_7$ has character values in $[-2,7]$ for all $g \\in G_2(2)$, thus this is the restriction $\\varrho_1$ of the seven-dimensional fundamental representation $\\rho_1$ of $G_2$ to $G_2(2)$.\nThe McKay graph for $\\varrho_1$ is given in Figure \\ref{Fig-McKay_Graph-IP5-rho1}.\nThis graph is a $\\mathbb{Z}_2$-orbifold of the McKay graph $\\mathcal{G}^{\\varrho_1}_{PU(3;3)}$ for $PU(3;3)$.\nNote that the McKay graph given in \\cite[Figure 2]{he:2003} is not that for $\\varrho_1$, but rather for the other irreducible seven-dimensional representation $\\Sigma_7'$ of $G_2(2)$.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=70mm]{Fig-McKay_Graph-IP5-rho1}\\\\\n \\caption{The McKay graphs $\\mathcal{G}^{\\varrho_1}_{G_2(2)}$.} \\label{Fig-McKay_Graph-IP5-rho1}\n\\end{center}\n\\end{figure}\n\nThe decomposition of the Kronecker square of $\\varrho_1$ into irreducibles is given by\n$$\\varrho_1^2 = \\mathrm{id} + \\varrho_1 + \\Sigma_{14} + \\Sigma_{27},$$\nthus the fourteen-dimensional representation $\\varrho_2$ is given by the irreducible representation $\\Sigma_{14}$.\nWe note that the eigenvalues of the representation matrices for $C_4, C_4^{\\prime}, C_4^{\\prime\\prime}$ all square to those for $C_2$, those for $C_6$ square to those for $C_3$, those for $C_6^{\\prime}$ square to those for $C_3^{\\prime}$, those for $C_8$ square to those for $C_4$, those for $C_8^{\\prime}$ square to those for $C_4^{\\prime\\prime}$, and those for $C_{12}, C_{12}^{\\prime}, C_{12}^{\\prime\\prime}$ all square to those for $C_6$.\n\n\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:Character_table-IP5} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n\\begin{align*}\n\\varsigma_{m,n} & = \\frac{1}{12096} \\Omega^W(0,0) + \\frac{56}{12096} \\Omega^W(1\/3,2\/3) + \\frac{63+252}{12096} \\Omega^W(0,1\/2) + \\frac{672}{12096} \\Omega^W(0,1\/3) \\\\\n& \\quad + \\frac{126+252}{12096} \\Omega^W(0,1\/4) + \\frac{378}{12096} \\Omega^W(1\/4,1\/2) + \\frac{504}{12096} \\Omega^W(1\/6,1\/3) \\\\\n& \\quad + \\frac{1728}{12096} \\Omega^W(1\/7,3\/7) + \\frac{1512}{12096} \\Omega^W(1\/8,1\/2) + \\frac{1512}{12096} \\Omega^W(1\/8,3\/8) \\\\\n& \\quad + \\frac{1008+1008+1008}{12096} \\Omega^W(1\/12,5\/12) + \\frac{2016}{12096} \\Omega^W(1\/6,1\/2),\n\\end{align*}\nwhere $\\Omega^W(\\theta_1,\\theta_2)$ is as in Section \\ref{sect:II1}.\nThe measures given by these points have all appeared in the previous sections.\nNote however that $12(\\Omega^W(1\/8,1\/2) + \\Omega^W(1\/8,3\/8)) = 8 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}_8 \\times \\mathrm{d}_8$, since the Jacobian $J=0$ along the boundaries of the orbit of the fundamental domain, whilst $J^2(g(1\/8,1\/2))\/64\\pi^4 = J^2(g(1\/8,3\/8))\/64\\pi^4 = 8$ for all $g \\in D_{12}$.\nThus we obtain:\n\n\\begin{Thm}\nThe joint spectral measure (over $\\mathbb{T}^2$) for all embeddings of $G_2(2)$ into the fundamental representations of $G_2$ is\n\\begin{equation}\n\\begin{split}\n\\mathrm{d}\\varepsilon & = \\frac{1}{768\\pi^4} J^2 \\, \\mathrm{d}_8 \\times \\mathrm{d}_8 + \\frac{1}{1344\\pi^4} J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7 + \\frac{1}{1152\\pi^4} J^2 \\, \\mathrm{d}_6 \\times \\mathrm{d}_6 + \\frac{1}{12} \\, \\mathrm{d}_4 \\times \\mathrm{d}_4 \\\\\n& \\quad + \\frac{1}{12} \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 - \\frac{1}{72} \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 - \\frac{1}{252} \\, \\mathrm{d}_1 \\times \\mathrm{d}_1 + \\frac{1}{768\\pi^4} J^2 \\, \\mathrm{d}^{(4)} + \\frac{1}{12} \\, \\mathrm{d}^{(2)} - \\frac{1}{24} \\, \\mathrm{d}^{(1)},\n\\end{split}\n\\end{equation}\nwhere $\\mathrm{d}_m$ is the uniform measure over $m^{\\mathrm{th}}$ roots of unity and $\\mathrm{d}^{(k+4)}$ is the uniform measure on the points in $C_k^W$.\n\\end{Thm}\n\n\n\n\n\n\n\n\n\n\\bigskip \\bigskip\n\n\\begin{footnotesize}\n\\noindent{\\it Acknowledgement.}\n\nThe second author was supported by the Coleg Cymraeg Cenedlaethol.\n\\end{footnotesize}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $S$ be a complex manifold and $D$ be a (reduced) hypersurface $D$, referred to as a \\emph{divisor} in the sequel. \nIn the landmark paper \\cite{Sai80}, Kyoji Saito introduced the sheaves of $\\O_S$-modules of logarithmic differential forms and logarithmic vector fields on $S$ along $D$.\nLogarithmic vector fields are tangent to $D$ at any smooth point of $D$; logarithmic differential forms have simple poles and form a complex under the usual differential.\nSaito's clean algebraically flavored definition encodes deep geometric, topological, and representation theoretic information on the singularities that is yet only partly understood.\nThe precise target for his theory was the Gau\\ss-Manin connection on the base $S$ of the semiuniversal deformation of isolated hypersurface singularities, a logarithmic connection along the discriminant $D$.\nSaito developed mainly three aspects of his logarithmic theory in loc.~cit.: Free divisors, logarithmic stratifications, and logarithmic residues.\nMany fascinating developments grew out of Saito's paper, a few of which we highlight in the following brief overview.\n\nA divisor is called free if the sheaf of logarithmic vector fields, or its dual, the sheaf of logarithmic $1$-forms, is a vector bundle; in particular, normal crossing divisors are free.\nNot surprisingly, discriminants of isolated hypersurface singularities are free divisors (see \\cite[(3.19)]{Sai80}). \nSimilar results were shown for isolated complete intersection singularities (see \\cite[\\S6]{Loo84}) and space curve singularities (see \\cite{vST95}).\nBoth the reflection arrangements and discriminants associated with finite unitary reflection groups are free divisors (see \\cite{Ter80b}).\nMore recent examples include discriminants in certain prehomogeneous vector spaces (see \\cite{GMS11}) whose study led to new constructions such as a chain rule for free divisors (see \\cite[\\S4]{BC12}).\nFree divisors can be seen as the extreme case opposite to isolated singularities: Unless smooth, free divisors have Cohen--Macaulay singular loci of codimension $1$.\nThe freeness property is closely related to the complement of the divisor being a $K(\\pi,1)$-space (see \\cite[(1.12)]{Sai80}, \\cite{Del72}), although these two properties are not equivalent (see \\cite{ER95}). \nEven in special cases, such as that of hyperplane arrangements, freeness is not fully understood yet. \nFor instance, Terao's conjecture on the combinatorial nature of freeness for arrangements is one of the central open problems in arrangement theory.\n\nSaito's second topic, the so-called logarithmic stratification of $S$, consists of immersed integral manifolds of logarithmic vector fields along $D$.\nContrary to what the terminology suggests, the resulting decomposition of $S$ is not locally finite, in general.\nSaito attached the term holonomic to this additional feature: a point in $S$ is holonomic if a neighborhood meets only finitely many logarithmic strata.\nAlong any logarithmic stratum, the pair $(D,S)$ is analytically trivial which turns holonomicity into a property of logarithmic strata.\nThe logarithmic vector fields are tangent to the strata of the canonical Whitney stratification; the largest codimension up to which all Whitney strata are (neccessarily holonomic) logarithmic strata is called the holonomic codimension (see \\cite[p.~221]{DM91}).\nHolonomic free divisors were later called Koszul--free divisors.\n\nIn case of a normal crossing divisor, the complex of logarithmic differential forms computes the cohomology of the complement of $D$ in $S$, an ingredient of Deligne's mixed Hodge structure (see \\cite{Del71}).\nThe natural question, for which free divisors the same holds true for the complex of logarithmic differential forms is referred to as the logarithmic comparison theorem, or, for short, by the LCT (see \\cite{Tor07} for a survey).\nFor free divisors, this property turned out to be related to homogeneity properties of the singularities. \nIndeed, an explicit class of hypersurfaces for which the LCT holds true is that of (weakly) locally quasihomogeneous divisors (see \\cite{CNM96,Nar08}).\nMoreover, it is conjectured that the LCT implies strong Euler homogeneity of $D$, which has been proved only for Koszul free divisors, and in dimension $\\dim S\\le3$ (see \\cite{CMNC02,GS06}).\nFor strongly Koszul--free divisors $D$ (see \\cite[Def.~7.1]{GS10}), the logarithmic comparison theorem holds true exactly if $D$ is strongly Euler homogeneous and $-1$ is the minimal integer root of all local $b$-functions (see \\cite[Cor.~4.3]{CN05} and \\cite[Cor.~1.8]{Tor04}).\nFor isolated quasihomogeneous singularities, the LCT is equivalent to the vanishing of certain graded parts of the Milnor algebra (cee \\cite{HM98}); there are related Hodge-theoretic properties in the non-quasihomogeneous case (\\cite{Sch10}).\nThe study of the LCT lead to a variant of $D$-module theory over the ring of logarithmic differential operators along a free divisor (see \\cite{CU02,CN05,Nar08}).\nA key player in this context is the $D$-module $M^{\\log D}$ defined by the ideal of logarithmic vector fields, considered as differential operators of order one (see \\cite{CU02}); Saito-holonomicity of $D$ implies holonomicity of $M^{\\log D}$ in the $D$-module-sense; but the converse is false.\n\nMuch less attention has been devoted to Saito's logarithmic residues, the main topic in this paper.\nIt was Poincar\\'e who first defined a residual $1$-form of a rational differential $2$-form on $\\mathds{C}^2$ (see \\cite{Poi87}).\nLater, the concept was generalized by de Rham and Leray to residues of closed meromorphic $p$-forms with simple poles along a smooth divisor $D$; these residues are holomorphic $(p-1)$-form on $D$ (see \\cite{Ler59}).\nThe construction of Deligne's mixed Hodge structure uses, again holomorphic, residues of logarithmic differential forms along normal crossing divisors (see \\cite{Del71}).\nNotably, in Saito's generalization to arbitrary singular divisors $D$, the residue of a logarithmic $p$-form becomes a \\emph{meromorphic} $(p-1)$-form on $D$, or on its normalization $\\widetilde D$. \nUsing work of Barlet~\\cite{Bar78}, Aleksandrov linked Saito's construction to Grothendieck duality theory: The image of Saito's logarithmic residue map is the module of regular differential forms on $D$ (see \\cite[\\S4, Thm.]{Ale88} and \\cite{Bar78}).\nWith Tsikh, he suggested a generalization for complete intersection singularities based on multilogarithmic differential forms depending on a choice of generators of the defining ideal (see \\cite{AT01}). \nRecently, he approached the natural problem of describing the mixed Hodge structure on the complement of an LCT divisor in terms of logarithmic differential forms (see \\cite{Ale12}).\nIn Dolgachev's work, one finds a different sheaf of logarithmic differential forms which is a vector bundle exactly for normal crossing divisors and whose reflexive hull is Saito's sheaf of logarithmic differential forms (see \\cite{Dol07}).\nAlthough his approach to logarithmic residues using adjoint ideals has a similar flavor to ours, he does not reach the conclusion of our main Theorem~\\ref{10} (see Remark~\\ref{49b}).\n\n\\medskip\n\nWhile most constructions in Saito's logarithmic theory and its generalizations have a dual counterpart, a notion of a dual logarithmic residue associated to a vector field was not known to the authors.\nThe main motivation and fundamental result of this article is the construction of a dual logarithmic residue (see Section~\\ref{13}).\nThis turned out to have surprising applications including a proof of a conjecture of Saito, that was open for more than 30 years.\nSaito's conjecture is concerned with comparing logarithmic residues of $1$-forms, that is, certain meromorphic functions on $\\widetilde D$, with holomorphic functions on $\\widetilde D$.\nThe latter can also be considered as a weakly holomorphic function on $D$, that is, functions on the complement of the singular locus $Z$ of $D$, locally bounded near points of $Z$.\nWhile any such weakly holomorphic function is the residue of some logarithmic $1$-form, the image of the residue map can contain functions which are not weakly holomorphic.\nThe algebraic condition of equality was related by Saito to a geometric and a topological property as follows (see \\cite[(2.13)]{Sai80}).\n\n\\begin{thm}[Saito]\\label{28}\nFor a divisor $D$ in a complex manifold $S$, consider the following conditions:\n\\begin{enumerate}[(A)]\n\\item\\label{28a} The local fundamental groups of the complement $S\\backslash D$ are Abelian.\n\\item\\label{28b} In codimension $1$, that is, outside of an analytic subset of codimension at least $2$ in $D$, $D$ is normal crossing.\n\\item\\label{28c} The residue of any logarithmic $1$-form along $D$ is a weakly holomorphic function on $D$.\n\\end{enumerate}\nThen the implications \\eqref{28a} $\\Rightarrow$ \\eqref{28b} $\\Rightarrow$ \\eqref{28c} hold true.\n\\end{thm}\n\nSaito asked whether the the converse implications in Theorem~\\ref{28} hold true.\nThe first one was later established by L\\^e and Saito~\\cite{LS84}; it generalizes the Zariski conjecture for complex plane projective nodal curves proved by Fulton and Deligne (see \\cite{Ful80,Del81}).\n\n\\begin{thm}[L\\^e--Saito]\nThe implication \\eqref{28a} $\\Leftarrow$ \\eqref{28b} in Theorem~\\ref{28} holds true.\n\\end{thm}\n\nOur duality of logarithmic residues turns out to translate condition \\eqref{28c} in Theorem~\\ref{28} into the more familiar equality of the Jacobian ideal and the conductor ideal.\nA result of Ragni Piene~\\cite{Pie79} proves that such an equality forces $D$ to have only smooth components if it has a smooth normalization. \nThis is a technical key point which leads to a proof of the missing implication in Theorem~\\ref{28}.\n\n\\begin{thm}\\label{10}\nThe implication \\eqref{28b} $\\Leftarrow$ \\eqref{28c} in Theorem~\\ref{28} holds true: If the residue of any logarithmic $1$-form along $D$ is a weakly holomorphic function on $D$ then $D$ is normal crossing in codimension $1$.\n\\end{thm}\n\n\\begin{rmk}\\label{49b}\nSaito~\\cite[(2.11)]{Sai80} proved Theorem~\\ref{10} for plane curves.\nIf $D$ has holonomic codimension at least $1$ (as defined above), this yields the general case by analytic triviality along logarithmic strata (see \\cite[\\S3]{Sai80}).\nUnder this latter hypothesis, Theorem~\\ref{10} follows also from a result of Dolgachev (see \\cite[Cor.~2.2]{Dol07}).\nHowever, for example, the equation $xy(x+y)(x+yz)=0$ defines a well-known free divisor with holonomic codimension $0$.\n\\end{rmk}\n\nThe preceding results and underlying techniques serve to address two natural questions:\nThe algebraic characterization of condition~\\eqref{28c} through Theorem~\\ref{10} raises the question about the algebraic characterizations of normal crossing divisors.\nEleonore Faber was working on this question at the same time as the results presented here were developed.\nShe considered freeness as a first approximation for being normal crossing and noticed that normal crossing divisors satisfy an extraordinary condition:\nThe ideal of partial derivatives of a defining equation is radical. \nShe proved the following converse implications (see \\cite{Fab11,Fab12}).\n\n\\begin{thm}[Faber]\nConsider the following condition:\n\\begin{enumerate}[(A)]\\setcounter{enumi}{3}\n\\item\\label{28e} At any point $p\\in D$, there is a local defining equation $h$ for $D$ such that the ideal $\\mathcal{J}_h$ of partial derivatives is radical.\n\\item\\label{28f} $D$ is normal crossing.\n\\end{enumerate}\nThen the following holds:\n\\begin{asparaenum}\n\\item If $D$ is free then condition~\\eqref{28e} decends to all irreducible components of $D$.\n\\item Conditions~\\eqref{28e} and \\eqref{28f} are equivalent if $D$ is locally a plane curve or\na hyperplane arrangement, or \nif its singular locus is Gorenstein.\n\\end{asparaenum}\n\\end{thm}\n\nMotivated by Faber's problem we prove the following \n\n\\begin{thm}\\label{16}\nExtend the list of conditions in Theorem~\\ref{28} as follows:\n\\begin{enumerate}[(A)]\\setcounter{enumi}{5}\n\\item\\label{28d} The Jacobian ideal $\\mathcal{J}_D$ of $D$ is radical.\n\\item\\label{28g} The Jacobian ideal $\\mathcal{J}_D$ of $D$ equals the conductor ideal $\\mathcal{C}_D$ of the normalization $\\tilde D$.\n\\end{enumerate}\nThen condition~\\eqref{28d} implies condition~\\eqref{28b}.\nIf $D$ is a free divisor then conditions~\\eqref{28b}, \\eqref{28d} and \\eqref{28g} are equivalent.\n\\end{thm}\n\n\\begin{rmk}\\label{9}\nNote that $\\mathcal{J}_h$ is an $\\O_S$-ideal sheaf depending on a choice of local defining equation whereas its image $\\mathcal{J}_D$ in $\\O_D$ is intrinsic to $D$.\nIn particular, condition~\\eqref{28e} implies condition~\\eqref{28d}.\n\\end{rmk}\n\nWe obtain the following algebraic characterization of normal crossing divisors.\n\n\\begin{thm}\\label{38}\nFor a free divisor with smooth normalization, any one of the conditions~\\eqref{28a}, \\eqref{28b}, \\eqref{28c}, \\eqref{28d}, or \\eqref{28g} implies condition \\eqref{28f}.\n\\end{thm}\n\n\\begin{rmk}\nThe implication \\eqref{28d} $\\Rightarrow$ \\eqref{28f} in Theorem~\\ref{38} improves Theorem A in \\cite{Fab12} (see Remark~\\ref{9}), which is proved using \\cite{Pie79} like in the proof of our main result.\nProposition~C in \\cite{Fab12} is the implication \\eqref{28c} $\\Rightarrow$ \\eqref{28f} in Theorem~\\ref{38}, for the proof of which Faber uses our arguments.\n\\end{rmk}\n\nAs remarked above, free divisors are characterized by their singular loci being (empty or) maximal Cohen--Macaulay.\nIt is natural to ask when the singular locus of a divisor is Gorenstein.\nThis question is answered by the following\n\n\\begin{thm}\\label{40}\nA divisor $D$ has Gorenstein singular locus $Z$ of codimension $1$ if and only if $D$ is locally the product of a quasihomogeneous plane curve and a smooth space.\nIn particular, $D$ is locally quasihomogeneous and $Z$ is locally a complete intersection.\n\\end{thm}\n\n\\begin{rmk}\nTheorem~\\ref{40} complements a result of Kunz--Waldi \\cite[Satz~2]{KW84} saying that a Gorenstein algebroid curve has Gorenstein singular locus if and only if it is quasihomogeneous.\n\\end{rmk}\n\n\\section{Logarithmic modules and fractional ideals}\\label{30}\n\nIn this section, we review Saito's logarithmic modules, the relation of freeness and Cohen--Macaulayness of the Jacobian ideal, and the duality of maximal Cohen--Macaulay fractional ideals.\nWe switch to a local setup for the remainder of the article.\n \nLet $D$ be a reduced effective divisor defined by $\\mathcal{I}_D=\\O_S\\cdot h$ in the smooth complex analytic space germ $S=(\\mathds{C}^n,0)$.\nDenote by $h\\colon S\\to T=(\\mathds{C},0)$ a function germ generating the ideal $\\mathcal{I}_D=\\O_S\\cdot h$ of $D$.\nWe abbreviate by\n\\[\n\\Theta_S:=\\Der_\\mathds{C}(\\O_S)=\\Hom_{\\O_S}(\\Omega^1_S,\\O_S)\n\\]\nthe $\\O_S$-module of vector fields on $S$.\nRecall Saito's definition \\cite[\\S1]{Sai80} of the $\\O_S$-modules of logarithmic differential forms and of logarithmic vector fields.\n\n\\begin{dfn}[Saito]\\label{33}\n\\begin{align*}\n\\Omega^p(\\log D)&:=\\{\\omega\\in\\Omega^p_S(D)\\mid d\\omega\\in\\Omega_S^{p+1}(D)\\}\\\\\n\\Der(-\\log D)&:=\\{\\delta\\in\\Theta_S\\mid dh(\\delta)\\in\\mathcal{I}_D\\}\n\\end{align*}\n\\end{dfn}\n\nThese modules are stalks of analogously defined coherent sheaves of $\\O_S$-modules (see \\cite[(1.3),(1.5)]{Sai80}).\nIt is obvious that each of these sheaves $\\L$ is torsion free and normal, and hence reflexive (see \\cite[Prop.~1.6]{Har80}).\nMore precisely, $\\Omega^1(\\log D)$ and $\\Der(-\\log D)$ are mutually $\\O_S$-dual (see \\cite[(1.6)]{Sai80}).\nNormality of a sheaf $\\L$ means that $\\L=i_*i^*\\L$ where $i\\colon S\\setminus Z\\hookrightarrow S$ denotes the inclusion of the complement of the singular locus of $D$.\nIn case of $\\L=\\Der(-\\log D)$, this means that $\\delta\\in\\Der(-\\log D)$ if and only if $\\delta$ is tangent to $D$ at all smooth points.\nIn addition, $\\Omega^\\bullet(\\log D)$ is an exterior algebra over $\\O_S$ closed under exterior differentiation and $\\Der(-\\log D)$ is closed under the Lie bracket.\n\n\\begin{dfn}\nA divisor $D$ is called free if $\\Der(-\\log D)$ is a free $\\O_S$-module.\n\\end{dfn}\n\nThe definition of $\\Der(-\\log D)$ can be rephrased as a short exact sequence of $\\O_S$-modules\n\\begin{equation}\\label{1}\n\\SelectTips{cm}{}\\xymatrix{\n0&\\mathcal{J}_D\\ar[l]&\\Theta_S\\ar[l]_-{dh}&\\Der(-\\log D)\\ar[l]&0\\ar[l]\n}\n\\end{equation}\nwhere the Jacobian ideal $\\mathcal{J}_D$ of $D$ is defined as the Fitting ideal \n\\[\n\\mathcal{J}_D:=\\mathcal{F}^{n-1}_{\\O_D}(\\Omega_D^1)=\\ideal{\\frac{\\partial h}{\\partial x_1},\\dots,\\frac{\\partial h}{\\partial x_n}}\\subset{\\O_D}.\n\\]\nNote that $\\mathcal{J}_D$ is an ideal in $\\O_D$ and pulls back to $\\ideal{h,\\frac{\\partial h}{\\partial x_1},\\dots,\\frac{\\partial h}{\\partial x_n}}$ in $\\O_S$.\nWe shall consider the singular locus $Z$ of $D$ equipped with the structure defined by $\\mathcal{J}_D$, that is,\n\\begin{equation}\\label{21}\n\\O_Z:=\\O_D\/\\mathcal{J}_D.\n\\end{equation}\nNote that $Z$ might be non-reduced.\nThere is the following intrinsic characterization of free divisors in terms of their singular locus (see \\cite[\\S1 Thm.]{Ale88} or \\cite[Prop.~2.4]{Ter80a}).\n\n\\begin{thm}\\label{19}\nThe following are equivalent:\n\\begin{enumerate}\n\\item\\label{19a} $D$ is a free divisor.\n\\item\\label{19b} $\\mathcal{J}_D$ is a maximal Cohen--Macaulay $\\O_D$-module.\n\\item\\label{19c} $D$ is smooth or $Z$ is Cohen--Macaulay of codimension $1$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nIf $dh(\\Theta_S)$ does not minimally generate $\\mathcal{J}_D$, then $D\\cong D'\\times(\\mathds{C}^k,0)$, $k>0$, by the triviality lemma~\\cite[(3.5)]{Sai80}.\nBy replacing $D$ by $D'$, we may therefore assume that \\eqref{1} is a minimal resolution of $\\mathcal{J}_D$ as $\\O_S$-module.\nThus, the equivalence of \\eqref{19a} and \\eqref{19b} is due to the Auslander--Buchsbaum formula.\nBy Lemma~\\ref{25} below, $\\mathcal{J}_D$ has height at least $1$ and the implication~\\eqref{19b} $\\Leftrightarrow$ \\eqref{19c} is proved in \\cite[Satz~4.13]{HK71}.\n\\end{proof}\n\n\\begin{cor}\\label{27}\nAny $D$ is free in codimension $1$.\n\\end{cor}\n\n\\begin{proof}\nBy Theorem~\\ref{19}, the non-free locus of $D$ is contained in $Z$ and equals\n\\[\n\\{z\\in Z\\mid\\depth\\O_{Z,z}<n-2\\}\\subset D.\n\\]\nBy Scheja~\\cite[Satz~5]{Sch64}, this is an analytic set of codimension at least $2$ in $D$.\n\\end{proof}\n\nWe denote by $Q(-)$ the total quotient ring.\nThen $\\mathcal{M}_D:=Q(\\O_D)$ is the ring of meromorphic functions on $D$.\n\n\\begin{dfn}\nA fractional ideal (on $D$) is a finite $\\O_D$-submodule of $\\mathcal{M}_D$ which contains a non-zero divisor.\n\\end{dfn}\n\n\\begin{lem}\\label{25}\n$\\mathcal{J}_D$ is a fractional ideal.\n\\end{lem}\n\n\\begin{proof}\nBy assumption $D$ is reduced, so $\\O_D$ satisfies Serre's condition $R_0$.\nThis means that $Z\\subset D$ has codimension at least $1$.\nIn other words, $\\mathcal{J}_D\\not\\subseteq\\mathfrak{p}$ for all $\\mathfrak{p}\\in\\Ass(\\O_D)$ where the latter denotes the set of (minimal) associated primes of $\\O_D$.\nBy prime avoidance, $\\mathcal{J}_D\\not\\subseteq\\bigcup_{\\mathfrak{p}\\in\\Ass\\O_D}\\mathfrak{p}$.\nBut the latter is the set of zero divisors of $\\O_D$ and the claim follows.\n\\end{proof}\n\n\\begin{prp}\\label{18}\nThe $\\O_D$-dual of any fractional ideal $\\mathcal{I}$ is again a fractional ideal $\\mathcal{I}^\\vee=\\{f\\in\\mathcal{M}_D\\mid f\\cdot\\mathcal{I}\\subseteq\\O_D\\}$.\nThe duality functor \n\\[\n-^\\vee=\\Hom_{\\O_D}(-,\\O_D)\n\\]\nreverses inclusions. \nIt is an involution on the class of maximal Cohen--Macaulay fractional ideals.\n\\end{prp}\n\n\\begin{proof}\nSee \\cite[Prop.~(1.7)]{JS90a}.\n\\end{proof}\n\nFor lack of reference, we prove the following \n\n\\begin{lem}\\label{39}\nLet $A$ be a Gorenstein ring whose normalization $B$ is a finite $A$-module. \nThen $B$ is a reflexive $A$-module.\n\\end{lem}\n\n\\begin{proof}\nWe apply a standard characterization of reflexive modules (see \\cite[Prop.~1.4.1.(b)]{BH93}).\nTo this end let $\\mathfrak{p}\\in\\Spec A$ and let $\\mathfrak{q}\\in\\Spec B$ with $\\mathfrak{q}\\cap A=\\mathfrak{p}$.\nBy incomparability for integral extensions, $\\mathfrak{q}$ induces a maximal ideal in $B_\\mathfrak{p}$ and a minimal prime ideal in $B_\\mathfrak{p}\/\\mathfrak{p} B_\\mathfrak{p}$.\nIn particular, $B_\\mathfrak{p}$ has finitely many maximal ideals, the localization at one of them being $B_\\mathfrak{q}$.\nBy the Chinese remainder theorem, $B_\\mathfrak{q}\/\\mathfrak{p} B_\\mathfrak{q}$ is a direct factor of $B_\\mathfrak{p}\/\\mathfrak{p} B_\\mathfrak{p}$, and hence $B_\\mathfrak{q}$ is a finite $A_\\mathfrak{p}$-module by Nakayama's lemma.\nBy incomparability and going up for integral extensions, $\\dim B_\\mathfrak{q}=\\dim A_\\mathfrak{p}$.\nBy hypothesis, $B$ satisfies Serre's conditions $R_1$ and $S_2$.\n\nAfter these preparations, two cases need to be considered (see loc.~cit.):\n\\begin{asparaenum}\n\n\\item If $\\depth A_\\mathfrak{p}\\ge2$ then $\\dim B_\\mathfrak{q}=\\dim A_\\mathfrak{p}\\ge 2$.\nHence, $\\depth_{A_\\mathfrak{p}}B_\\mathfrak{q}=\\depth_{B_\\mathfrak{q}} B_\\mathfrak{q}\\ge 2$ by $S_2$ for $B$ and invariance of depth under finite extensions (see \\cite[IV-18, Prop.~12]{Ser65}).\nBut then also $\\depth_{A_\\mathfrak{p}}B_\\mathfrak{p}\\ge2$.\n\n\\item If $\\depth A_\\mathfrak{p}\\le1$ then also $\\dim A_\\mathfrak{p}\\le 1$ by $S_2$ for $A$.\nThus, $\\dim B_\\mathfrak{q}\\le 1$ and $B_\\mathfrak{q}$ is regular by $R_1$ for $B$.\nIn particular, $B_\\mathfrak{q}$, and hence $B_\\mathfrak{p}$, is a maximal Cohen--Macaulay ring.\nBy invariance of Cohen--Macaulayness under finite extensions (see \\cite[IV-18, Prop.~11]{Ser65})\\footnote{The proof in loc.~cit.~applies verbatim to the case where $B$ is semilocal.}, $B_\\mathfrak{p}$ is then also a maximal Cohen--Macaulay $A_\\mathfrak{p}$-module.\nAs $A_\\mathfrak{p}$ is Gorenstein, this implies that $B_\\mathfrak{p}$ is reflexive.\n\n\\end{asparaenum}\n\\end{proof}\n\n\\section{Logarithmic residues and duality}\\label{13}\n\nIn this section, we develop the dual picture of Saito's residue map and apply it to find inclusion relations of certain natural fractional ideals and their duals.\n\nLet $\\pi\\colon\\widetilde D\\to D$ denote the normalization of $D$.\nThen $\\mathcal{M}_D=\\mathcal{M}_{\\widetilde D}:=Q(\\O_{\\widetilde D})$ and $\\O_{\\widetilde D}$ is the ring of weakly holomorphic functions on $D$ (see \\cite[Exc.~4.4.16.(3), Thm.~4.4.15]{JP00}).\nLet\n\\[\n\\SelectTips{cm}{}\\xymatrix{\n\\Omega^p(\\log D)\\ar[r]^-{\\rho^p_D}&\\Omega_D^{p-1}\\otimes_{\\O_D}\\mathcal{M}_D\n}\n\\]\nbe Saito's residue map \\cite[\\S2]{Sai80} which is defined as follows:\nBy \\cite[(1.1)]{Sai80}, any $\\omega\\in\\Omega^p(\\log D)$ can be written as\n\\begin{equation}\\label{4}\n\\omega=\\frac{dh}{h}\\wedge\\frac{\\xi}{g}+\\frac{\\eta}{g},\n\\end{equation}\nfor some $\\xi\\in\\Omega_S^{p-1}$, $\\eta\\in\\Omega^p_S$, and $g\\in\\O_S$, which restricts to a non-zero divisor in $\\O_D$.\nThen \n\\begin{equation}\\label{34}\n\\rho_D^p(\\omega):=\\frac{\\xi}{g}\\vert_D\n\\end{equation}\nis well defined by \\cite[(2.4)]{Sai80}.\nWe shall abbreviate $\\rho_D:=\\rho_D^1$ and denote its image by \n\\[\n\\mathcal{R}_D:=\\rho_D(\\Omega^1(\\log D)).\n\\]\nUsing this notation, condition~\\eqref{28c} in Theorem~\\ref{28}, that the residue of any $\\omega\\in\\Omega^1(\\log D)$ is weakly holomorphic, can be written as $\\O_{\\widetilde D}=\\mathcal{R}_D$.\n\nThe following result of Saito~\\cite[(2.9)]{Sai80} can be considered as a kind of approximation of our main result Theorem~\\ref{10}; in fact, it shall be used in its proof.\nCombined with his freeness criterion~\\cite[(1.8)]{Sai80} it yields Faber's characterization of normal crossing divisors in Proposition~B of \\cite{Fab12}.\n\n\\begin{thm}\\label{60}\nLet $D_i=\\{h_i=0\\}$, $i=1,\\dots,k$, denote irreducible components of $D$.\nThen the following conditions are equivalent:\n\\begin{enumerate}\n\\item\\label{60a} $\\Omega^1(\\log D)=\\sum_{i=1}^k\\O_S\\frac{dh_i}{h_i}+\\Omega^1_S$.\n\\item\\label{60b} $\\Omega^1(\\log D)$ is generated by closed forms.\n\\item\\label{60d} The $D_i$ are normal and intersect transversally in codimension $1$.\n\\item\\label{60c} $\\mathcal{R}_D=\\bigoplus_{i=1}^k\\O_{D_i}$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{exa}\nThe Whitney umbrella (which is not a free divisor) satisfies the conditions in Theorem~\\ref{28}, but not those in Theorem~\\ref{60}. \n\\end{exa}\n\nThe implication~\\eqref{60d} $\\Leftarrow$ \\eqref{60c} in Theorem~\\ref{60} will be used in the proof of Theorem~\\ref{10} after a reduction to nearby germs with smooth irreducible components. \nIts proof essentially relies on parts \\eqref{48b} and \\eqref{48c} of the following \n\n\\begin{exa}\\label{48}\\\n\\begin{asparaenum}\n\n\\item\\label{48a} Let $D=\\{xy=0\\}$ be a normal crossing of $2$ irreducible components.\nThen $\\frac{dx}{x}\\in\\Omega^1(\\log D)$ and \n\\[\n(x+y)\\frac{dx}{x}=y\\frac{d(xy)}{xy}+dx-dy\n\\]\nshows that\n\\[\n\\rho_D\\left(\\frac{dx}{x}\\right)=\\frac{y}{x+y}\\Big\\vert_D.\n\\]\nOn the components $D_1=\\{x=0\\}$ and $D_2=\\{y=0\\}$ of the normalization $\\widetilde D=D_1\\coprod D_2$, this function equals $1$ and $0$ respectively and is therefore not in $\\O_D$.\nBy symmetry, this yields\n\\[\n\\mathcal{R}_D=\\O_{\\widetilde D}=\\O_{D_1}\\times\\O_{D_2}=\\mathcal{J}_D^\\vee\n\\]\nsince $\\mathcal{J}_D=\\ideal{x,y}_{\\O_D}$ is the maximal ideal in $\\O_D=\\mathds{C}\\{x,y\\}\/\\ideal{xy}$.\nThis observation will be generalized in Proposition~\\ref{22}.\n\n\\item\\label{48b} Conversely, assume that $D_1=\\{h_1=x=0\\}$ and $D_2=\\{h_2=x+y^m=0\\}$ are two smooth irreducible components of $D$.\nConsider the logarithmic $1$-form\n\\[\n\\omega=\\frac{ydx-mxdy}{x(x+y^m)}=y^{1-m}\\left(\\frac{dh_1}{h_1}-\\frac{dh_2}{h_2}\\right)\\in\\Omega^1(\\log(D_1+D_2))\\subset\\Omega^1(\\log D).\n\\]\nIts residue $\\rho_D(\\omega)\\vert_{D_1}=y^{1-m}\\vert_{D_1}$ has a pole along $D_1\\cap D_2$ unless $m=1$.\nThus, if $\\O_{\\widetilde D}=\\mathcal{R}_D$ then $D_1$ and $D_2$ must intersect transversally.\n\n\\item\\label{48c} Assume that $D$ contains $D'=D_1\\cup D_2\\cup D_3$ with $D_1=\\{x=0\\}$, $D_2=\\{y=0\\}$, and $D_3=\\{x-y=0\\}$.\nConsider the logarithmic $1$-form\n\\[\n\\omega=\\frac{1}{x-y}\\left(\\frac{dx}{x}-\\frac{dy}{y}\\right)\\in\\Omega^1(\\log D')\\subset\\Omega^1(\\log D).\n\\]\nIts residue $\\rho_D(\\omega)\\vert_{D_1}=-\\frac{1}{y}\\vert_{D_1}$ has a pole along $D_1\\cap D_2\\cap D_3$ and hence $\\O_{\\widetilde D}\\subsetneq\\mathcal{R}_D$.\n\n\\end{asparaenum}\n\\end{exa}\n\n\\medskip\n\nAfter these preparations, we shall now approach the construction of the dual logarithmic residue.\nBy definition, there is a short exact residue sequence\n\\begin{equation}\\label{2}\n\\SelectTips{cm}{}\\xymatrix{\n0\\ar[r]&\\Omega^1_S\\ar[r]&\\Omega^1(\\log D)\\ar[r]^-{\\rho_D}&\\mathcal{R}_D\\ar[r]&0.\n}\n\\end{equation}\nApplying $\\Hom_{\\O_S}(-,\\O_S)$ to \\eqref{2} gives an exact sequence\n\\begin{equation}\\label{44}\n\\SelectTips{cm}{}\\xymatrix@C-1em{\n0&\\Ext^1_{\\O_S}(\\Omega^1(\\log D),\\O_S)\\ar[l]&\\Ext^1_{\\O_S}(\\mathcal{R}_D,\\O_S)\\ar[l]&\\Theta_S\\ar[l]&\\Der(-\\log D)\\ar[l]&0\\ar[l].\n}\n\\end{equation}\nThe right end of this sequence extends to the short exact sequence \\eqref{1}.\nFor the hypersurface ring $\\O_D$, the change of rings spectral sequence\n\\begin{equation}\\label{41}\nE^{p,q}_2=\\Ext_{\\O_D}^p(-,\\Ext^q_{\\O_S}(\\O_D,\\O_S))\\underset{p}\\Rightarrow\\Ext_{\\O_S}^{p+q}(-,\\O_S)\n\\end{equation}\ndegenerates because $E^{p,q}_2=0$ if $q\\neq 1$ and hence \n\\begin{equation}\\label{43}\n\\Ext_{\\O_S}^1(-,\\O_S)\\cong E^{0,1}_2\\cong\\Hom_{\\O_D}(-,\\O_D)\\cong-^\\vee.\n\\end{equation}\nTherefore, the second term in the sequence~\\eqref{44} is $\\mathcal{R}_D^\\vee$.\nThis motivates the following key technical result of this paper, describing the dual of Saito's logarithmic residue.\n\n\\begin{prp}\\label{22}\nThere is an exact sequence\n\\begin{equation}\\label{36}\n\\SelectTips{cm}{}\\xymatrix{\n0&\\Ext^1_{\\O_S}(\\Omega^1(\\log D),\\O_S)\\ar[l]&\\mathcal{R}_D^\\vee\\ar[l]&\\Theta_S\\ar[l]_-{\\sigma_D}&\\Der(-\\log D)\\ar[l]&0\\ar[l]\n}\n\\end{equation}\nsuch that $\\sigma_D(\\delta)(\\rho_D(\\omega))=dh(\\delta)\\cdot\\rho_D(\\omega)$.\nIn particular, $\\sigma_D(\\Theta_S)=\\mathcal{J}_D$ as fractional ideals.\nMoreover, $\\mathcal{J}_D^\\vee=\\mathcal{R}_D$ as fractional ideals.\n\\end{prp}\n\n\\begin{proof}\nThe spectral sequence \\eqref{41} applied to $\\mathcal{R}_D$ is associated with\n\\[\n\\RHom_{\\O_S}(\\Omega_S^1\\hookrightarrow\\Omega^1(\\log D),h\\colon\\O_S\\to\\O_S).\n\\]\nExpanding the double complex $\\Hom_{\\O_S}(\\Omega_S^1\\hookrightarrow\\Omega^1(\\log D),h\\colon\\O_S\\to\\O_S)$, we obtain the following diagram of long exact sequences:\n\\begin{equation}\\label{3}\\small\n\\SelectTips{cm}{}\\xymatrix@C-2em{\n&0\\ar[d]&0\\ar[d]\\\\\n\\Ext_{\\O_S}^1(\\mathcal{R}_D,\\O_S)\\ar[d]^-{0}&\\Hom_{\\O_S}(\\Omega^1_S,\\O_S)\\ar[l]\\ar[d]^-{h}&\\Hom_{\\O_S}(\\Omega^1(\\log D),\\O_S)\\ar[l]\\ar[d]^-{h}&0\\ar[l]\\\\\n\\Ext_{\\O_S}^1(\\mathcal{R}_D,\\O_S)&\\Hom_{\\O_S}(\\Omega^1_S,\\O_S)\\ar[l]\\ar[d]&\\Hom_{\\O_S}(\\Omega^1(\\log D),\\O_S)\\ar[l]\\ar[d]&0\\ar[l]\\ar[d]\\\\\n&\\Hom_{\\O_S}(\\Omega^1_S,\\O_D)\\ar[d]&\\Hom_{\\O_S}(\\Omega^1(\\log D),\\O_D)\\ar[l]\\ar[d]&\\mathcal{R}_D^\\vee\\ar[l]_-{\\rho_D^\\vee}\\ar[d]^-\\alpha&0\\ar[l]\\\\\n&0&\\Ext^1_{\\O_S}(\\Omega^1(\\log D),\\O_S)\\ar[l]&\\Ext^1_{\\O_S}(\\mathcal{R}_D,\\O_S)\\ar[l]\\ar[d]\\\\\n&&&0\n}\n\\end{equation}\nWe can define a homomorphism $\\sigma_D$ from the upper left $\\Hom_{\\O_S}(\\Omega^1_S,\\O_S)$ to the lower right $\\mathcal{R}_D^\\vee$ by a diagram chasing process and we find that $\\delta\\in\\Theta_S=\\Hom_{\\O_S}(\\Omega^1_S,\\O_S)$ maps to \n\\[\n\\sigma_D(\\delta)=\\ideal{h\\delta,\\rho_D^{-1}(-)}\\vert_D\\in\\mathcal{R}_D^\\vee\n\\]\nand that \\eqref{36} is exact.\n\nBy comparison with the spectral sequence, one can check that $\\alpha$ is the change of rings isomorphism \\eqref{43} applied to $\\mathcal{R}_D$, and that $\\alpha\\circ\\sigma_D$ coincides with the connecting homomorphism of the top row of the diagram, which is the same as the one in \\eqref{44}.\n \nLet $\\rho_D(\\omega)\\in\\mathcal{R}_D$ where $\\omega\\in\\Omega^1(\\log D)$.\nFollowing the definition of $\\rho_D$ in \\eqref{34}, we write $\\omega$ in the form \\eqref{4}.\nThen we compute\n\\begin{align}\n\\nonumber\\sigma_D(\\delta)(\\rho_D(\\omega))&=\\\\\\label{5}\n\\ideal{h\\delta,\\omega}\\vert_D&=\ndh(\\delta)\\cdot\\frac{\\xi}{g}\\vert_D+h\\cdot\\frac{\\ideal{\\delta,\\eta}}{g}\\vert_D=\ndh(\\delta)\\cdot\\rho_D(\\omega)\n\\end{align}\nwhich proves the first two claims.\n\nFor the last claim, we consider the diagram dual to \\eqref{3}:\n\\[\\small\n\\SelectTips{cm}{}\\xymatrix@C-2em{\n&&0\\ar[d]&0\\ar[d]\\\\\n&0\\ar[r]&\\Hom_{\\O_S}(\\Theta_S,\\O_S)\\ar[r]\\ar[d]^-{h}&\\Hom_{\\O_S}(\\Der(-\\log D),\\O_S)\\ar[r]\\ar[d]^-{h}&\\Ext_{\\O_S}^1(\\mathcal{J}_D,\\O_S)\\ar[d]^-{0}\\\\\n&0\\ar[r]&\\Hom_{\\O_S}(\\Theta_S,\\O_S)\\ar[r]\\ar[d]&\\Hom_{\\O_S}(\\Der(-\\log D),\\O_S)\\ar[r]\\ar[d]&\\Ext_{\\O_S}^1(\\mathcal{J}_D,\\O_S)\\\\\n0\\ar[r]&\\mathcal{J}_D^\\vee\\ar[d]^\\beta\\ar[r]^-{dh^\\vee}&\\Hom_{\\O_S}(\\Theta_S,\\O_D)\\ar[r]\\ar[d]&\\Hom_{\\O_S}(\\Der(-\\log D),\\O_D)\\\\\n&\\Ext_{\\O_S}^1(\\mathcal{J}_D,\\O_S)\\ar[r]&0\n}\n\\]\nAs before, we construct a homomorphism $\\rho'_D$ from the upper right $\\Hom_{\\O_S}(\\Der(-\\log D),\\O_S)$ to the lower left $\\mathcal{J}_D^\\vee$ such that $\\beta\\circ\\rho'_D$ coincides with the connecting homomorphism of the top row of the diagram, where $\\beta$ is the change of rings isomorphism \\eqref{43} applied to $\\mathcal{J}_D$.\nBy the diagram, $\\omega\\in\\Omega^1(\\log D)=\\Hom_{\\O_S}(\\Der(-\\log D),\\O_S)$ maps to\n\\[\n\\rho'_D(\\omega)=\\ideal{h\\omega,dh^{-1}(-)}\\vert_D\\in \\mathcal{J}_D^\\vee\n\\]\nwhich gives a short exact sequence\n\\begin{equation}\\label{6}\n\\SelectTips{cm}{}\\xymatrix{\n0\\ar[r]&\\Omega^1_S\\ar[r]&\\Omega^1(\\log D)\\ar[r]^-{\\rho'_D}&\\mathcal{J}_D^\\vee\\ar[r]&0\n}\n\\end{equation}\nsimilar to the sequence \\eqref{2}.\nUsing \\eqref{4} and \\eqref{5}, we compute \n\\[\n\\rho'_D(\\omega)(\\delta(h))=\n\\rho'_D(\\omega)(dh(\\delta))=\n\\ideal{h\\omega,\\delta}\\vert_D=\n\\rho_D(\\omega)\\cdot dh(\\delta)=\n\\rho_D(\\omega)\\cdot \\delta(h)\n\\]\nfor any $\\delta(h)\\in \\mathcal{J}_D$ where $\\delta\\in\\Theta_S$.\nHence, $\\rho'_D=\\rho_D$ and the last claim follows using \\eqref{2} and \\eqref{6}.\n\\end{proof}\n\n\\begin{cor}\\label{37}\nThere is a chain of fractional ideals\n\\[\n\\mathcal{J}_D\\subseteq\\mathcal{R}_D^\\vee\\subseteq\\mathcal{C}_D\\subseteq\\O_D\\subseteq\\O_{\\widetilde D}\\subseteq\\mathcal{R}_D\n\\]\nin $\\mathcal{M}_D$ where $\\mathcal{C}_D=\\O_{\\widetilde D}^\\vee$ is the conductor ideal of $\\pi$.\nIn particular, $\\mathcal{J}_D\\subseteq\\mathcal{C}_D$.\n\\end{cor}\n\n\\begin{proof}\nBy Lemma~\\ref{25}, $\\mathcal{J}_D$ is a fractional ideal contained in $\\mathcal{R}_D^\\vee$ by Proposition~\\ref{22}.\nBy \\cite[(2.7),(2.8)]{Sai80}, $\\mathcal{R}_D$ is a finite $\\O_D$-module containing $\\O_{\\widetilde D}$ and hence a fractional ideal.\nThe remaining inclusions and fractional ideals are then obtained using Proposition~\\ref{18}.\n\\end{proof}\n\n\\begin{cor}\\label{24}\nIf $D$ is free then $\\mathcal{J}_D=\\mathcal{R}_D^\\vee$ as fractional ideals.\n\\end{cor}\n\n\\begin{proof}\nIf $D$ is free then the Ext-module in the exact sequence \\eqref{36} disappears and $\\sigma_D$ becomes surjective.\nThen the claim is part of the statement of Proposition~\\ref{22}.\n\\end{proof}\n\nBy Corollary~\\ref{37}, the inclusion $\\O_{\\widetilde D}\\subset R_D$ always holds.\nFor a free divisor $D$, the case of equality is translated into more familiar terms by the following\n\n\\begin{cor}\\label{42}\nIf $D$ is free then $\\mathcal{R}_D=\\O_{\\widetilde D}$ is equivalent to $\\mathcal{J}_D=\\mathcal{C}_D$.\n\\end{cor}\n\n\\begin{proof}\nBy the preceding Corollary~\\ref{24} and the last statement of Proposition~\\ref{22}, the freeness of $D$ implies that $\\mathcal{R}_D$ and $\\mathcal{J}_D$ are mutually $\\O_D$-dual.\nBy Lemma~\\ref{39} and the definition of the conductor, the same holds true for $\\O_{\\widetilde D}$ and $\\mathcal{C}_D$. \nThe claim follows.\n\\end{proof}\n\n\\section{Algebraic normal crossing conditions}\n\nIn this section, we prove our main Theorem~\\ref{10} settling the missing implication in Theorem~\\ref{28}.\nWe begin with some general preparations.\n\n\\begin{lem}\\label{35}\nAny map $\\phi\\colon Y\\to X$ of analytic germs with $\\Omega^1_{Y\/X}=0$ is an immersion.\n\\end{lem}\n\n\\begin{proof}\nThe map $\\phi$ can be embedded in a map $\\Phi$ of smooth analytic germs:\n\\[\n\\SelectTips{cm}{}\\xymatrix{\nY\\ar@{^(->}[r]\\ar[d]^-\\phi&T\\ar[d]^-\\Phi\\\\\nX\\ar@{^(->}[r]&S.\n}\n\\]\nSetting $\\Phi_i=x_i\\circ\\Phi$ and $\\phi_i=\\Phi_i+\\mathcal{I}_Y$ for coordinates $x_1,\\dots,x_n$ on $S$ and $\\mathcal{I}_Y$ the defining ideal of $Y$ in $T$, we can write $\\Phi=(\\Phi_1,\\dots,\\Phi_n)$ and $\\phi=(\\phi_1,\\dots,\\phi_n)$ and hence\n\\begin{equation}\\label{26}\n\\Omega^1_{Y\/X}=\\frac{\\Omega^1_Y}{\\sum_{i=1}^n\\O_Yd\\phi_i}=\\frac{\\Omega^1_T}{\\O_Td\\mathcal{I}_Y+\\sum_{i=1}^n\\O_Td\\Phi_i}.\n\\end{equation}\nWe may choose $T$ of minimal dimension so that $\\mathcal{I}_Y\\subseteq\\mathfrak{m}_T^2$ and hence $d\\mathcal{I}_Y\\subseteq\\mathfrak{m}_T\\Omega^1_T$.\nNow \\eqref{26} and the hypothesis $\\Omega^1_{Y\/X}=0$ show that $\\Omega^1_T=\\sum_{i=1}^n\\O_Td\\Phi_i+\\mathfrak{m}_T\\Omega^1_T$ which implies that $\\Omega^1_T=\\sum_{i=1}^n\\O_Td\\Phi_i$ by Nakayama's Lemma.\nBut then $\\Phi$ and hence $\\phi$ is a closed embedding as claimed.\n\\end{proof}\n\n\\begin{lem}\\label{7}\nIf $\\mathcal{J}_D=\\mathcal{C}_D$ and $\\widetilde D$ is smooth then $D$ has smooth irreducible components.\n\\end{lem}\n\n\\begin{proof}\nBy definition, the ramification ideal of $\\pi$ is the Fitting ideal\n\\[\n\\mathcal{R}_\\pi:=\\mathcal{F}^0_{\\O_{\\widetilde D}}(\\Omega^1_{\\widetilde D\/D}).\n\\]\nAs a special case of a result of Ragnie Piene~\\cite[Cor.~1, Prop.~1]{Pie79} (see also \\cite[Cor.~2.7]{OZ87}), \n\\[\n\\mathcal{C}_D\\mathcal{R}_\\pi=\\mathcal{J}_D\\O_{\\widetilde D}\n\\]\nBy hypothesis, this becomes\n\\[\n\\mathcal{C}_D\\mathcal{R}_\\pi=\\mathcal{C}_D\n\\]\nsince $\\mathcal{C}_D$ is an ideal in both $\\O_D$ and $\\O_{\\widetilde D}$.\nBy Nakayama's lemma, it follows that that $\\mathcal{R}_\\pi=\\O_{\\widetilde D}$ and hence that $\\Omega^1_{\\widetilde D\/D}=0$.\n\nSince $\\widetilde D$ is normal, irreducible and connected components coincide.\nBy localization to a connected component $\\widetilde D_i$ of $\\widetilde D$ and base change to $D_i=\\pi(\\widetilde D_i)$ (see \\cite[Ch.~II, Prop.~8.2A]{Har77}), we obtain $\\Omega^1_{\\widetilde D_i\/D_i}=0$.\nThen the normalization $\\widetilde D_i\\to D_i$ is an immersion by Lemma~\\ref{35} and hence $D_i=\\widetilde D_i$ is smooth.\n\\end{proof}\n\nWe are now ready to prove our main results.\n\n\\begin{proof}[Proof of Theorem~\\ref{10}]\nIn codimension $1$, $D$ is free by Corollary~\\ref{27} and hence $\\mathcal{J}_D=\\mathcal{C}_D$ by Corollary~\\ref{42} and our hypothesis.\nMoreover, $\\widetilde D$ is smooth in codimension $1$ by normality. \nBy our language convention, this means that there is an analytic subset $A\\subset D$ of codimension at least $2$ such that, for $p\\in D\\setminus A$, $\\mathcal{J}_{D,p}=\\mathcal{C}_{D,p}$ and $\\widetilde D$ is smooth above $p$.\nFrom Lemma~\\ref{7} we conclude that the local irreducible components $D_i$ of the germ $(D,p)$ are smooth. \nThe hypothesis $\\mathcal{R}_D=\\O_{\\widetilde D}$ at $p$ then reduces to the equality $\\mathcal{R}_{D,p}=\\bigoplus\\O_{D_i}$.\nThus, the implication \\eqref{60d} $\\Leftarrow$ \\eqref{60c} in Theorem~\\ref{60} yields the claim. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{16}]\nIn order to prove that \\eqref{28d} implies \\eqref{28b}, we may assume that $Z$ is smooth and reduce to the case of a plane curve as in the proof of Theorem~\\ref{40}. \nThen the Mather--Yau theorem~\\cite{MY82} applies (see \\cite[Prop.~9]{Fab12} for details).\n\nNow assume that $D$ is free and normal crossing in codimension $1$. \nBy the first assumption and Theorem~\\ref{19}, $Z$ is Cohen--Macaulay of codimension $1$ and, in particular, satisfies Serre's condition $S_1$.\nBy the second assumption, $Z$ also satisfies Serre's condition $R_0$.\nThen $Z$ is reduced, and hence $\\mathcal{J}_D$ is radical, by Serre's reducedness criterion.\nThis proves that \\eqref{28b} implies \\eqref{28d} for free $D$.\n\nThe last equivalence then follows from Theorems~\\ref{28} and \\ref{10} and Corollary~\\ref{42}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{38}]\nBy Theorems~\\ref{28}, \\ref{10}, and \\ref{16}, we may assume that $\\mathcal{J}_D=\\mathcal{C}_D$.\nThen Lemma~\\ref{7} shows that the irreducible components $D_i=\\{h_i=0\\}$, $i=1,\\dots,m$, of $D$ are smooth, and hence normal.\nIt follows that \n\\[\n\\mathcal{R}_D=\\O_{\\widetilde D}=\\bigoplus_{i=1}^m\\O_{\\widetilde D_i}=\\bigoplus_{i=1}^m\\O_{D_i}.\n\\]\nBy the implication \\eqref{60a} $\\Leftarrow$ \\eqref{60c} in Theorem~\\ref{60}, this is equivalent to \n\\begin{equation}\\label{51}\n\\Omega^1(\\log D)=\\sum_{i=1}^m\\O_S\\frac{dh_i}{h_i}+\\Omega^1_S.\n\\end{equation}\nOn the other hand, Saito's criterion \\cite[(1.8) i)]{Sai80} for freeness of $D$ reads\n\\begin{equation}\\label{52}\n\\bigwedge^n\\Omega^1(\\log D)=\\Omega^n_S(D).\n\\end{equation}\nCombining \\eqref{51} and \\eqref{52}, it follows immediately that $D$ is normal crossing (see also \\cite[Prop.~B]{Fab12}):\n\nAs $\\O_S$-module and modulo $\\Omega_S^n$, the left hand side of \\eqref{52} is, due to \\eqref{51}, generated by expressions \n\\begin{gather}\\label{53}\n\\frac{d h_{i_1}\\wedge\\dots\\wedge d h_{i_k}\\wedge dx_{j_1}\\wedge\\dots\\wedge dx_{j_{n-k}}}{h_{i_1}\\cdots h_{i_k}},\\\\ \n\\nonumber1\\le i_1<\\cdots<i_k\\le m, \\quad 1\\le j_1<\\cdots<j_{n-k}\\le n\\ge k\n\\end{gather}\nwhereas the right hand side of \\eqref{52} is generated by\n\\[\n\\frac{d x_1\\wedge\\dots\\wedge d x_n}{h_1\\cdots h_m}.\n\\]\nIn order for an instance of \\eqref{53} to attain the denominator of this latter expression, it must satisfy $k=m$, and, in particular, $m\\le n$.\nFurther, comparing numerators, $d h_1\\wedge\\dots\\wedge d h_m\\wedge dx_{j_1}\\wedge\\dots\\wedge dx_{j_{n-m}}$ must be a unit multiple of $d x_1\\wedge\\dots\\wedge d x_n$.\nIn other words, choosing $i_1,\\dots,i_m$ such that $\\{i_1,\\dots,i_m,j_1,\\dots,j_{n-m}\\}=\\{1,\\dots,n\\}$, \n\\[\n\\frac{\\partial(h_1,\\dots,h_m)}{\\partial(x_{i_1},\\dots,x_{i_m})}\\in\\O_S^*.\n\\]\nBy the implicit function theorem, $h_1,\\dots,h_m,x_{j_1},\\dots,x_{j_{n-m}}$ is then a coordinate system and hence $D$ is a normal crossing divisor as claimed.\n\\end{proof}\n\n\\section{Gorenstein singular locus}\n\nIn this section, we describe the canonical module of the singular locus $Z$ in terms of the module $\\mathcal{R}_D$ of logarithmic residues.\nThen, we prove Theorem~\\ref{40}.\n\nThe complex of logarithmic differential forms along $D$ relative to the map $h\\colon S\\to T:=(\\mathds{C},0)$ defining $D$ is defined as $\\Omega^\\bullet(\\log h):=\\Omega^\\bullet(\\log D)\/\\frac{dh}h\\wedge\\Omega^{\\bullet-1}(\\log D)$ (see \\cite[\\S22]{GMS09} or \\cite[Def.~2.7]{DS12}).\n\n\\begin{prp}\\label{20}\nThe $\\O_Z$-module $\\mathcal{R}_h:=\\mathcal{R}_D\/\\O_D$ fits into an exact square\n\\[\n\\SelectTips{cm}{}\\xymatrix{\n&0\\ar[d]&0\\ar[d]&0\\ar[d]\\\\\n0\\ar[r]&\\O_S\\ar[r]^-h\\ar[d]^-{dh}&\\O_S\\ar[r]\\ar[d]^-{\\frac{dh}{h}}&\\O_D\\ar[r]\\ar[d]&0\\\\\n0\\ar[r]&\\Omega^1_S\\ar[r]\\ar[d]&\\Omega^1(\\log D)\\ar[r]^-{\\rho_D}\\ar[d]&\\mathcal{R}_D\\ar[r]\\ar[d]&0\\\\\n0\\ar[r]&\\Omega^1_{S\/T}\\ar[r]\\ar[d]&\\Omega^1(\\log h)\\ar[r]\\ar[d]&\\mathcal{R}_h\\ar[r]\\ar[d]&0\\\\\n&0&0&0\n}.\n\\]\nIf $D$ is free then $Z$ is Cohen--Macaulay with canonical module $\\omega_Z=\\mathcal{R}_h$; in particular, $Z$ is Gorenstein if and only if $\\mathcal{R}_D$ is generated by $1$ and one additional generator.\n\\end{prp}\n\n\\begin{proof}\nWe set $\\omega_\\emptyset:=0$ in case $D$ is smooth and assume $Z\\ne\\emptyset$ in the following.\nThe exact square arises from the residue sequence \\eqref{2} using the Snake Lemma.\nDualizing the short exact sequence\n\\[\n\\SelectTips{cm}{}\\xymatrix{\n0\\ar[r]&\\mathcal{J}_D\\ar[r]&\\O_D\\ar[r]&\\O_Z\\ar[r]&0,\n}\n\\]\none computes\n\\begin{equation}\\label{29}\n\\Ext_{\\O_D}^1(\\O_Z,\\O_D)=\\mathcal{J}_D^\\vee\/\\O_D=\\mathcal{R}_D\/\\O_D=\\mathcal{R}_h.\n\\end{equation}\nwhich shows that $\\mathcal{R}_h$ is an $\\O_Z$-module.\nIf $D$ is free then, by Theorem~\\ref{19}, $Z$ is Cohen--Macaulay of codimension $1$ and $\\omega_Z=\\mathcal{R}_h$ by \\eqref{29}.\n\\end{proof}\n\nPart~\\eqref{56e} of the following proposition can also be found in Faber's thesis (see \\cite[Prop.~1.29]{Fab11}).\n\n\\begin{prp}\\label{56}\nLet $D$ be a free divisor.  \nThen the following statements hold true:\n\\begin{asparaenum} \n\\item\\label{56e} $\\frac{dh}h$ is part of an $\\O_S$-basis of $\\Omega^1(\\log D)$ if and only if $D$ is Euler homogeneous.\n\\item\\label{56d} $\\frac{dh}h$ is part of an $\\O_S$-basis of $\\Omega^1(\\log D)$ if and only if $1$ is part of a minimal set of $\\O_D$-generators of $\\mathcal{R}_D$.\n\\item\\label{56g} $\\mathcal{R}_D$ is a cyclic $\\O_D$-module if and only if $D$ is smooth.\n\\end{asparaenum}\n\\end{prp}\n\n\\begin{proof}\\\n\\begin{asparaenum} \n\n\\item This is immediate from the existence of a dual basis and the fact that $\\chi\\in\\Der(-\\log D)$ is an Euler vector field exactly if $\\ideal{\\chi,\\frac{dh}h}=\\frac{\\chi(h)}h=1$. \nThen $\\chi$ can be chosen as a member of some basis.\n\n\\item We may assume that $D\\cong D'\\times S'$ with $S'=(\\mathds{C}^r,0)$ implies $r=0$.\nIndeed, a basis of $\\Omega^1(\\log D)$ is the union of bases of $\\Omega^1(\\log D')$ and $\\Omega_{S'}^1$.\nThis assumption is equivalent to $\\Der(-\\log D)\\subseteq\\mathfrak{m}_S\\Theta_S$, where $\\mathfrak{m}_S$ denotes the maximal ideal of $\\O_S$. \nDually, this means that no basis element of $\\Omega^1(\\log D)$ can lie in $\\Omega_S^1$.\n\nConsider a basis $\\omega_1,\\dots,\\omega_n$ of $\\Omega^1(\\log D)$ with $\\omega_1=\\frac{dh}h$ and set $\\rho_i:=\\rho_D(\\omega_i)$ for $i=1,\\dots,n$.\nIf $\\rho_1=1$ is not a member of some minimal set of generators of $\\mathcal{R}_D$ then $\\rho_1=\\sum_{i=2}^na_i\\rho_i$ with $a_i\\in\\mathfrak{m}_S$. \nThus, the form $\\omega_1':=\\omega_1-\\sum _{i=2}^na_i\\omega_i$ can serve as a replacement for $\\omega_1$ in the basis. \nBut, by construction, $\\rho_D(\\omega'_1)=0$ which implies that $\\omega'_1\\in\\Omega_S$ by \\eqref{2}.\nThis contradicts our assumption on $D$.\n \nConversely, suppose that $\\frac{dh}h$ is not a member of any $\\O_S$-basis $\\omega_1,\\dots,\\omega_n$ of $\\Omega^1(\\log D)$.\nThen, by Nakayama's lemma, there are $a_i\\in\\mathfrak{m}_S$ such that $\\frac{dh}h=\\sum_{i=1}^na_i\\omega_i$.\nApplying $\\rho_D$, this gives $1=\\rho_D(\\frac{dh}h)=\\sum_{i=1}^na_i\\rho_i\\in\\mathfrak{m}_D\\mathcal{R}_D$.\nAgain by Nakayama's lemma, this means that $1$ is not a member of any minimal set of $\\O_D$-generators of $\\mathcal{R}_D$.\n\n\\item If $\\mathcal{R}_D\\cong\\O_D$ then also $\\mathcal{J}_D\\cong\\O_D$ by Corollary~\\ref{24} then $D$ must be smooth by Lipman's criterion~\\cite{Lip69}.\nAlternatively, it follows that $\\mathcal{J}_h+\\O_S\\cdot h=\\O_S$ and hence also $\\mathcal{J}_h=\\O_S$ (see Remark~\\ref{9}); so $D$ is smooth by the Jacobian criterion.\n\n\\end{asparaenum} \n\\end{proof}\n\nAs observed by Faber~\\cite[Rmk.~54]{Fab12}, condition~\\eqref{56d} of Proposition~\\ref{56} is satisfied given $\\mathcal{R}_D=\\O_{\\widetilde D}$ and one obtains\n\n\\begin{cor}[Faber]\nAny free divisor $D$ with $\\mathcal{R}_D=\\O_{\\widetilde D}$ is Euler homogeneous.\n\\end{cor}\n\n\\begin{proof}[Proof of Theorem~\\ref{40}]\nAssume that $Z$ is Gorenstein of codimension $1$ in $D$.\nThe preimage $\\mathcal{J}'_D$ of $\\mathcal{J}_D$ in $\\O_S$ is then a Gorenstein ideal of height $2$.\nAs such, it is a complete intersection ideal by a theorem of Serre (see \\cite[Cor.~21.20]{Eis95}), and hence generated by two of the generators $h,\\partial_1(h),\\dots,\\partial_n(h)$.\nThen the triviality lemma~\\cite[(3.5)]{Sai80} shows that $D$ is either smooth or $D\\cong C\\times(\\mathds{C}^{n-2},0)$ and $C\\subset(\\mathds{C}^2,0)$ a plane curve.\nThen also $C$ has Gorenstein singular locus and is hence quasihomogeneous by \\cite{KW84}.\nAlternatively, the last implication follows from Propositions~\\ref{20} and \\ref{56} using that quasihomogeneity of $C$ follows from Euler homogeneity of $C$ by Saito's quasihomogeneity criterion~\\cite{Sai71} for isolated singularities.\nFinally, $D$ is quasihomogeneous and $Z$ is a complete intersection.\nThe converse is trivial.\n\\end{proof}\n\n\\section*{Acknowledgements}\n\nWe are grateful to Eleonore Faber and David Mond for helpful discussions and comments.\nThe foundation for this paper was laid during a ``Research in Pairs'' stay at the ``Mathematisches Forschungsinstitut Oberwolfach'' in summer 2011.\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure}[!t]\n\\centering\n\\begin{center}\n\\begin{tabular}{C{5.0cm}C{5.0cm}}\n    \\includegraphics[height=3.8cm]{figs\/framework_img0_a} &\n    \\includegraphics[height=3.8cm]{figs\/framework_img0_b} \\\\\n    \\centering (a) FCN & (b) DilatedFCN \\\\ \\\\\n    \\includegraphics[height=3.8cm]{figs\/framework_img0_c} &\n    \\includegraphics[height=3.8cm]{figs\/framework_img0_d} \\\\\n    \\centering (c) Encoder-Decoder & (d) EfficientFCN \n\\end{tabular}\n\\end{center}\n\\caption{Different architectures for semantic segmentation. (a) the original FCN with output stride (OS)=32. (b). DilatedFCN based methods sacrifice efficiency and exploit the dilated convolution with stride 2 and 4 in the last two stages to generate high-resolution feature maps. (c)Encoder-Decoder methods employ the U-Net structure to recover the high-resolution feature maps. (d) Our proposed EfficientFCN with codebook generation and codeword assembly for high-resolution feature upsampling in semantic segmentation.}\n\\label{fig:framework_img0}\n\\end{figure}\n\nSemantic segmentation or scene parsing is the task of assigning one of the pre-defined class \nlabels to each pixel of an input image. It is a fundamental yet challenging task in computer vision.\nThe Fully Convolutional Network (FCN) \\cite{long2015fully}, as shown in\nFig.~\\ref{fig:framework_img0}(a), for the first time demonstrates the success of exploiting a fully convolutional network in semantic segmentation, which adopts a DCNN as the feature encoder (\\textit{i}.\\textit{e}., ResNet\\cite{he2016deep}) to extract high-level semantic feature maps and then applies a convolution layer to generate the dense prediction. \nFor the semantic segmentation, high-resolution feature maps are critical for achieving accurate\nsegmentation performance since they contain fine-grained structural information to delineate\ndetailed boundaries of various foreground regions. In addition, due to the lack of large-scale\ntraining data on semantic segmentation, transferring the weights pre-trained on ImageNet can greatly\nimprove the segmentation performance. Therefore, most state-of-the-art semantic segmentation methods\nadopt classification networks as the backbone to take full advantages of ImageNet pre-training. The resolution of feature maps in the original\nclassification model is reduced with consecutive pooling and strided\nconvolution operations to learn high-level\nfeature representations. The output stride of the final feature map is 32\n(OS=32), where the fine-grained structural information is discarded. \nSuch low-resolution feature maps cannot fully meet the requirements of semantic segmentation where detailed spatial information is needed. \nTo tackle this problem, many works exploit dilated convolution (or atrous convolution) to enlarge the receptive field (RF) while maintaining the resolution of high-level feature maps.\nState-of-the-art dilatedFCN based\nmethods\\cite{yu2017dilated,chen2017deeplab,Zhang_2018_CVPR,he2019adaptive,Zhang_2019_CVPR} (shown in\nFig.~\\ref{fig:framework_img0}(b)) have demonstrated that removing the downsampling\noperation and replacing convolution with the dilated convolution\nin the later blocks can achieve superior performance, resulting in \nfinal feature maps of output stride 8 (OS=8). Despite the superior performance and no extra parameters introduced by dilated convolution,  the high-resolution feature representations require high computational complexity and memory consumption. \nFor instance, for an input image with 512$\\times$512 and the ResNet101 as the backbone encoder, the computational complexity of the encoder increases from 44.6 GFlops to 223.6 GFlops when adopting the dilated convolution with the strides 2 and 4 into the last two blocks.\n\n\n\nAlternatively, as shown in Fig.~\\ref{fig:framework_img0}(c), the encoder-decoder based methods (\n\\textit{e}.\\textit{g}.\\ \\cite{ronneberger2015u}) exploit using a decoder to gradually upsample and generate the high-resolution feature maps by aggregating multi-level feature representations from the backbone (or the encoder). These encoder-decoder based methods can obtain high-resolution\nfeature representations efficiently. However, on one hand, the fine-grained structural details are already lost in the topmost high-level feature maps of OS=32. Even with the skip connections, lower-level high-resolution feature maps cannot provide abstractive enough features for achieving high-performance segmentation. \nOn the other hand, existing decoders mainly utilize the bilinear upsampling or deconvolution operations to increase the resolution of the high-level feature maps. These operations are conducted in a\nlocal manner. The feature vector at each location of the upsampled\nfeature maps is recovered from a limited receptive filed. Thus, although the encoder-decoder models are generally faster and more memory friendly than dilatedFCN based methods, their performances generally cannot compete with those of the dilatedFCN models.\n\nTo tackle the challenges in both types of models, we propose the EfficienFCN\n(as shown in Fig.~\\ref{fig:framework_img0}(d)) with the Holistically-guided Decoder (HGD) to bridge the\ngap between the dilatedFCN based methods and the encoder-decoder based methods. Our network can adopt any widely\nused classification model without dilated convolution as the encoder (such as ResNet models) to\ngenerate low-resolution high-level feature maps (OS=8). Such an encoder is both computationally and\nmemory efficient than those in DilatedFCN model.\nGiven the multi-level feature maps from the last three blocks of the encoder, the proposed holistically-guided decoder takes the advantages of both high-level but low-resolution (OS=32) and also mid-level high-resolution feature maps (OS=8, OS=16) for achieving high-level feature upsampling with semantic-rich features. Intuitively, the higher-resolution feature maps contain more fine-grained structural information, which is beneficial for spatially guiding the feature upsampling process; the lower-resolution feature maps contain more high-level semantic information, which are more suitable to encode the global context effectively. Our HGD therefore generates a series of holistic codewords in a codebook to summarize different global and high-level aspects of the input image from the low-resolution feature maps (OS=32). Those codewords can be properly assembled in a high-resolution grid to form the upsampled feature maps with rich semantic information. Following this principle, the HGD generates assembly coefficients from the mid-level high-resolution feature maps (OS=8, OS=16) to guide the linear assembly of the holistic codewords at each high-resolution spatial location to achieve feature upsampling.\nOur proposed EfficientFCN with holistically-guided decoder achieves high  segmentation accuracy on three popular public benchmarks, which demonstrate the efficiency and effectiveness of our proposed decoder.\n\nIn summary, our contributions are as follows.\n\\begin{itemize}\n    \\item We propose a novel holistically-guided decoder, which can efficiently generate the high-resolution feature maps considering holistic contexts of the input image.\n   \n\\item Because of the light weight and high performance of the proposed holistically-guided decoder, our EfficientFCN can adopt the encoder without any dilated convolution but still achieve superior performance.   \n        \n   \\item Our EfficientFCN achieves competitive (or better) results compared\n       with the state-of-the-art dilatedFCN based methods on the PASCAL\n       Context, PASCAL VOC, ADE20K datasets, with 1\/3 fewer FLOPS.\n\\end{itemize}\n\n\n\n\n\\section{Related Work}\nIn this section, we review recent FCN-based methods for semantic segmentation. Since the successful demonstration of FCN \\cite{long2015fully} on semantic segmentation, many methods were proposed to improve the performance the FCN-based methods, which mainly include two categories of methods: the dilatedFCN-based methods and the encoder-decoder architectures.\n\n\\noindent \\textbf{DilatedFCN.} \nThe Deeplab V2 \\cite{chen2017deeplab,chen2017rethinking} proposed to exploit dilated convolution in\nthe backbone to learn a high-resolution feature map, which increases the\noutput stride from 32 to 8. However, the dilated convolution in the last two layers of the backbone adds huge extra computation and leaves large\nmemory footprint. Based on the dilated convolution backbone, many works \\cite{Zhang_2019_CVPR,fu2019dual,Fu_2019_ICCV,he2019dynamic} continued to apply different strategies as the segmentation heads to acquire the context-enhanced feature maps. PSPNet \\cite{zhao2017pyramid} utilized the Spatial Pyramid Pooling (SPP) module to increase the receptive field. EncNet \\cite{Zhang_2018_CVPR} proposed an encoding layer to predict a feature re-weighting vector from the global context and selectively high-lights \nclass-dependent feature maps. \nCFNet \\cite{Zhang_2019_CVPR} exploited an aggregated co-occurrent feature (ACF) module to\naggregate the co-occurrent context by the pair-wise similarities in the feature space.\nGated-SCNN\\cite{takikawa2019gated} proposed to use a new gating mechanism to connect the intermediate layers and a new loss function that exploits the duality\nbetween the tasks of semantic segmentation and semantic\nboundary prediction.\nDANet \\cite{fu2019dual} proposed to use two attention modules with the self-attention mechanism to aggregate features from\nspatial and channel dimensions respectively.\nACNet \\cite{Fu_2019_ICCV} applied a dilated ResNet as the backbone and combined the encoder-decoder strategy for the \nobservation that the global context from high-level features\nhelps the categorization of some large semantic confused\nregions, while the local context from lower-level visual features helps to generate sharp boundaries or clear details.\nDMNet \\cite{he2019dynamic} generated a set of dynamic filters of different\nsizes from the multi-scale neighborhoods for handling the scale variations of objects for semantic segmentation.\nAlthough these works further improve the performances on different benchmarks, these proposed heads \nstill adds extra computational costs to the already burdensome encoder.\n\n\n\\noindent \\textbf{Encoder-Decoder.} \nAnother type of methods focus on efficiently acquire the high-resolution semantic feature maps via the encoder-decoder architectures. Through\nthe upsampling operations and the skip connections, the encoder-decoder\narchitecture \\cite{ronneberger2015u} can gradually recover the high-resolution feature maps for \nsegmentation. DUsampling \\cite{tian2019decoders} designed a data-dependent upsampling module based on fully \nconnected layers for constructing the high-resolution feature maps from the low-resolution \nfeature maps. FastFCN \\cite{wu2019fastfcn} proposed a Joint Pyramid Upsampling (JPU) method via multiple \ndilated convolution to generate the high-resolution feature maps.\nOne common drawback of these methods is that the feature at each location of the upsampled high-resolution feature maps is \nconstructed via only local feature fusion. Such a property limited limits their performance in \nsemantic segmentation, where global context is important for the final performance.\n\n\n\\section{Proposed Method}\nIn this section, We firstly give a thorough analysis of the classical\nencoder-decoder based methods. \nThen, to tackle the challenges in the classical encoder-decoder methods, \nwe propose the EfficientFCN, which is based on the traditional ResNet as the\nencoder backbone \nnetwork for semantic segmentation. In our EfficientFCN, the\nHolistically-guided Decoder (HGD)\nis designed to recover the high-resolution (OS=8) feature maps from three\nfeature maps in the last three blocks from the ResNet encoder backbone.\n\\begin{figure}[tb]\n    \\centering\n    \\subfloat{\\includegraphics[width=11.0cm]{figs\/framework_img1}}\n\\caption{Illustration of the proposed EfficientFCN model. It consists of three main components. Multi-scale feature fusion fuses multi-scale features to obtain OS=8 and OS=32 multi-scale feature maps. Holistic codebook generation results in a series of holistic codewords summarizing different aspects of the global context. High-resolution feature upsampling can be achieved by codeword assembly.}\n\\label{fig:framework_img1}\n\\end{figure}\n\\subsection{Overview}\nIn state-of-the-art DCNN backbones, the high-resolution mid-level feature maps\nat earlier stages can better encode fine-grained structures, while the\nlow-resolution high-level features at the later stages are generally more\ndiscriminative for category prediction. \nThey are essential and complementary information sources for achieving\naccurate semantic segmentation.\nTo combine the strengths of both features, encoder-decoder structures were\ndeveloped to reconstruct high-resolution feature maps that have semantic-rich\ninformation from the multi-scale feature maps. \nTo generate the final feature map $\\tilde{f}_8$ of OS=8 for mask prediction, a\nconventional three-stage encoder-decoder first upsamples the deepest encoder\nfeature map $f_{32}$ of OS=32 to generate OS=16 feature maps $f_{16}$. The\nOS=16 feature maps $e_{16}$ of the same size from the encoder is either\ndirectly concatenated as $[f_{16}; e_{16}]$ (U-Net \\cite{ronneberger2015u}) or summed as\n$f_{16}+e_{16}$ followed by some $1\\times 1$ convolution to\ngenerate the upsampled OS=16 feature maps, $\\tilde{f}_{16}$. The same\nupsampling + skip connection procedure repeats again for $\\tilde{f}_{16}$ to\ngenerate $\\tilde{f}_{8}$. The upsampled features $\\tilde{f}_8$ contain both\nmid-level and high-level information to some extent and can be used to\ngenerate the segmentation masks.\nHowever, since bilinear upsampling and deconvolution layer in the classical decoder are local operations\nwith limited receptive field. We argue that they are incapable of exploring\nimportant global context of the input image, which are crucial for achieving\naccurate segmentation. \nAlthough there were some existing attempts \\cite{Zhang_2018_CVPR,hu2018squeeze} on using\nglobal context to re-weight the contributions of different channels of the\nfeature maps either in the backbone \\cite{Zhang_2018_CVPR} or in the upsampled feature\nmaps \\cite{hu2018squeeze}. This strategy only scales each feature channel but\nmaintains the original spatial size and structures. Therefore, it is incapable\nof generating high-resolution semantic-rich feature maps to improve the\nrecovery of fine-grained structures.\nTo solve this drawback, we propose a novel Holistically-guided Decoder (HGD),\nwhich decomposes the feature upsampling task into the generation of a series\nof holistic codewords from high-level feature maps to capture global contexts,\nand linearly assembling codewords at each spatial location for semantic-rich\nfeature upsampling. Such a decoder can exploit the global contextual information\nto effectively guide the feature upsampling process and is able to recover\nfine-grained details. \nBased on the proposed HGD, we present the EfficientFCN (as shown in\nFig.~\\ref{fig:framework_img0}(d)) with an efficient encoder free of dilated\nconvolution for efficient semantic segmentation. \n\\subsection{Holistically-guided Decoder for Semantic-rich Feature Upsampling}\nTo take advantages of both low-resolution high-level feature maps of size\nOS=32 and the high-resolution mid-level feature maps of sizes OS=8 and OS=16,\nsince the high-level feature maps have already lost most structural details\nbut are semantic-rich to encode categorical information, we argue that\nrecovering detailed structures from them is quite challenging and also\nnon-necessary. Instead, we propose to generate a series of holistic codewords\nwithout any spatial order from the high-level feature maps to capture\ndifferent aspects of the global context. On the other hand, the mid-level\nhigh-resolution feature maps have maintained abundant image structural\ninformation. But they are from relatively shallower layers and cannot encode\naccurate enough categorical features  for final mask prediction. However, they\nwould still be representative enough for guiding the linear assembly of the\nsemantic-rich codewords for high-resolution feature upsampling. Our proposed\nHolistically-guided Decoder therefore contains three main components:\nmulti-scale feature fusion, holistic codebook generation, and codeword\nassembly for high-resolution feature upsampling.\n\n\\noindent \\textbf{Multi-scale features fusion.}\nGiven the multi-scale feature maps from the encoder, although we can directly\nencode the holistic codewords from the high-level OS=32 feature maps and also\ndirectly generate the codeword combination coefficients from the mid-level\nOS=16 and OS=8 feature maps, we observe the fusion of multi-scale feature maps\ngenerally result in better performance. \nFor the OS=8, OS=16, OS=32 feature maps from the encoder, we first adopt\nseparate $1\\times 1$ convolutions to compress each of their channels to 512\nfor reducing the follow-up computational complexity, obtaining $e_{8}, e_{16},\ne_{32}$, respectively. The multi-scale fused OS=32 feature maps $m_{32}$ are\nthen obtained by downsampling $e_{8}, e_{16}$ to the size of OS=32 and\nconcatenating them along the channel dimension with $e_{32}$ as $m_{32} = [\n    e_{8}^\\downarrow;  e_{16}^\\downarrow; e_{32}] \\in \\mathbb{R}^{1536\\times\n(H\/32) \\times (W\/32)}$, where $^\\downarrow$ represents bilinear downsampling,\n$[\\cdot;\\cdot]$ denotes concatenation along the channel dimension, and $H$ and\n$W$ are the input image's height and width, respectively. We can also obtain\nthe multi-scale fused OS=8 feature maps, $m_8 = [ e_{8}; e_{16}^\\uparrow;\ne_{32}^\\uparrow] \\in \\mathbb{R}^{1536\\times (H\/8) \\times (W\/8)}$, in a similar\nmanner.\n\n\\noindent \\textbf{Holistic codebook generation.}\nAlthough the mutli-scale fused feature maps $m_{32}$ are created to integrate\nboth high-level and mid-level features, their small resolutions make them lose\nmany structural details of the scene. On the other hand, because $e_{32}$ is\nencoded from the deepest layer, $m_{32}$ is able to encode rich categorical\nrepresentations of the image. We therefore propose to generate a series of\nunordered holistic codewords from $m_{32}$ to implicitly model different\naspects of the global context.\nTo generate $n$ holistic codewords, a codeword base map $B \\in\n\\mathbb{R}^{1024 \\times (H\/32)\\times}$ $^{(W\/32)}$ and $n$ spatial weighting\nmaps $A \\in \\mathbb{R}^{n \\times (H\/32)\\times (W\/32)}$ are first computed from\nthe fused multi-scale feature maps $m_{32}$ by two separate $1\\times 1$\nconvolutions. \nFor the bases map $B$, we denote $B(x,y) \\in \\mathbb{R}^{1024}$ as the 1024-d\nfeature vector at location $(x,y)$; for the spatial weighting maps $A$, we use\n$A_i \\in \\mathbb{R}^{(H\/32) \\times (W\/32)}$ to denote the $i$th weighting map.\nTo ensure the weighting maps $A$ are properly normalized, the softmax function is adopted to \nnormalize all spatial locations of each channel $i$ (the $i$-th spatial feature map) as\n\\begin{align}\n            \\tilde{A}_i(x,y) = \\frac{\\exp(A_i(x,y))}{\\sum_{p,q}\n            \\exp(A_i(p,q))}.\n\\end{align}\nThe $i$-th codeword $c_i\\in \\mathbb{R}^{1024}$ can be obtained as the weighted\naverage of all codeword bases $B(x,y)$, \\textit{i}.\\textit{e}.,\n\\begin{align}\n            c_i = \\sum_{p,q} \\tilde{A}_i(p,q) B(p,q).\n\\end{align}\nIn other words, each spatial weighting map $\\tilde{A}_i$ learns to linearly\ncombine all codeword bases $B(x,y)$ from all spatial locations to form a\nsingle codeword, which captures certain aspect of the global context. The $n$\nweighting maps eventually result in $n$ holistic codewords $C = [c_1, \\cdots,\nc_n] \\in \\mathbb{R}^{1024 \\times n}$ to encode high-level global features.\n\n\\noindent \\textbf{Codeword assembly for high-resolution feature upsampling.} \nThe holistic codewords can capture various global contexts of the input image.\nThey are perfect ingredients for reconstructing the high-resolution\nsemantic-rich feature maps as they are encoded from the high-level features\n$m_{32}$. However, since their structural information have  been mostly\nremoved during codeword encoding, we turn to use the OS=8 multi-scale fused\nfeatures $m_8$ to predict the linear assembly coefficients of the $n$\ncodewords at each spatial location for creating a high-resolution feature map.\nMore specifically, we first create a raw codeword assembly guidance feature\nmap $G \\in \\mathbb{R}^{1024\\times (H\/8)\\times (W\/8)}$ to predict the assembly\ncoefficients at each spatial location, which are obtained by applying a\n$1\\times 1$ convolution on the multi-scale fused features $m_8$. However, the\nOS=8 fused features $m_8$ have no information on the holistic codewords as they\nare all generated from $m_{32}$. We therefore consider the general codeword\ninformation as the global average vector of the codeword based map $\\bar{B} \\in\n\\mathbb{R}^{1024}$ and location-wisely add it to the raw assembly guidance\nfeature map to obtain the novel guidance feature map $\\bar{G} = G \\oplus \\bar{B}$, where $\\oplus$ represents\nlocation-wise addition. Another $1\\times 1$ convolution applied on the\nguidance feature map $\\bar{G}$ generates the linear assembly weights of the $n$\ncodewords $W \\in \\mathbb{R}^{n \\times (H\/8) \\times (W\/8)}$ for all $(H\/8)\n\\times (W\/8)$ spatial locations. By reshaping the weighting map $W$ as an $n\n\\times (HW\/8^2)$ matrix, the holistically-guided upsampled feature\n$\\tilde{f}_8$ can be easily obtained as\n\\begin{align}\n            \\tilde{f}_8 = W^\\top C.\n\\end{align}\nGiven the holistically-guided upsampled feature map $\\tilde{f}_8$, we reconstruct the final \nupsampled feature map  $\\hat{f}_8$ by concatenating the feature map $\\tilde{f}_8$ with the\nguidance feature map $G$. \nSuch an upsampled feature map $\\hat{f}_8$ takes advantages of both $m_8$\nand $m_{32}$, and contains semantic-rich and also structure-preserved features\nfor achieving accurate segmentation. \n\n\\noindent {\\bf Final segmentation mask.} Given the upsampled feature map\n$\\hat{f}_8$, a $1\\times 1$ convolution can output a segmentation map of\nOS=8, which is further upsampled back to the original resolution $H\n\\times W$ as the final segmentation mask.\n\n\n\n\n\n\\section{Experiments}\nIn this section, we introduce the implementation details, training strategies and evaluation metrics of our experiments.\nTo evaluate our proposed EfficientFCN model, we conduct comprehensive experiments\non three public datasets PASCAL Context \\cite{mottaghi2014role}, PASCAL VOC 2012 \\cite{everingham2010pascal}\nand ADE20K \\cite{zhou2017scene}. To further evaluate the contributions of individual components in our model, we conduct detailed ablation studies on the PASCAL Context dataset. \n\\subsection{Implementation Details}\n\\noindent\n\\textbf{Network Structure.}\\\nDifferent with the dilatedFCN based methods, which remove the stride of the last two blocks of\nthe backbone networks and adopt the dilated convolution with the dilation rates $2$ and $4$,\nwe use the original ResNet \\cite{he2016deep} as our encoder backbone network. \nThus the size of the output feature maps from the last ResBlock is $32\\times$ smaller than that of\nthe input image. After feeding the encoder feature maps into our proposed holistic-guided decoder,\nthe classification is performed on the output upsampled feature map $\\hat{f}_8$. \nThe ImageNet \\cite{russakovsky2015imagenet} pre-trained weights are utilized to initialize the encoder network.\n\n\\noindent\n\\textbf{Training Setting.}\\\nA poly learning rate policy \\cite{chen2017deeplab} is used in our experiments. We set the initial learning rates as $0.001$\nfor PASCAL Context \\cite{mottaghi2014role}, $0.002$ for PASCAL VOC 2012\n\\cite{everingham2010pascal} and \nADE20K \\cite{zhou2017scene}. The power of poly learning rate policy is set as $0.9$. The optimizer\nis stochastic gradient descent (SGD) \\cite{bottou2010large} with momentum $0.9$ and weight\ndecay $0.0001$.\nWe train our EfficientFCN for $120$ epochs on PASCAL Context, $80$ epochs on PASCAL 2012 and $120$ epochs on ADE20K, respectively. We set the crop size to $512\\times 512$ on PASCAL Context and PASCAL 2012. Since the average image size is larger than other two datasets, we use $576 \\times 576$ as the crop size on ADE20K. For data augmentation, we only randomly flip the input image and scale it randomly in the range $[0.5, 2.0]$.\n\n\\noindent\n\\textbf{Evaluation Metrics.}\\\nWe choose the standard evaluation metrics of pixel accuracy (pixAcc) and mean Intersection of Union (mIoU) as the evaluation metrics in our experiments. Following the best practice \\cite{Zhang_2018_CVPR,he2019adaptive,fu2019dual}, \nwe apply the strategy of averaging the network predictions in multiple scales for evaluation. For\neach input image, we first randomly resize the input image with a scaling factor sampled uniformly\nfrom [0.5, 2.0] and also randomly horizontally flip the image. These predictions are then averaged to generate the final prediction.\n\\begin{table}[tb]\n\\centering\n\\centering\n\\caption{Comparisons with classical encoder-decoder methods.}\n\\label{table:ablation_encoder_decoder}\n\\setlength{\\tabcolsep}{0.5mm}{\n\\begin{tabular}{llllll}\n\\hline\n\\textbf{Method} & \\textbf{Backbone} & \\textbf{OS} &\\textbf{mIoU\\%} &\n\\textbf{Parameters (MB)} &\\textbf{GFlops (G)}\\\\\\hline\\hline\nFCN-32s  &  ResNet101 & 32 & 43.3 & 54.0 & 44.6 \\\\\ndilatedFCN-8s  & dilated-ResNet101 & 8 & 47.2 &54.0 & 223.6 \\\\\nUNet-Bilinear  & ResNet101 & 8 & 49.3 & 60.7 & 87.9 \\\\\nUNet-Deconv  & ResNet101 & 8  & 49.1 & 62.8 & 93.2\\\\\n\\hline\nEfficientFCN & ResNet101 & 8  & 55.3 & 55.8 & 69.6\\\\\n\\hline\n\\end{tabular}}\n\\end{table}\n\\subsection{Results on PASCAL Context}\nThe PASCAL Context dataset consists of 4,998 training images and 5,105 testing images for scene parsing. It is a complex and challenging dataset based on PASCAL VOC 2010 with more annotations and fine-grained scene classes, which includes 59 foreground classes and one background class. \nWe take the same experimental settings and evaluation strategies following previous works \\cite{Zhang_2018_CVPR,Zhang_2019_CVPR,fu2019dual,Fu_2019_ICCV,he2019dynamic}.\nWe first conduct ablation studies on this dataset to demonstrate the\neffectiveness of each individual module design of our proposed EfficientFCN\nand then compare our model with state-of-the-art methods. The ablation studies are conducted with a ResNet101 encoder backbone.\n\\begin{table}[tb]\n\\centering\n\\begin{minipage}[t]{0.44\\textwidth}\n\\caption{Results of using different numbers of scales for multi-scale fused feature $m_{32}$ to generate the holistic codewords.}\n\\label{table:ablation_ms_codebook}\n\\setlength{\\tabcolsep}{1mm}{\n\\begin{tabular}{c|cccc}\n\\hline\n        {} & {$\\{32\\}$} & {$\\{16, 32\\}$} & {$\\{8, 16, 32\\}$} \\\\ \\hline\\hline\n        pixAcc &80.0 & 80.1 & 80.3  \\\\\n        mIoU & 54.8 & 55.1 & 55.3   \\\\\n\\hline\n\\end{tabular}}\n\\end{minipage}\n\\begin{minipage}[t]{0.11\\textwidth}\n\\end{minipage}\n\\begin{minipage}[t]{0.44\\textwidth}\n\\centering\n\\caption{Results of using different numbers of scales for multi-scale fused feature $m_8$ to estimate codeword assembly coefficients.}\n\\label{table:ablation_ms_assembly}\n\\setlength{\\tabcolsep}{1mm}{\n\\begin{tabular}{c|cccc}\n\\hline\n        {} & {$\\{8\\}$} & {$\\{8, 16\\}$} & {$\\{8,16,32\\}$} \\\\ \\hline\\hline\n        pixAcc &78.9 & 80.0 & 80.3  \\\\\n        mIoU & 47.9 & 52.1 & 55.3   \\\\\n\\hline\n\\end{tabular}}\n\\end{minipage}\n\\end{table}\n\n\\noindent \\textbf{Comparison with the classical encoder-decoders.} For the classical encoder-decoder based methods, the feature upsampling is achieved via either bilinear interpolation or\ndeconvolution. We implement two classical encoder-decoder based methods, which include the\nfeature upsampling operation (bilinear upsampling or deconvolution) and the skip-connections. \nTo verify the effectiveness of our proposed HGD,  these two methods are trained and tested \non the PASCAL Context dataset with the same training setting as our model.\nThe results are shown in Table \\ref{table:ablation_encoder_decoder}. Although the classical \nencoder-decoder methods have similar computational complexities, their performances \nare generally far inferior than our EfficientFCN. The key reason is that their upsampled \nfeature maps are recovered in a local manner. The simple bilinear interpolation or \ndeconvolution cannot effectively upsample the OS=32 feature maps even with the skip-connected \nOS=8 and OS=16 feature maps. In contrast, our proposed HGD can effectively upsample the \nhigh-resolution semantic-rich feature maps not only based on the fine-grained structural \ninformation in the OS=8 and OS=16 feature maps but also from the holistic semantic information from the OS=32 feature maps.  \n\n\\noindent \\textbf{Multi-scale features fusion.} We conduct two experiments on multi-scale features fusion to verify their effects on semantic codebook generation and codeword assembly for feature upsampling. In our holistically-guided decoder, the semantic codewords are generated based on the OS=32 multi-scale fused feature maps $m_{32}$ and the codewords assembly coefficients are predicted from the OS=8 multi-scale fused feature maps $m_8$.\nFor the codeword generation, we conduct three experiments to generate the semantic codebook from\nmulti-scale fused features with different numbers of scales. As shown in Table\n\\ref{table:ablation_ms_codebook}, when reducing the number of fusion scales from 3 to 2 and \nfrom 2 to 1, the performances of our EfficientFCN slightly decrease. The phenomenon is reasonable as the deepest feature maps contain more categorical information than the OS=8 and OS=16 feature maps. \nFor the codeword assembly coefficient estimation, the similar experiments are conducted, where results are shown in Table \\ref{table:ablation_ms_assembly}. However, different from the above results, when fewer scales of feature maps are used to form the multi-scale fused OS=8 feature map $m_8$, the performances of our EfficientFCN show significant drops.\nThese results demonstrate that although the OS=8 feature maps contain more fine-grained structural information, the semantic features from higher-level feature maps are essential for guiding the recovery of the semantic-rich high-resolution feature maps.\n\\begin{table}[!t]\n\\centering\n\\caption{Ablation study of the number of the semantic codewords.}\n\\label{table:ablation_n_codewords}\n\\setlength{\\tabcolsep}{2mm}{\n\\begin{tabular}{c|cccccc}\n\\hline\n       \n        {} & {32} & {64} & {128}  & {256} &{512} &{1024}\\\\ \\hline\\hline\n        pixAcc &79.9 & 80.1 & 80.1 & 80.3 & 80.3 &80.1 \\\\\n        mIoU & 54.5 & 54.9 & 55.0 &55.3 & 55.5 & 55.1  \\\\\n        GFLOPS & 67.9 & 68.1 & 68.6 & 69.6 & 72.1 & 78.9 \\\\\n\\hline\n\\end{tabular}}\n\\end{table}\n\\begin{table}[!t]\n\\centering\n\\caption{Segmentation results of state-of-the-art methods on PASCAL Context and ADE20K validation dataset.}\n\\label{table:pascal-context}\n\\centering\n\\begin{tabular}{lllll}\n\\hline\n    \\textbf{Method} & \\textbf{Backbone} &\\begin{tabular}{c} \n        \\textbf{mIoU\\%}\\\\ (PASCAL Context) \\end{tabular} & \\begin{tabular}{c} \n        \\textbf{mIoU\\%} \\\\ (ADE20K) \\end{tabular} & \\textbf{GFlops} \\\\\\hline\\hline\n    DeepLab-v2 \\cite{chen2017deeplab} &  Dilated-ResNet101-COCO & 45.7 & - & $>$223\\\\\n    RefineNet \\cite{RefineNet} &  Dilated-ResNet152 & 47.3 & - & $>$223\\\\\n    MSCI \\cite{MSCI} &  Dilated-ResNet152 & 50.3 & - & $>$223\\\\\n    PSPNet \\cite{zhao2017pyramid} & Dilated-ResNet101 & - & 43.29 & $>$223 \\\\\n    SAC \\cite{zhang2017scale} & Dilated-ResNet101 & - & 44.30 & $>$223  \\\\\n    EncNet \\cite{Zhang_2018_CVPR} &  Dilated-ResNet101 & 51.7 & 44.65 & 234\\\\\n    DANet \\cite{fu2019dual} &  Dilated-ResNet101 & 52.6 & - & $>$223 \\\\ \n    APCNet \\cite{he2019adaptive} &  Dilated-ResNet101 & 54.7 & 45.38 & 245\\\\ \n    CFNet \\cite{Zhang_2019_CVPR} &  Dilated-ResNet101 & 54.0 & 44.89 & $>$223 \\\\ \n    ACNet \\cite{Fu_2019_ICCV} & Dilated-ResNet101 & 54.1 & \\textbf{45.90} & $>$223\\\\ \n    APNB \\cite{zhu2019asymmetric} & Dilated-ResNet101 & 52.8 & 45.24 & $>$223  \\\\ \n    DMNet \\cite{he2019dynamic} & Dilated-ResNet101 & 54.4 & 45.50 & 242\\\\\n\\hline\n    Ours & ResNet101 & \\textbf{55.3} & {45.28} & \\textbf{70} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{comment}\n\\begin{table}[!t]\n\\caption{Segmentation results of state-of-the-art methods on ADE20K validation set.}\n\\label{table:ade20k}\n\\centering\n\\begin{tabular}{lll}\n\\hline\n\\textbf{Method} & \\textbf{Backbone} & \\textbf{mIoU\\%} \\\\\\hline\\hline\nPSPNet \\cite{zhao2017pyramid} & Dilated-ResNet101 & 43.29 \\\\\nEncNet \\cite{Zhang_2018_CVPR} & Dilated-ResNet101 & 44.65 \\\\\nSAC \\cite{zhang2017scale} & Dilated-ResNet101 & 44.30  \\\\\nDSSPN \\cite{DSSPN} & Dilated-ResNet101 & 43.68 \\\\\nAPCNet \\cite{he2019adaptive} & Dilated-ResNet101 & 45.38 \\\\\nCFNet \\cite{Zhang_2019_CVPR} & Dilated-ResNet101 & 44.89 \\\\\nDMNet \\cite{he2019dynamic} & Dilated-ResNet101 & 45.50  \\\\ \nACNet \\cite{Fu_2019_ICCV} & Dilated-ResNet101 & 45.90 \\\\ \nAPCNet \\cite{he2019adaptive} & Dilated-ResNet101 & 45.38 \\\\\nCCNet \\cite{Huang_2019_ICCV} & Dilated-ResNet101 & 45.22 \\\\ \nAPNB \\cite{zhu2019asymmetric} & Dilated-ResNet101 & 45.24 \\\\ \n\\hline\nOurs & ResNet101 & \\textbf{45.28} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\end{comment}\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\begin{center}\n\\begin{tabular}{C{2.2cm}C{2.2cm}C{2.2cm}C{2.2cm}C{2.2cm}}\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig0_a} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig0_b} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig0_c} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig0_d} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig0_e} \\\\\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig1_a} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig1_b} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig1_c} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig1_d} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig1_e} \\\\\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig2_a} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig2_b} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig2_c} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig2_d} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig2_e} \\\\\n   \n   \n   \n   \n   \n    \\centering (a) & (b) & (c) & (d) & (e) \\\\\n\\end{tabular}\n\\end{center}\n\\caption{(a) Input images from the PASCAL Context and ADE20K dataset. (b-e) Different weighting maps $\\tilde{A}_i$ for creating the holistic codewords.}\n\\label{fig:weighting_maps}\n\n\\begin{center}\n\\begin{tabular}{C{2.50cm}C{2.50cm}C{2.50cm}C{2.50cm}}\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004877_img} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004877_gt} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004877_fcn} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004877_our} \\\\\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2008_004203_img} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2008_004203_gt} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2008_004203_fcn} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2008_004203_our} \\\\\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004980_img} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004980_gt} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004980_fcn} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004980_our} \\\\\n    \\centering (a) Image & (b) GT & (c) Baseline & (d) EfficientFCN \\\\\n\\end{tabular}\n\\end{center}\n\\caption{Visualization results from the PASCAL Context dataset.}\n\\label{fig:vis}\n\\end{figure}\n\n\\noindent \\textbf{Number of holistic codewords.} We also conduct  experiments to survey the\neffectiveness of the number of codewords in our predicted semantic codebook for feature upsampling.\nAs shown in Table \\ref{table:ablation_n_codewords}, as the number of the semantic codewords\nincreases from 32 to 512, the performance improves 1\\% in terms of mIoU on PASCAL Context.\nHowever, when the number of the semantic codewords further increases from 512 to 1024, the performance\nhas a slight drop, which might be caused by the additional parameters. The larger model capacity\nmight cause model to overfit the training data. In addition, since the assembly coefficients of the\nsemantic codewords are predicted from the OS=8 multi-scale fused feature $m_8$, the increased number\nof the semantic codewords also leads to significantly more extra computational cost. Thus, to balance\nthe performance and also the efficiency, we set the number of the holistic codewords as 256 for the\nPASCAL Context and PASCAL VOC 2012 datasets. Since PASCAL Context only has 60 classes and we observe\nthe number of codewords needed is approximately 4 times than the number of classes. We therefore set\nthe number of codewords as 600 for ADE20K, which has 150 classes.\n\n\\noindent \\textbf{Importance of the codeword information transfer for accurate assembly coefficient estimation.} \nThe key of our proposed HGD is how to linearly assemble holistic codewords at each spatial location\nto form high-resolution upsampled feature maps based on the feature maps $m_8$. In our HGD, although\nthe OS=8 features have well maintained structural image information, we argue that directly using OS=8 features to predict codeword assembly coefficients are less effective since they have no information about the codewords. \nWe propose to transfer the codeword information as the average codeword basis,\nwhich is location-wisely added to the OS=8 feature maps. To verify this\nargument, we design an experiment that removes the additive information\ntransfer, and only utilizes two $1\\times 1$ convolutions with the same output\nchannels on the OS=8 feature maps $m_8$ for directly predicting assembly\ncoefficients. The mIoU of this implementation is 54.2\\%, which has a clear performance drop if there\nis no codeword information transfer from the codeword generation branch to the codeword coefficient prediction branch.\n\n\n\\noindent \\textbf{Visualization of the weighting maps and example results.} \nTo better interpret the obtained holistic codewords, we visualize the weighting maps $\\tilde{A}$ for\ncreating the holistic codewords in Fig.~\\ref{fig:weighting_maps}, where each column shows one\nweighting map $\\tilde{A}_i$ for generating one holistic codeword. Some weighting maps focus on summarizing  foreground objects or regions to create holistic codewords, while some other weighting maps pay attention to summarizing background contextual regions or objects as the holistic codewords. The visualization shows that the learned codewords implicitly capture different global contexts from the scenes.\nIn Fig.~\\ref{fig:vis}, we also visualize some predictions by the baseline\nDilatedFCN-8s and by our EfficientFCN, where our model significantly improves the visualized results with the proposed HGD.\n\n\n\n\n\\begin{table*}[!t]\n\\small\n\\centering\n\\caption{Results of each category on PASCAL VOC 2012 test set. Our\n        EfficientFCN obtains 85.4 \\% without MS COCO dataset pre-training and 87.6\\% with MS COCO dataset pre-training. (For each\n    columns, the best two entries are filled in gray color. )}\n\\label{table:pascal_voc_2012}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l|cccccccccccccccccccc|c}\n\\hline\n\\textbf{Method}    & \\textbf{aero} & \\textbf{bike} & \\textbf{bird} & \\textbf{boat} & \\textbf{bottle} & \\textbf{bus}  & \\textbf{car}  & \\textbf{cat}  & \\textbf{chair} & \\textbf{cow}  & \\textbf{table} & \\textbf{dog}  & \\textbf{horse} & \\textbf{mbike} & \\textbf{person} & \\textbf{plant} & \\textbf{sheep} & \\textbf{sofa} & \\textbf{train} & \\textbf{tv}   & \\textbf{mIoU\\%} \\\\ \\hline\\hline\n\\textbf{FCN} \\cite{long2015fully}       & 76.8          & 34.2          & 68.9\n& 49.4          & 60.3            & 75.3          & 74.7          & 77.6\n& 21.4           & 62.5          & 46.8           & 71.8          & 63.9\n& 76.5           & 73.9            & 45.2           & 72.4           & 37.4\n& 70.9           & 55.1          & 62.2   \\\\\n\\textbf{DeepLabv2} \\cite{chen2017deeplab} & 84.4          & 54.5          & 81.5\n& 63.6          & 65.9            & 85.1          & 79.1          & 83.4\n& 30.7           & 74.1          & 59.8           & 79.0            & 76.1           & 83.2           & 80.8            & 59.7           & 82.2           & 50.4          & 73.1           & 63.7          & 71.6  \\\\\n\\textbf{CRF-RNN} \\cite{CRF-RNN}   & 87.5          & 39.0          & 79.7          & 64.2          & 68.3            & 87.6          & 80.8          & 84.4          & 30.4           & 78.2          & 60.4           & 80.5          & 77.8           & 83.1           & 80.6            & 59.5           & 82.8           & 47.8          & 78.3           & 67.1          & 72.0  \\\\\n\\textbf{DeconvNet} \\cite{DeconvNet} & 89.9          & 39.3          & 79.7          & 63.9          & 68.2            & 87.4          & 81.2          & 86.1          & 28.5           & 77.0          & 62.0           & 79.0          & 80.3           & 83.6           & 80.2            & 58.8           & 83.4           & 54.3          & 80.7           & 65.0            & 72.5   \\\\\n\\textbf{DPN} \\cite{DPN}       & 87.7          & 59.4          & 78.4          & 64.9          & 70.3            & 89.3          & 83.5          & 86.1          & 31.7           & 79.9          & 62.6           & 81.9          & 80.0           & 83.5           & 82.3            & 60.5           & 83.2           & 53.4          & 77.9           & 65.0            & 74.1   \\\\\n\\textbf{Piecewise} \\cite{Piecewise} & 90.6          & 37.6          & 80.0\n& 67.8          & 74.4            & 92            & 85.2          & 86.2\n& 39.1           & 81.2          & 58.9           & 83.8          & 83.9\n& 84.3           & 84.8            & 62.1           & 83.2           & 58.2\n& 80.8           & 72.3          & 75.3    \\\\\n\\textbf{ResNet38} \\cite{ResNet38}  & 94.4          & 72.9          & 94.9\n& 68.8          & 78.4            & 90.6          & 90.0          & 92.1\n& 40.1           & 90.4          & 71.7           & 89.9          & 93.7\n& \\bgGray 91.0           & 89.1            & 71.3           & 90.7           & 61.3          & 87.7           & 78.1          & 82.5     \\\\\n\\textbf{PSPNet} \\cite{zhao2017pyramid}    & 91.8          & 71.9          & 94.7          & 71.2          & 75.8            & 95.2          & 89.9          & 95.9          & 39.3           & 90.7          & 71.7           & 90.5          & 94.5           & 88.8           & 89.6            & 72.8           & 89.6           & \\bgGray {64.0}          & 85.1           & 76.3          & 82.6   \\\\\n\\textbf{EncNet} \\cite{Zhang_2018_CVPR}    & 94.1          & 69.2          & \\bgGray\\textbf{96.3} & \\bgGray 76.7          & \\bgGray \\textbf{86.2}   & 96.3          & 90.7          & 94.2          & 38.8           & 90.7          & 73.3           & 90.0          & 92.5           & 88.8           & 87.9            & 68.7           & 92.6           & 59.0          & 86.4           & 73.4          & 82.9            \n            \\\\\n            \\textbf{APCNet} \\cite{he2019adaptive}      & 95.8 &\\bgGray 75.8 & 84.5  & 76.0 & 80.6 &\n            \\bgGray 96.9 & 90.0 & 96.0 & \\bgGray\\textbf{42.0} & \\bgGray 93.7\n            &\\bgGray 75.4 & 91.6 & 95.0 & 90.5 &\n             89.3 &  75.8 & 92.8 & 61.9 &  88.9 & \\bgGray 79.6 & 84.2\n            \\\\\n            \\textbf{CFNet} \\cite{Zhang_2019_CVPR}      & 95.7 & 71.9 &\\bgGray\n            95.0  &\\bgGray 76.3 & \\bgGray 82.8 &\n            94.8 & 90.0 & 95.9 & 37.1 & 92.6 & 73.0 & \\bgGray 93.4 & 94.6 & 89.6 &\n            88.4 & 74.9 & \\bgGray \\textbf{95.2} &  63.2 & \\bgGray \\textbf{89.7} & 78.2 & 84.2\n            \\\\\n            \\textbf{DMNet} \\cite{he2019dynamic}        & \\bgGray 96.1 &\n            \\bgGray\\textbf{77.3} & 94.1 & 72.8 & 78.1 & \n            \\bgGray\\textbf{97.1} & \\bgGray \\textbf{92.7} & \\bgGray 96.4 & 39.8 & 91.4 & \\bgGray 75.5 & 92.7 & \\textbf{95.8} &\n            \\bgGray {91.0} & \\bgGray {90.3} & \\bgGray {76.6} & \\bgGray 94.1 & 62.1 & 85.5 & 77.6 & \\bgGray 84.4 \n            \\\\ \\hline\n            \\textbf{Ours}      & \\bgGray \\textbf{96.4} & {74.1} &\n            92.8 & \\bgGray 75.6 & 81.9 &\\bgGray 96.9 \n            & \\bgGray {92.6}  & \\bgGray \\textbf{97.1} & \\bgGray 41.6 & \\bgGray \\textbf{95.4} \n            & 72.9 & \\bgGray \\textbf{93.9} & \\bgGray \\textbf{95.9} \n            & {90.6} &\\bgGray \\textbf{ 90.6} &\\bgGray \\textbf{77.2}  & 94.0 &\n            67.5 &\\bgGray 89.3 &\n            \\bgGray \\textbf{79.8} & \\bgGray \\textbf{85.4} \\\\\n             \\hline\n            \\multicolumn{22}{c}{\\textbf{With COCO Pre-training}}\\\\\n            \\hline\n            \n            \\textbf{CRF-RNN}~\\cite{CRF-RNN} & 90.4 & 55.3 & 88.7 & 68.4 & 69.8 & 88.3 & 82.4 & 85.1 & 32.6 & 78.5 & 64.4 & 79.6 & 81.9 & 86.4 & 81.8 & 58.6 & 82.4 & 53.5 & 77.4 & 70.1 & 74.7 \\\\\n           \n            \\textbf{Piecewise}~\\cite{Piecewise} & 94.1 & 40.7 & 84.1 & 67.8 & 75.9 & 93.4 & 84.3 & 88.4 & 42.5 & 86.4 & 64.7 & 85.4 & 89.0 & 85.8 & 86.0 & 67.5 & 90.2 & 63.8 & 80.9 & 73.0 & 78.0 \\\\\n           \n           \n            \\textbf{DeepLabv2}~\\cite{chen2017deeplab} & 92.6 & 60.4 & 91.6 & 63.4 & 76.3 & 95.0 & 88.4 & 92.6 & 32.7 & 88.5 & 67.6 & 89.6 & 92.1 & 87.0 & 87.4 & 63.3 & 88.3 & 60.0 & 86.8 & 74.5 & 79.7  \\\\\n            \\textbf{RefineNet}\\cite{RefineNet}  & 95.0 & 73.2 & 93.5 & 78.1 & 84.8 & 95.6 & 89.8 & 94.1 & 43.7 & 92.0 & 77.2 & 90.8 & 93.4 & 88.6 & 88.1 & 70.1 & 92.9 & 64.3 & 87.7 & 78.8 & 84.2 \\\\\n            \\textbf{ResNet38}\\cite{ResNet38}  &  96.2 & 75.2 & \\cellcolor[gray]{0.92} \n            95.4 & 74.4 & 81.7 & 93.7 & 89.9 & 92.5 & \\cellcolor[gray]{0.92}  48.2 & 92.0 & 79.9\n            & 90.1 & 95.5 & 91.8 & 91.2 &  73.0 & 90.5 & 65.4 & 88.7 & 80.6 & 84.9\\\\\n            \\textbf{PSPNet}~\\cite{zhao2017pyramid} & {95.8} & {72.7} & {95.0}\n            & {78.9} & {84.4} & 94.7 & \\cellcolor[gray]{0.92} {92.0} & {95.7} & {43.1} &\n            {91.0} & \\bf \\cellcolor[gray]{0.85}{80.3} & {91.3} & {96.3} & {92.3} & {90.1} & {71.5}\n            & \\cellcolor[gray]{0.92} {94.4} & \\cellcolor[gray]{0.92} {66.9} & {88.8} & \\bf \\cellcolor[gray]{0.85}{82.0} & {85.4} \\\\\n            \\textbf{DeepLabv3}\\cite{chen2017rethinking} & \\cellcolor[gray]{0.92}  96.4 & 76.6\n            & 92.7 & 77.8 & \\cellcolor[gray]{0.92} {87.6} & 96.7 & 90.2 & 95.4 &  47.5 &\n            \\cellcolor[gray]{0.92}  93.4 & 76.3 & 91.4 & \\bf \\cellcolor[gray]{0.85}{97.2} &  91.0 & \\bf \\cellcolor[gray]{0.85}{92.1} &\n            71.3 & 90.9 & \\cellcolor[gray]{0.92} {68.9} & \\cellcolor[gray]{0.92} {90.8} & 79.3 & 85.7 \\\\ \n            \\textbf{EncNet}\\cite{Zhang_2018_CVPR}  & 95.3 & 76.9 & 94.2 &\n            \\cellcolor[gray]{0.92}  80.2 & 85.2 & 96.5 & 90.8 & 96.3 &  47.9\n            &  93.9 & \\cellcolor[gray]{0.92}  80.0 & 92.4 & \\cellcolor[gray]{0.92}  96.6 & 90.5 &  91.5 & 70.8 &  93.6 & 66.5 & 87.7 & 80.8 & 85.9 \n           \n            \\\\ \n            \\textbf{CFNet} \\cite{Zhang_2019_CVPR}      &\\bf \\cellcolor[gray]{0.85} 96.7 &\\cellcolor[gray]{0.92}  79.7 &\n            94.3  & 78.4 & 83.0 & \\bf \\cellcolor[gray]{0.85} 97.7 & 91.6 &\\cellcolor[gray]{0.92}  96.7 &\\bf \\cellcolor[gray]{0.85} 50.1\n            &\\cellcolor[gray]{0.92}  95.3 & 79.6 & \\bgGray 93.6 &\\bf \\cellcolor[gray]{0.85} 97.2 &\\cellcolor[gray]{0.92} \n            94.2 &\\cellcolor[gray]{0.92}  91.7 & \\bgGray {78.4} &\\bf \\cellcolor[gray]{0.85}  95.4 & \\bgGray\n            \\textbf{69.6} & 90.0 & 81.4 & \\cellcolor[gray]{0.92}  87.2\n            \\\\ \\hline\n            \\textbf{Ours} & \\bgGray {96.6} & \\bf \\cellcolor[gray]{0.85} {80.6} &\n            \\bf \\cellcolor[gray]{0.85} 96.1 & \\bf \\cellcolor[gray]{0.85}{82.3} &\\bf \\cellcolor[gray]{0.85} 87.8 &\\bf \\cellcolor[gray]{0.85} 97.7 \n            & \\bf \\cellcolor[gray]{0.85} {94.4}  & \\bf \\cellcolor[gray]{0.85} {97.3} &  47.1 & \\bgGray \\textbf{96.3} \n            & {77.9} & \\bgGray \\textbf{94.8} & \\bgGray\n            \\textbf{97.2} \n            &\\bgGray \\textbf{94.3} & 91.1 & \\bf \\cellcolor[gray]{0.85} 81.0  & 94.3 & 61.5\n            &\\bf \\cellcolor[gray]{0.85} 91.6 &\\bf \\cellcolor[gray]{0.85} {83.5} & \\bgGray \\textbf{87.6}\n            \\\\ \\hline\n\\end{tabular}}\n\\end{table*}\n\n\\noindent\n\\textbf{Comparison with state-of-the-art methods.} \nTo further demonstrate the effectiveness of our proposed EffectiveFCN with the holistically-guided decoder, the comparisons with state-of-the-art methods are shown in Table \\ref{table:pascal-context}. The dilatedFCN based methods dominate semantic segmentation. However, our work is still able to achieve the best results compared to the dilatedFCN based methods on the PASCAL Context validation set without using any dilated convolution and has significantly less computational cost. Because of the efficient design of our HGD, our EfficientFCN only has 1\/3 of the computational cost of state-of-the-arts methods but can still achieve the best performance. \n\n\\subsection{Results on PASCAL VOC}\nThe original PASCAL VOC 2012 dataset consists of 1,464 images for training, 1,449 for validation, and 1,456 for testing, which is a major benchmark dataset for semantic object segmentation. It includes 20 foreground objects classed and one background class. The augmented training set of 10,582 images, namely train-aug, is adopted as the training set following the previous experimental set in \\cite{Zhang_2019_CVPR}.\nTo further demonstrate the effectiveness of our proposed HGD. We adopt all the best strategies of HGD design and compare it with state-of-the-art methods on the test set of PASCAL-VOC 2012, which is evaluated on the official online server. As shown in Table \\ref{table:pascal_voc_2012}, the dilatedFCN based methods dominate the top performances on the PSCAL VOC benchmark. However, our EfficientFCN with a backbone having no dilated convolution can still achieve the best results among all the ResNet101-based methods. \n\\subsection{Results on ADE20K}\nThe ADE20K dataset consists of 20K images for training, 2K images for\nvalidation, and 3K images for testing, which were used for ImageNet Scene\nParsing Challenge 2016. This dataset is more complex and challenging with 150 labeled classes and more diverse scenes. As shown in Table \\ref{table:pascal-context}, our\nEfficientFCN achieves the competitive performance than the dilatedFCN based\nmethods but has only 1\/3 of their computational cost.\n\\section{Conclusions}\nIn this paper, we propose the EfficientFCN model with the holistically-guied decoder for achieving efficient and accurate semantic segmentation. The novel decoder is able to reconstruct the high-resolution semantic-rich feature maps from multi-scale feature maps of the encoder. \nBecause of the superior feature upsampling performance of the HGD, our EfficientFCN, with much fewer parameters and less computational cost, achieves competitive or even better performance compared with state-of-the-art dilatedFCN based methods. \n\n\n\\subsection*{Acknowledgements}\nThis work is supported in part by SenseTime Group Limited, in part by the General Research Fund through the Research Grants Council of Hong Kong under Grants CUHK 14202217 \/ 14203118 \/ 14205615 \/ 14207814 \/ 14213616 \/ 14208417 \/ 14239816, in part by CUHK Direct Grant.\n\n\\clearpage\n\n\n\n{\\small\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Introduction}\n\nThe discovery and characterization of primeval galaxies constitute some of the biggest challenges in current observational and theoretical cosmology\\footnote{In the following we assume cosmological parameters compatible with \\emph{Planck} results, i.e. a $\\Lambda$CDM model with total matter, vacuum and baryonic densities in units of the critical density $\\Omega_{\\Lambda}= 0.692$, $\\Omega_{m}= 0.308$, $\\Omega_{b}= 0.0481$, Hubble constant $\\rm H_0=100\\,{\\rm h}\\,{\\rm km}\\,{\\rm s}^{-1}\\,{\\rm Mpc}^{-1}$ with ${\\rm h}=0.678$, spectral index $n=0.967$, $\\sigma_{8}=0.826$ \\citep[][]{planck:2013_xvi_parameters}.}.\n\nDeep optical\/near infrared (IR) surveys \\citep{Dunlop13,Madau14,Bouwens:2015} have made impressive progresses in identifying galaxies well within the Epoch of Reionization ($z\\simeq6$). Such surveys yield key information about the star formation (SF) of hundreds of galaxies in the early Universe. They also allow to statistically characterize galaxies in terms of their UltraViolet (UV) luminosity up to $z\\sim10$ \\citep{Bouwens:2015}. However -- using these surveys broad band alone -- little can be learned about other properties as their gas and dust content, metallicity, interactions with the surrounding environment \\citep[e.g.][]{Barnes:2014PASP}, feedback \\citep[e.g.][]{Dayal14}, and outflows \\citep{gallerani:2016outflow}. \n\nTo obtain a full picture of these systems, optical\/IR surveys must be complemented with additional probes. Information on the metal content and energetics of the interstellar medium (ISM) can be obtained with observations of Far IR (FIR) fine structure lines, and in particular the \\hbox{[C~$\\scriptstyle\\rm II $]}~{\\small$\\left(^{2}P_{3\/2} \\rightarrow\\,^{2}P_{1\/2}\\right)$} line at 157.74~$\\mu$m. The \\hbox{[C~$\\scriptstyle\\rm II $]}~line is the dominant coolant of the ISM being excited in different ISM phases, as the diffuse cold neutral medium (CNM), warm neutral medium (WNM), high density photodissociation regions (PDRs), and -- to a lower extent -- ionized gas \\citep[][]{Tielens:1985ApJ,Wolfire:1995ApJ,Abel:2006MNRAS,Vallini:2013MNRAS}. As \\hbox{[C~$\\scriptstyle\\rm II $]}~emission can be enhanced by shocks, it has been suggested as a good outflow tracer\\\\ (e.g. \\citealt{maiolino:2012,kreckel:2014apj,cicone:2015aa,janssen:2016arxiv}), and can thus in general be used to study feedback processes in galaxies.\n\nObservationally, the \\hbox{[C~$\\scriptstyle\\rm II $]}~line is a promising probe as it is often the brightest among FIR emission lines, accounting for up to $\\sim1\\%$ of the total IR luminosity of galaxies \\citep[e.g.][]{Crawford:1985ApJ,Madden:1997ApJ}. It has been successfully used to probe the low-$z$ ISM \\citep[e.g.][]{delooze:2014aa}. The unprecedented sensitivity of the Atacama Large Millimeter\/Submillmeter Array (ALMA) makes it possible for the first time to use \\hbox{[C~$\\scriptstyle\\rm II $]}~emission to characterize high-$z$ galaxies. Before the ALMA advent, in fact, detections were limited to a handful of QSO host galaxies, and rare galaxies with extreme SF rates \\citep[$SFR\\simeq10^3{\\rm M}_{\\odot}\\,{\\rm yr}^{-1}$, e.g.][]{maiolino:2005AA,debreuck:2011,Carilli:2013ARA&A,gallerani:2012aa,cicone:2015aa}.\n\nHowever, for \\quotes{normal} star forming galaxies ($\\lsim10^{2}{\\rm M}_{\\odot}\\,{\\rm yr}^{-1}$) at $z\\sim 6-7$ early ALMA searches for \\hbox{[C~$\\scriptstyle\\rm II $]}~lines have mostly yielded upper limits (e.g. \\citealt{ouchi2013} \\citealt{kanekar2013}; \\citealt{ota:2014apj,schaerer:2015}). The situation has changed recently with a number of robust \\hbox{[C~$\\scriptstyle\\rm II $]}~detections (e.g. \\citealt{maiolino:2015arxiv,capak:2015arxiv}; \\citealt{Willott:2015arXiv15,knudsen:2016arxiv}).\n\nIn many cases the high-$z$ \\hbox{[C~$\\scriptstyle\\rm II $]}~line luminosity is fainter than expected from the \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$ relation found in local galaxies \\citep{delooze:2014aa}. To explain such \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$~\\emph{deficit}, some efforts have been devoted to model the \\hbox{[C~$\\scriptstyle\\rm II $]}~emission from high-$z$ galaxies \\citep{nagamine:2006ApJ,Vallini:2013MNRAS,munoz:2014MNRAS,vallini:2015,olsen:2015apj}. In brief, these theoretical works show that the \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$~deficit can be ascribed to different effects:\n\\begin{itemize}\n\\item[(a)] Lower metallicity of high-$z$ galaxies \\citep{Vallini:2013MNRAS,munoz:2014MNRAS,vallini:2015}, in particular supported by observations of lensed galaxies \\citep{knudsen:2016arxiv}.\n\\item[(b)] Suppression of \\hbox{[C~$\\scriptstyle\\rm II $]}~line around star forming regions \\citet{Vallini:2013MNRAS}, typically observed as a displacement of the \\hbox{[C~$\\scriptstyle\\rm II $]}~ with respect to the UV emitting region, as seen e.g. in BDF3299 \\citep{maiolino:2015arxiv} and in some of the \\citet{capak:2015arxiv} galaxies. This would be a signature of stellar feedback heating\/ionizing the putative \\hbox{[C~$\\scriptstyle\\rm II $]}-emitting gas.\n\\item[(c)] Suppression of \\hbox{[C~$\\scriptstyle\\rm II $]}~line by the increased CMB temperature in the WNM\/CNM component \\citep[][]{pallottini:2015_cmb,vallini:2015}, similarly to what observed for dust emission \\citep{dacunha:2013apj}.\n\\end{itemize}\n\nSimulating the ISM of early galaxies at sufficient resolution and including feedback effects might shed light on these questions. Feedback prescriptions are particularly important as such process regulates the amount of (dense) gas likely to radiate most of the power detected with FIR lines. Several studies have explored optimal strategies to include feedback in galaxy simulations.\n\nFor some works, the interest is in the comparison between different kind of stellar feedback prescription, as modelled via thermal and\/or kinetic energy deposition in the gas from supernovae (SN), winds \\citep[][]{agertz:2012arxiv,fire:2014mnras,barai:2015mnras,agertz:2015apj}, and radiation pressure \\citep[][]{wise:2012radpres,ceverino:2014}; other analyses focus on implementing complex chemical networks in simulations \\citep{tomassetti:2015MNRAS,maio:2015,bovino:2015arxiv,richings:2016,grassi_dust:2016}, radiative transfer effect \\citep{petkova:2012mnras,roskar:2014,rosdahl:2015mnras,maio:2016mnras}, or aim at removing tensions between different coding approaches \\citep[][]{agora:2013arxiv}.\n\nThus, we can improve galaxy simulations by providing theoretical expectations for \\hbox{[C~$\\scriptstyle\\rm II $]}~that should be compared with state-of-the-art data. Such a synergy between theory and observations, in turn, can guide the interpretation of upcoming ALMA data and drive future experiments of large\nscale \\hbox{[C~$\\scriptstyle\\rm II $]}~mapping \\citep{Gong:2012ApJ,silva:2015apj,bin:2015mapping, pallottini:2015_cmb}, which would led to a statistical characterization of the high-$z$ galaxy population. In the present work we simulate a $z\\sim6$ galaxy typically detected in \\hbox{[C~$\\scriptstyle\\rm II $]}~with ALMA current observations.\n\nThe paper is structured as follows. In Sec. \\ref{sec_numerical} we detail the numerical model used to set-up the zoom-in simulation, and describe the adopted ${\\rm {H_2}}$~star formation prescription (Sec. \\ref{sec_model_sf}), mass and energy inputs from the stellar populations (Sec. \\ref{sec_stellar_inputs}) and feedback (including SN, winds and radiation pressure Sec. \\ref{sezione_blast} -- see also App. \\ref{app_rad_press} and App. \\ref{app_blastwave}). The results are discussed in Sec. \\ref{sec_result}, where we analyze star formation history and feedback effects in relation to ISM thermodynamics (Sec. \\ref{sec_sfr_result}) and its structural properties. The expected \\hbox{[C~$\\scriptstyle\\rm II $]}~emission and other observational properties of high-$z$ galaxies are discussed in Sec. \\ref{sec_final_results}. Conclusions are given in Sec. \\ref{sec_conclusioni}.\n\n\n\\section{Numerical simulations}\\label{sec_numerical}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ccccccc}\n\\hline\n~ & $m_{dm}$ & $m_{b}$ & $\\Delta_{x}^{\\rm max}$ & $\\Delta_{x}^{\\rm min}$ & $\\Delta_{x}^{\\rm min}$ at $z=6$\\\\\n~ & \\multicolumn{2}{c}{${\\rm M}_{\\odot}\/{\\rm h}$} & \\multicolumn{2}{c}{${\\rm kpc}\/{\\rm h}$} & pc\\\\\n\\hline\n{\\tt cosmo} & $3.4\\times 10^{7}$ & $-$                & $78.1$ & $78.1$ & $2.5\\times10^3$\\\\\n{\\tt zoom}  & $6.7\\times 10^{4}$ & $1.2\\times 10^{4}$ & $9.7$  & $0.1$  & $32.1$\\\\\n\\end{tabular}\n\\caption{Resolution set-up for the cosmological run ({\\tt cosmo}) and subsequent zoom-in ({\\tt zoom}) simulation. $m_{dm}$ and $m_{b}$ are in units of ${\\rm M}_{\\odot}\/{\\rm h}$ and indicate the dark matter (DM) and baryon mass resolution, respectively; $\\Delta_{x}^{\\rm max}$ and $\\Delta_{x}^{\\rm min}$ indicate the coarse grid and minimum available refinement scale, respectively. Both scales are reported in comoving ${\\rm kpc}\/{\\rm h}$. For $\\Delta_{x}^{\\rm min}$ we also report also the physical pc scale at $z=6$. For the {\\tt cosmo} run, no refinement is used, and for the {\\tt zoom}, we indicate the increased resolution of the zoomed halo due to the multi-mass approach and the AMR.\n\\label{tagella_res}}\n\\end{table}\n\nWe carry out our simulation using a customized version of the adaptive mesh refinement (AMR) code \\textlcsc{ramses} \\citep[][]{Teyssier:2002}. \\textlcsc{ramses} is an octree-based code that uses Particle Mesh N-body solver for the dark matter (DM) and an unsplit 2nd-order MUSCL\\footnote{MUSCL: Monotone Upstream-centred Scheme for Conservation Laws} scheme for the baryons. Gravity is accounted by solving the Poisson equation on the AMR grid via a multi-grid scheme with Dirichlet boundary conditions on arbitrary domains \\citep{guillet:2011Jcoph}. For the present simulation we choose a refinement based on a Lagrangian mass threshold-based criterion.\n\nChemistry and heating\/cooling processes of the baryons are implemented with \\textlcsc{grackle} 2.1\\footnote{See also \\url{https:\/\/grackle.readthedocs.org\/}} \\citep{bryan:2014apjs}, the standard library of thermo-chemical processes of the {\\tt AGORA} project \\citep{agora:2013arxiv}. Via \\textlcsc{grackle}, we follow the \\hbox{H}~and \\hbox{He}~primordial network and tabulated metal cooling and photo-heating rates calculated with \\textlcsc{cloudy} \\citep{cloudy:2013}. Cooling includes also inverse Compton off the cosmic microwave background (CMB), and heating from a redshift-dependent ionizing UV background \\citep[][UVB]{Haardt:2012}. Since ${\\rm {H_2}}$~gas phase formation is not accounted for, we do not include the cooling contribution of such species.\n\nBecause of stellar feedback (Sec \\ref{sec_stellar_inputs} and \\ref{sezione_blast}), the gas can acquire energy both in thermal and kinetic form. The distinction is considered by following the gas evolution of the standard thermal energy and a \\quotes{non-thermal} energy \\citep{agertz:2012arxiv}. Such approach is one of the possible scheme used to solve the over-cooling problem that affect galaxy-scale simulations \\citep[see][ and references therein]{dale:2015new}. The non-thermal energy mimics turbulence, i.e. it is not affected by cooling. The non-thermal energy variation is due to gas advection ($v\\nabla v$), work ($PdV$), and dissipation \\citep{agertz:2015apj}. Following \\citet{maclow1999turb} we assume a dissipation time scale proportional to the size of the cell (injection scale) and inversely proportional to the Mach number\\footnote{While the distinction in thermal and non-thermal is similar to previous works \\citep[e.g.][]{agertz:2015apj}, we note that usually the time scale for dissipation is fixed to $10\\,\\rm Myr$.}. Since the dynamical time is essentially set by the free-fall time, the dissipation time can be written as $t_{\\rm diss} = 9.785 (l_{\\rm cell}\/100\\,{\\rm pc})\/(v_{\\rm turb}\/10\\,{\\rm km}\\,{\\rm s}^{-1}) \\rm Myr$. Then, the non-thermal energy loss due to dissipation can be written as $\\dot{e}_{\\rm nth} = -e_{\\rm nth}\/t_{\\rm diss}$ \\citep[][see eq. 2]{teyssier:2013mnras}. As noted in \\citet{teyssier:2013mnras}, such scheme for non-thermal energy and its dissipation gives results qualitatively similar to a delayed cooling approach \\citep{stinson:2006mnras}.\n\n\\subsection{Initial conditions}\n\nThe initial conditions (IC) used for the suite are generated with \\textlcsc{music} \\citep{hahn:2011mnras}. \\textlcsc{music} produces IC on nested grid using a real-space convolution approach \\citep[cf.][]{bertschinger:1995astro}. The adopted Lagrangian perturbation theory scheme is perfectly suited to produce IC for multi-mass simulations and -- in particular -- zoom simulations. To generate the ICs, the transfer functions are taken from \\citep{eisenstein:1998apj}.\n\nTo set-up the zoom-in simulation, we start by carrying out a cosmological DM-only run. The simulation evolves a volume $V^{\\rm cosmo}=(20\\,{\\rm Mpc}\/{\\rm h})^{3}$ from $z=100$ to $z=6$ with DM mass resolution of $m_{dm}^{\\rm cosmo} = 3.4\\times 10^{7} \/{\\rm h}\\,{\\rm M}_{\\odot}$. The resolution of the coarse grid is $\\Delta x^{\\rm cosmo} = 78.1 \/{\\rm h}\\,{\\rm kpc}$, and we do not include additional levels of refinement. Using \\textlcsc{hop} \\citep{eisenstein_hop_1998apj} we find the DM halo catalogue at $z=6$. The cumulative halo mass function extracted from the catalogue is in agreement with analytical expectations \\citep[e.g.][]{sheth:1999mnras}, within the precision of halo-finder codes \\citep[e.g.][]{knebe:2013arxiv}.\n\nFrom the catalogue we select a halo with DM mass $M_{\\rm h} \\simeq 10^{11}\/{\\rm h}\\,{\\rm M}_{\\odot}$ (resolved by $\\simeq5\\times10^{4}$ DM particles), whose virial radius is $r_{\\rm vir}\\simeq 15\\,{\\rm kpc}$ at $z=6$. Using \\textlcsc{hop} we select the minimum ellipsoid enveloping $10\\,r_{\\rm vir}$, and trace it back to $z=100$. As noted in \\citet[][]{onorbe:2014mnras}, this is usually sufficient to avoid contamination\\footnote{A posteriori, we have checked that the halos in the zoom-in region have a contamination level $\\lsim0.1\\%$.}. At $z=100$ the trace back volume is $V^{\\rm zoom}\\simeq(2.1\\,{\\rm Mpc}\/{\\rm h})^{3}$. Using \\textlcsc{music} we recalculate the ICs, by generating 3 additional level of refinement. For such multi-mass set-up, the finer DM resolution is $m_{dm}^{\\rm zoom} = 6.7\\times 10^{4} \/{\\rm h}\\,{\\rm M}_{\\odot}$, that corresponds to a spatial resolution of $\\Delta x^{\\rm zoom} = 9.7 \/{\\rm h}\\,{\\rm kpc}$. We note that because of the traced back volume, our simulation is expected to probe not only the target halo, but also its satellites and environment, similar to other works (e.g. \\citealt{fiacconi:2015}, where the target halo is chosen at $z\\simeq 3$).\n\nIn the zoom-in simulation $\\Delta x^{\\rm zoom}$ corresponds to our coarse grid resolution, and we allow for 6 additional refinement levels, based on a Lagrangian mass threshold-based criterion. At $z=6$, the baryonic component of the selected halo has a mass resolution of $m_{b} = 1.8\\times 10^{4}{\\rm M}_{\\odot}$ and a physical resolution of $\\Delta x^{\\rm min} = 31.9\\,{\\rm pc}$. For convenience, a summary of the resolution outline can be found in Tab. \\ref{tagella_res}. Note that the refined cell of our simulations have mass and size typical of molecular clouds \\citep[MC, e.g.][]{gorti:2002apj,federrath:2013}.\n\nIn the present paper we refer to metallicity ($Z$) as the sum of all the heavy element species without differentiating among them, and assume solar abundance ratios \\citep{asplund:2009ara&a}. In the IC, the gas is characterized by a mean molecular weight $\\mu = 0.59$, and has metallicity floor $Z=Z_{\\rm floor}>0$. The metallicity floor mimics the pre-enrichment of the halo at high-$z$, when we do not have the resolution to follow precisely star formation and gas enrichment. We set $Z_{\\rm floor}=10^{-3}{\\rm Z}_{\\odot}$, a level that is compatible with the metallicity found at high-$z$ in cosmological simulations for diffuse enriched gas \\citep{dave:2011mnras,pallottini:2014_sim,maio:2015}. Note that such low metallicity only marginally affects the gas cooling time, but is above the critical metallicity for formation of Population III stars. Additionally, a posteriori, we have found that the metallicity floor contribute for only $\\lsim 0.2\\%$ of the total metal mass produced by stars by $z=6$ in the refined region.\n\n\\subsection{Star formation model}\\label{sec_model_sf}\n\nWe model star formation (SF) by assuming a ${\\rm {H_2}}$~dependent Schmidt-Kennicutt relation \\citep{schmidt:1959apj,kennicutt:1998apj}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/kmt09_test.pdf}\n\\caption{\n${\\rm {H_2}}$~fraction ($f_{\\rm H2}$) as a function of gas density ($n$) obtained using the \\citetalias{krumholz:2009apj} model (eqs. \\ref{eq_fh2_full}). Different solid lines correspond to different metallicity ($Z$) of the gas. Horizontal dotted grey lines mark $f_{\\rm H2}$ values of $0.5$ and $1$. Vertical dashed lines indicate the critical density $n_{c}$ where $f_{\\rm H2}=0.5$ for different $Z$; these critical density values are obtained as a fit (eq. \\ref{eq_critical_density}) to the \\citetalias{krumholz:2009apj} model (eq. \\ref{eq_fh2_full}). See the text for details.\n\\label{fig_kmt_test}}\n\\end{figure}\n\\begin{subequations}\\label{eq_sfr_tot}\n\\be\\label{eq_sfr1}\n\\dot{\\rho}_{\\star}=f_{\\rm H2} \\rho\/t_{\\rm sf}\\,\n\\ee\nwhere $\\dot{\\rho}_{\\star}$ is the local SF rate ($SFR$) density, $f_{\\rm H2}$ the molecular hydrogen fraction, $\\rho$ the gas density and $t_{\\rm sf}$ the SF time scale. In eq. \\ref{eq_sfr1} we assume the SF time scale to be proportional to the free-fall time, i.e.\n\n\\be\\label{eq_sfr2}\nt_{\\rm sf} = \\zeta_{\\rm sf}^{-1} \\sqrt{3\\pi\/(32\\,G\\rho)} \\,,\n\\ee\n\\end{subequations}\nwhere $\\zeta_{\\rm sf}$ describes the SF efficiency and it is treated as a parameter in the present work \\citep[cf.][see discussion in Sec. \\ref{sez_sfr_efficiency}]{semenov:2015}. To calculate $f_{\\rm H2}$ we adopt the \\citetalias{krumholz:2009apj} model \\citep{krumholz:2008apj,krumholz:2009apj,mckee:2010apj}. Such model considers ${\\rm {H_2}}$~formation on dust grains by computing radiative transfer on a idealized MC and assumes equilibrium between formation and dissociation rate of ${\\rm {H_2}}$. The solution for $f_{\\rm H2}$ can be approximated as\n\n\\begin{subequations}\\label{eq_fh2_full}\n\\begin{align}\nf_{\\rm H2} &= \\left(1 -0.75\\,s\/(1+0.25\\,s) \\right)\\Theta(2-s)\\\\\ns          &= \\ln\\left(1+0.6\\,\\chi +0.01\\chi^{2}\\right) \/0.6\\,\\tau_{\\rm uv}\\label{eq_dust_optical_depth}\\\\\n\\chi       &= 71\\, \\left(\\sigma_{d,21}\/\\mathcal{R}_{-16.5}\\right)\\,\\left((G\/G_{0})\/(n\/{\\rm cm}^{-3})\\right)\\,,\\label{eq_chi_full}\n\\end{align}\nwhere $\\Theta$ is the Heaviside function, $\\tau_{\\rm uv}$ the dust optical depth of the cloud, $\\sigma_{d}^{-21}=\\sigma_{d}\/10^{-21}{\\rm cm}^{-2}$ is the dust absorption cross section \\citep{li_draine:2001apj}, $\\mathcal{R}\/10^{-16.5}{\\rm cm}^{3}\\,{\\rm s}^{-1} $ is the formation rate coefficient of ${\\rm {H_2}}$~on dust grains \\citep{wolfire:2008apj}, $G$ is the FUV flux in the Habing band ($6-13.6\\,{\\rm eV}$) normalized to the average Milky Way (MW) value $G_{0}$ \\citep{habing:1968,draine:1978apjs}, and $n$ is the hydrogen number density. As in \\citetalias{krumholz:2009apj}, we calculate the dust optical depth by linearly rescaling the MW value, i.e. $\\tau_{\\rm uv} = 10^{-21}{\\rm cm}^{-2} N_{H}\\, Z\/{\\rm Z}_{\\odot} \/\\mu$, where $N_{H}$ is the hydrogen column density and $\\mu$ the mean molecular weight. In the simulation, the column density is calculated as $N_{H}= n\\,l_{\\rm cell}$; because of the mass threshold-based criterion used as a refinement in AMR, we expect $l_{\\rm cell} \\propto n^{-1\/3}$, thus $N_{H} \\propto n^{2\/3}$.\n\nNote that both $\\sigma_{d}$ and $\\mathcal{R}$ are proportional to the dust mass, that we assume to be proportional to the metallicity. Then the ratio between $\\sigma_{d}$ and $\\mathcal{R}$ is independent of $Z$. Additionally, eq. \\ref{eq_fh2_full} can be simplified by assuming pressure equilibrium between the CNM and WNM. In this case, eq. \\ref{eq_chi_full} turns out to be independent on $G\/G_{0}$ and can be written as \\citep{krumholz:2009apj}\n\\be\\label{eq_sfr_last}\n\\chi = 0.75\\,\\left(1+3.1\\,(Z\/{\\rm Z}_{\\odot})^{0.365}\\right)\\,.\n\\ee \n\\end{subequations}\nAs shown in \\citep{krumholz:2011apj}, for $Z\\gsim10^{-2}{\\rm Z}_{\\odot}$ such approximation gives ${\\rm {H_2}}$~fractions compatible with those resulting from a full non-equilibrium radiative transfer calculations.\n\nIn Fig. \\ref{fig_kmt_test} we plot $f_{\\rm H2}$ from the \\citetalias{krumholz:2009apj} model as a function of the gas density. Different solid lines refer to different metallicity. At a fixed metallicity, the molecular fraction as a function of density vanishes for low values of $n$; it steeply rises up to $f_{\\rm H2} \\sim 0.8$ in one density dex and asymptotically reaches $f_{\\rm H2} = 1$. The critical density where the gas can be considered molecular ($f_{\\rm H2}=0.5$) is roughly inversely proportional to the metallicity, i.e. $n_{c} \\sim 25 (Z\/{\\rm Z}_{\\odot})^{-1}{\\rm cm}^{-3}$ \\citep[see also][]{agertz:2012arxiv}. We note that when detailed chemistry calculations are performed, such critical density depends on the chemical network and the assumptions regarding gas shielding from external radiation and clumpiness. As a consequence, the actual critical density can be higher that the one predicted by the \\citetalias{krumholz:2009apj} model \\citep[e.g.][]{bovino:2015arxiv}.\n\nBecause of the particular shape of the $f_{\\rm H2}(n)$ relation, the adopted SF law (eqs. \\ref{eq_sfr_tot}--\\ref{eq_fh2_full}) is roughly equivalent to a prescription based on a density threshold criterion:\n\\begin{subequations}\\label{eqs_sfr_equivalence}\n\\be\n\\dot{\\rho}_{\\star}=\\Theta(n - n_{c}) m_p\\,n \/t_{\\rm sf}\\,,\n\\ee\nwhere $m_p$ is the proton mass and the critical density\n\\be\\label{eq_critical_density}\nn_{c} \\simeq 26.45 \\, (Z\/{\\rm Z}_{\\odot})^{-0.87} {\\rm cm}^{-3}\\,\n\\ee\n\\end{subequations}\nis calculated as a fit to the $f_{\\rm H2}$ \\citetalias{krumholz:2009apj} model. In Fig. \\ref{fig_kmt_test}, we show $n_c$ for various metallicities (dashed vertical lines).\n\nEqs. \\ref{eqs_sfr_equivalence} are not used to calculate the $SFR$ in the simulation. However, being simpler, such formulation can be used to enhance our physical intuition of the adopted SF law\\footnote{As a consequence of the rough equivalence, it is not necessary to manually prevent SF in underdense regions, by imposing that an overdensity $\\Delta>200$ is needed to form stars. At the start of the simulation ($z=100$), the mean density of the gas is $\\sim 0.1\\,m_p\\,{\\rm cm}^{-3}$, while the \\quotes{effective} SF threshold would be $n_{c} \\sim 10^4{\\rm cm}^{-3}$ for gas at $Z=Z_{\\rm floor}$.\\label{footnote_sfr_equivalence}} in analyzing the results. As noted in \\citet[][]{hopkins:2013arxiv}, the morphology of a galaxy is very sensitive to the minimum density of the cells that are able to form star.\n\nDuring the simulation, eqs. \\ref{eq_sfr_tot} are solved stochastically, by drawing the mass of the new star particles from a Poisson distribution \\citep{rasera:2006,dubois:2008,pallottini:2014_sim}. We impose that no more than half of a cell mass can be converted into a star particle in each event. This prescription ensures the numerical stability of the code \\citep{dubois:2008}. This is also consistent with the picture that nearly half of the mass in a MC is Jeans unstable \\citep{federrath:2013}.\n\nWe allow SF only if the mass of a new star particle is at least equal to the baryon mass resolution. This avoids numerical errors for the star particle dynamics and enables us to treat the particle as a stellar population with a well sampled initial mass function (IMF). Additionally, the SF law is driven by ${\\rm {H_2}}$~formation on dust grains, we do not allow gas to form stars if the dust temperature is larger than $\\simeq2\\times 10^{3}$, because of dust sublimation (see Sec. \\ref{sezione_blast} and App. \\ref{app_rad_press} for the details on the dust prescriptions).\n\nFor the present work we assume a SF efficiency $\\zeta_{\\rm sf}=10\\%$, in accordance with the average values inferred from MC observations \\citep[][see also \\citealt{agertz:2012arxiv}]{murray:2011apj}. Note that varying the parameters for the SF law should lead to similar $SFR$ once feedback are properly included, although the galaxy morphology can be different \\citep{hopkins:2013arxiv}.\n\n\\subsection{Mass and energy inputs from stars}\\label{sec_stellar_inputs}\n\nBecause of the finite mass resolution, it is necessary to introduce (according to eqs. \\ref{eq_sfr_tot}--\\ref{eq_sfr_last}) ``star particles'' to represent stellar populations. To this aim, we adopt a \\citet{kroupa:2001} IMF\n\\begin{subequations}\n\\begin{align}\n\\Phi(m)\\propto & \\left[m^{-\\alpha_{1}} \\Theta(m_{1}-m)\\right.\\label{eq_imf}\\\\\n+& \\left. m^{-\\alpha_{2}} \\Theta(m-m_{1}) m_{1}^{\\alpha_{2}-\\alpha_{1}} \\right]\\,,\\nonumber\n\\end{align}\nwhere $\\alpha_{1}= 1.3$, $\\alpha_{2}= 2.3$, $m_{1} = 0.5\\,{\\rm M}_{\\odot}$, and $m$ is in the range $[10^{-1}-10^{2}]{\\rm M}_{\\odot}$. The proportionality constant is chosen such that\n\\be\n\\int_{ 0.1\\,{\\rm M}_{\\odot}}^{100\\,{\\rm M}_{\\odot}} m\\Phi\\,{\\rm d}m=1\\, .\n\\ee\n\\end{subequations}\n\nOnce formed, stars affect the environment with chemical, mechanical and radiative feedback. These stellar inputs are parameterized by the cumulative fraction of the returned gas mass, metals and energy \\citep[e.g.][]{salvadori:2008mnras,debennassuti2014mnras,salvadori:2015}. Mass and energy inputs are conveniently expressed per unit stellar mass formed ($M_{\\star}$).\n\nChemical feedback depends on the return fraction ($R$) and the yield ($Y$):\n\\begin{subequations}\\label{eqs_stellar_inputs}\n\\begin{align}\\label{eqs_def_R_Y}\n  R(t_{\\star})\t=&\\int_{m(t_{\\star}) }^{100\\,{\\rm M}_{\\odot}} (m-w) \\Phi\\,{\\rm d}m\\\\\n  Y(t_{\\star})\t=&\\int_{m(t_{\\star}) }^{100\\,{\\rm M}_{\\odot}} m_{Z} \\Phi\\,{\\rm d}m\\,,\n\\end{align}\nwhere $w(m,Z_{\\star})$ and $m_{Z}(m,Z_{\\star})$ are the stellar remnant and the metal mass produced for a star of mass $m$ and metallicity $Z_{\\star}$ \\citep[e.g.][]{woosley:1995apjs,vandenhoek:1997a&as}, and $m(t_{\\star})$ is the minimum stellar mass with lifetime\\footnote{Stellar lifetimes are roughly independent of metallicity for $Z_{\\star}>10^{-4}{\\rm Z}_{\\odot}$ \\citep[][see eq. 3]{raiteri:1996eq3}.} shorter than $t_{\\star}$, the time elapsed from the creation of the stellar particle (i.e. the \\quotes{burst age}).\n\nThis approach is used both in zoom galaxy simulations \\citep[e.g.][]{agora:2013arxiv} and cosmological simulations \\citep[e.g.][hereafter \\citetalias{pallottini:2014_sim}]{pallottini:2014_sim}. Compared to cosmological simulations, though, zoom simulations have typically a better spatial and -- consequently -- time resolution (e.g. $\\Delta t\\sim 10^{-2}\\,\\rm Myr$ vs $\\Delta t\\sim \\rm Myr$). Thus, here we can follow the gradual release of both gas and metals in the ISM.\n\nThe mechanical energy input includes SN explosions and winds, either by OB or AGB stars in young ($< 40\\,\\rm Myr$) or evolved stellar populations:\n\\begin{align}\\label{eqs_def_mec_energy}\n  \\epsilon_{\\rm sn}(t_{\\star}) =&\\int_{m(t_{\\star})>8\\,{\\rm M}_{\\odot}}^{40\\,{\\rm M}_{\\odot} }\te_{\\rm sn}\\Phi\\,{\\rm d}m,\\\\\n  \\epsilon_{\\rm w}(t_{\\star})  =&\\int_{m(t_{\\star})}^{100\\,{\\rm M}_{\\odot} }\t \te_{\\rm w}\\Phi\\,{\\rm d}m\\,,\n\\end{align}\nwhere $e_{\\rm sn}=e_{\\rm sn}(m,Z)$ and $e_{\\rm w}=e_{\\rm w}(m,Z)$ are the energy released by SN and stellar winds in units of $10^{51}{\\rm erg}\\equiv{\\rm 1 foe}$; we have further assumed that only stars with $8 \\leq m\/{\\rm M}_{\\odot}\\leq40$ can explode as SN.\n\nRadiative energy inputs can be treated within a similar formalism. The cumulative energy $\\epsilon_{12}$ associated to the spectral range $(\\lambda_{1}, \\lambda_{2})$ can be written as\n\\begin{align}\\label{eqs_def_rad_energy}\n  \\epsilon_{\\rm 12}(t_{\\star}) \t=&\\int_0^{t_{\\star}}\\int_{m(t)}^{100\\,{\\rm M}_{\\odot} } L_{12}\\Phi\\,{\\rm d}m\\,{\\rm d}t\\\\\n  L_{\\rm 12}(t) \t\t=& \\int_{\\lambda_{1}}^{\\lambda_{2}}L_{\\lambda}{\\rm d}{\\lambda}\\,,\n\\end{align}\n\\end{subequations}\nwhere $L_{\\lambda}=L_{\\lambda}(m,Z_{\\star})$ is the luminosity per unit wavelength and mass. For convenience, we express the radiation energy in units of ${\\rm foe}$, as for the mechanical energy (eqs. \\ref{eqs_def_mec_energy}). In the following we specify $\\epsilon_{\\rm 12}$ in eq. \\ref{eqs_def_rad_energy}, by separately considering ionizing radiation ($\\lambda_{1}=0$, $\\lambda_{2}=912\\,\\textrm{A\\kern -1.3ex\\raisebox{0.6ex}{$^\\circ$}}$) denoted by $\\epsilon_{\\rm ion}$, and the soft UV band, $\\epsilon_{\\rm uv}$, defined as the range ($\\lambda_{1}=912\\,\\textrm{A\\kern -1.3ex\\raisebox{0.6ex}{$^\\circ$}}$, $\\lambda_{2}=4000\\,\\textrm{A\\kern -1.3ex\\raisebox{0.6ex}{$^\\circ$}}$).\n\nIn eqs. \\ref{eqs_stellar_inputs}, the quantities $w$, $m_{Z}$, $e_{\\rm sn}$, $e_{\\rm w}$, and $L_{\\lambda}$ can be calculated from stellar evolutionary models. We adopt the {\\tt padova} \\citep{padova:1994} stellar tracks for metallicities $Z_{\\star}\/{\\rm Z}_{\\odot} = 0.02,\\, 0.2,\\, 0.4,{\\rm and}\\, 1$ to compute the chemical\\footnote{Similarly to \\citet{agora:2013arxiv}, when computing the yields in eq. \\ref{eqs_def_R_Y}, we assume that the metal mass is linked to the oxygen and iron masses via $m_{Z}= 2.09\\,m_{\\rm O} + 1.06\\,m_{\\rm Fe}$, as appropriate for \\citet{asplund:2009ara&a} abundances.}, mechanical and radiative inputs using \\textlcsc{starburst99} \\citep{starburst99:1999,starburst99:2010apjs}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/stellar_inputs.pdf}\n\\caption{\nStellar inputs (cumulative fraction) as a function of stellar age ($t_{\\star}$). Shown are the return fraction ($R$), metal yield ($Y$), SN mechanical energy ($\\epsilon_{\\rm sn}$), wind mechanical energy ($\\epsilon_{\\rm w}$), ionizing radiation energy ($\\epsilon_{\\rm ion}$), and UV radiation energy ($\\epsilon_{\\rm uv}$).\nThe fractions are given per unit stellar mass formed; energies are expressed in units of $10^{51}{\\rm erg}\\equiv{\\rm foe}$.\nCumulative fractions are indicated with a different colours, as indicated in the legend: the shaded regions cover the $0.02\\leq Z_{\\star}\/{\\rm Z}_{\\odot}\\leq1$ metallicity range; dark lines denote single metallicity {\\tt padova} stellar tracks \\citep{padova:1994}.\nTo guide the eye, the SN explosion period is bracketed by vertical dashed lines; in the upper axis we report the value of $m(t_{\\star})$, the minimum stellar mass corresponding to the stellar lifetime $t_{\\star}$. For definitions, see eqs. \\ref{eqs_stellar_inputs}.\n\\label{fig_gamete_tables}}\n\\end{figure}\n\nIn Fig. \\ref{fig_gamete_tables} we plot $R$, $Y$, $\\epsilon_{\\rm sn}$, $\\epsilon_{\\rm w}$, $\\epsilon_{\\rm ion}$ and $\\epsilon_{\\rm uv}$ as a function of $t_{\\star}$. For each curve the shaded regions denote the $0.02\\leq Z_{\\star}\/{\\rm Z}_{\\odot}\\leq1$ metallicity range; single $Z_{\\star}$ tracks are indicated with dark lines. The time interval during which massive stars can explode as SN ($0.8 \\lsim \\log t_{\\star}\/\\rm Myr\\lsim 1.6$) is highlighted with vertical dashed lines, and the upper axis is labelled with the corresponding stellar mass.\n\nNote that the OB stars contribution ($\\log t_{\\star}\/\\rm Myr\\lsim 0.8$) to $\\epsilon_{\\rm w}$, $Y$ and $R$ is roughly proportional to $t_{\\star}$ and $Z_{\\star}$ (see also \\citealt{agertz:2012arxiv}, in particular eqs. 4). As in the simulation the metallicity floor is set to $Z_{\\rm floor}=10^{-3}{\\rm Z}_{\\odot}$, we slightly overestimate the wind contribution for low $Z_{\\star}$.\n\nFinally, note that the change of behavior of $\\epsilon_{\\rm uv}$ at $\\log t_{\\star}\/\\rm Myr\\lsim 2$ is due to the ionizing ($\\lambda\\leq\\,912\\,\\textrm{A\\kern -1.3ex\\raisebox{0.6ex}{$^\\circ$}}$) photon production suppression. At late times ($\\log t_{\\star}\/\\rm Myr\\gsim 1.6$), AGB stars give a negligible mechanical energy contribution ($\\epsilon_{\\rm w}\\simeq{\\rm constant}$) but return mass and metals to the gas ($R$, $Y$).\n\n\\subsection{Stellar feedback}\\label{sezione_blast}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/feedback_fractions.pdf}\n\\caption{\nExample of the adopted feedback model. Fractional energy evolution for a single SN explosion ($E_{0}=1\\,{\\rm foe}$) in a gas characterized by $n=1\\,{\\rm cm}^{-3}$ and $Z=10^{-3}{\\rm Z}_{\\odot}$ as a function of the time interval from the explosion $\\Delta t$. We plot the total ($f=f_{\\rm th}+f_{\\rm kn}$), thermal ($f_{\\rm th}$), and kinetic ($f_{\\rm th}$) energy fraction acquired by the gas (see eq. \\ref{sn_energy_gas_equation}) with solid black, dashed red and dotted blue lines, respectively.\nShaded regions indicate different stages of the SN evolution, i.e. the energy conserving Sedov-Taylor (ST) stage, shell formation (SF) stage, and pressure driven snowplow (PDS).\nIn the adopted formalism, the initial energy $E_{0}$ is a function of the stellar input (see Sec. \\ref{sec_stellar_inputs} and eq. \\ref{eqs_def_mec_energy}), e.g. $E_{0}=[\\epsilon_{\\rm sn}(t_{\\star}+\\Delta t)-\\epsilon_{\\rm sn}(t_{\\star})] M_{\\star}$. The full model is presented in App. \\ref{app_blastwave}.\n\\label{fig_blastwave_sketch}}\n\\end{figure}\n\nEqs. \\ref{eqs_stellar_inputs} provide us with the energy produced by stars in different forms. The next step is to understand what fraction of that energy is eventually deposited in the ISM. Consider a stellar population of initial mass $M_{\\star}$, metallicity $Z_{\\star}$ and age $t_{\\star}$ residing in a gas cell with volume $V_{\\rm cell}$. In our scheme, when the simulation evolves for a time $\\Delta t$, the chemical feedback act as follows:\n\n\\begin{subequations}\n\\begin{align}\n  \\rho \t&= \\rho + \\left[R(t_{\\star}+\\Delta t)-R(t_{\\star})\\right] M_{\\star}\/V_{\\rm cell}\\\\\n  Z    \t&= Z \t+ \\left[Y(t_{\\star}+\\Delta t)-Y(t_{\\star})\\right] M_{\\star}\/V_{\\rm cell}\\,,\n\\end{align}\nwhere $\\rho$ and $Z$ are the the gas density and metallicity and $R$ and $Y$ are taken from eqs. \\ref{eqs_def_R_Y}. Note that chemical enrichment is due both to the SN and AGB winds.\n\n\\subsubsection{Supernova explosions}\n\nFor the mechanical feedback, let us first consider the case of SNe. At each SN event the specific energy of the gas changes as\n\n\\begin{align}\\label{sn_energy_gas_equation}\n e_{\\rm th} &= e_{\\rm th}+ f_{\\rm th} \\left[\\epsilon_{\\rm sn}(t_{\\star}+\\Delta t)-\\epsilon_{\\rm sn}(t_{\\star})\\right] M_{\\star}\/V_{\\rm cell}\\\\\n e_{\\rm nth}&= e_{\\rm nth}+ f_{\\rm kn} \\left[\\epsilon_{\\rm sn}(t_{\\star}+\\Delta t)-\\epsilon_{\\rm sn}(t_{\\star})\\right] M_{\\star}\/V_{\\rm cell}\\,,\n\\end{align}\n\\end{subequations}\nwhere $e_{\\rm th}$ and $e_{\\rm nth}$ are the thermal and non-thermal energy densities, and $f_{\\rm th}$ and $f_{\\rm kn}$ are the fractions of thermal and kinetic energy deposited in the ISM. Thus, $e_{\\rm nth}$ accounts for the momentum injection by SN and $e_{\\rm th}$ for the thermal pressure part.\n\nIn the present work, we have developed a novel method to compute such quantities. The method derives $f_{\\rm th}$ and $f_{\\rm kn}$ from a detailed modelling of the subgrid blastwave evolution produced by the SN explosion. We calculate $f_{\\rm th}$ and $f_{\\rm kn}$ by evaluating the shock evolution at time $\\Delta t$, the time step of the simulation\\footnote{The underlying assumption is that the shock fronts exit the cell in $\\lsim\\Delta t$. This is quite consistent because the shock is expected to be supersonic, and the sound crossing time is larger or comparable with the simulation time step $\\Delta t$, dictated by the Courant-Friedrichs-Lewy conditions.}.\n\nThe adopted blastwave model is based on \\citet[][hereafter \\citetalias{ostriker:1988rvmp}]{ostriker:1988rvmp}, and it accounts for the evolution of the blast through its different evolutionary stages (energy conserving, momentum conserving, etc.). While each stage is self-similar, the passage from one stage to the next is determined by the cooling time. Thus, $f_{\\rm th}$ and $f_{\\rm kn}$ depends on the blastwave evolutionary stage. The latter, in turn depends on the gas density, cooling time, and the initial energy of the blast ($E_{0}=[\\epsilon_{\\rm sn}(t_{\\star}+\\Delta t)-\\epsilon_{\\rm sn}(t_{\\star})] M_{\\star}$, in eq. \\ref{eqs_def_mec_energy}).\n\nThe model details are presented in App. \\ref{app_blastwave}. As an example, in Fig. \\ref{fig_blastwave_sketch}, we show the energy evolution for a single SN explosion ($E_{0}=1\\,{\\rm foe}$) in a gas characterized by $n=1\\,{\\rm cm}^{-3}$ and $Z=10^{-3}{\\rm Z}_{\\odot}$. The total energy $E(t)$ is constant in the Sedov-Taylor (ST) stage, it decrease down to $0.5\\,E_{0}$ during the shell formation (SF) stage, and it evolves as $\\Delta t^{-2\/7}$ in the pressure driven snowplow (PDS) stage (see eq. \\ref{eq_energy_shock}). In the ST stage most of the energy is thermal, i.e. $f_{\\rm kn}\/f_{\\rm th}\\simeq 0.4$; however, in the SF stage $f_{\\rm kn}$ increases, since part of the thermal energy is radiated away and some is converted into kinetic form\\citep[e.g.][]{cox:1972apj,cioffi:1988apj}. Finally, during the PDS stage the ratio of thermal to kinetic is $f_{\\rm kn}\/f_{\\rm th}\\simeq 2$ (see eqs. 6.14 in \\citetalias{ostriker:1988rvmp}).\n\nIn this particular example -- a $1\\,{\\rm foe}$ SN exploding in a $n=1\\,{\\rm cm}^{-3}$ cell -- by assuming a simulation time step of $\\Delta t\\simeq 10^{-2}\\rm Myr$, we find that the blastwave is in the PDS stage, and the gas receives (via eqs. \\ref{sn_energy_gas_equation}) a fraction of energy $f_{\\rm th} \\simeq 8\\%$ and $f_{\\rm kn} \\simeq 16\\%$ in thermal and kinetic form, respectively. During $\\Delta t$, about $\\simeq 75$\\% of the initial SN energy has been either radiated away or lost to work done by the blastwave to exit the cell. The model is in broad agreement with other more specific numerical studies \\citep[e.g.][]{cioffi:1988apj,walch:2015mnras,martizzi:2015mnras}.\n\n\\subsubsection{Stellar winds}\n\nStellar winds are implemented in a manner paralleling the above scheme for SNe. The energy variation can be calculated via eq. \\ref{eqs_def_mec_energy}, where $\\epsilon_{\\rm sn}$ is substituted with $\\epsilon_{\\rm w}$, given in eqs. \\ref{sn_energy_gas_equation}. Then, $f_{\\rm th}$ and $f_{\\rm kn}$ for winds are calculated via a stage scheme similar to SN. The main difference in the efficiency factors calculation depends on the mode of energy production, i.e. impulsive for SNe, continuous for winds. The complete scheme is detailed in App. \\ref{app_blastwave}.\n\nThe efficiency of SN is greatly increased when the gas is pre-processed by stellar winds \\citep{walch:2015mnras,fierlinger:2016}, since the energy loss process is highly non-linear \\citep[][see Fig. 8]{fierlinger:2016}. For example, when a SN explodes in the lower density bubble produced by the stellar progenitor wind, the adiabatic phase lasts longer and consequently $f_{\\rm kn}$ and $f_{\\rm th}$ increase considerably. \n\n\\subsubsection{Radiation pressure}\\label{sec_rad_press}\n\nFinally, we account for radiation pressure from stars. The coupling of the gas with the radiation can be expressed in terms of $\\dot{p}_{rad}$, the rate of momentum injection \\citep{krumholz:2009radpress,hopkins:2011mnras,krumholz:2012radpress,wise:2012radpres,agertz:2012arxiv}, and accounts for the contribution from ionization, and from dust UV heating and IR-trapping\n\n\\begin{subequations}\n\\begin{align}\\label{eq_rad_moment_injection}\n\\dot{p}_{rad} =&  (L_{\\rm ion}\/c)(1-\\exp(-\\tau_{\\rm ion})) \\\\\n              +&   (L_{\\rm uv}\/c)((1-\\exp(-\\tau_{\\rm uv})) +f_{\\rm ir} )\\,,\\nonumber \n\\end{align}\nwhere $c$ is the speed of light, $\\tau_{\\rm ion}$ the hydrogen optical depth to ionizing radiation, and $f_{\\rm ir}$ is the term accounting for the IR-trapping. $L_{\\rm ion}$ and $L_{\\rm uv}$ are calculated by integration of the stellar tracks (eqs \\ref{eqs_def_rad_energy}). The calculation of $\\tau_{\\rm uv}$ is modelled in Sec. \\ref{sec_model_sf} (eq. \\ref{eq_dust_optical_depth} and related text). We compute $\\tau_{\\rm ion}$ and $f_{\\rm ir}$ according to the physical properties of the gas, as detailed in App. \\ref{app_rad_press}. Note that we do not assume, as sometimes done, $\\tau_{\\rm ion}\\sim \\tau_{\\rm uv}\\gg1$, i.e. we allow for the possibility that some LyC photons can escape.\n\nIn smoothed particle hydrodynamics (SPH) codes, radiation pressure (eq. \\ref{eq_rad_moment_injection}) can be implemented as a \\quotes{kick} \\citep[e.g.][]{hopkins:2011mnras,barai:2015mnras}. Namely, a velocity $\\Delta v = \\dot{p}_{rad}\\Delta t\/m_b$ is directly added to some of the SPH particles of mass $m_b$ near the photon source. The particles that receive kicks are statistically chosen according to a probability $\\mathcal{P}_{\\rm kick}$, and with kick direction $\\hat{v}$ that is sampled from a random distribution. Considering the specific kinetic energy of the SPH particles, we would have\n\\be\ne_k = 0.5\\,\\left\\langle m_b (\\mathbf{v} + \\Delta v \\mathcal{P}_{\\rm kick}\\mathbf{\\hat{v}} )^{2} \\right\\rangle\/V_{cell}\n\\ee\nwhere $\\mathbf{v}$ is the original particle velocity, the $\\langle\\,\\rangle$ operator indicates the particles sum weighted by the SPH kernel, and $V_{cell}$ is the kernel volume. Thus, because of the kick, the increase of energy density would be\\footnote{In eq. \\ref{eq_red_press_energy_increase}, when going from the first to the second line, note that first terms gives a null contribution, as $\\mathbf{v}$ is ordered motion, while the kicks are randomly oriented via $\\mathbf{\\hat{v}}$, and that, by definition, $\\langle \\mathcal{P}_{\\rm kick}\\rangle = 1$.}\n\n\\begin{align}\\label{eq_red_press_energy_increase}\n\\Delta e_k &= \\langle m_b\\, \\mathcal{P}_{\\rm kick} (\\Delta v\\mathbf{v}\\mathbf{\\hat{v}} + 0.5(\\Delta v)^{2} \\rangle\/V_{cell} \\nonumber\\\\\n           &= 0.5\\, m_b \\, (\\Delta v)^2\/V_{cell} \\\\\n           &= 0.5\\,(\\dot{p}_{rad}\\Delta t)^2\/(m_b\\,V_{cell}) \\nonumber\\,,\n\\end{align}\n\\end{subequations}\nwhere $\\dot{p}_{rad}$ can be calculated via eq. \\ref{eq_rad_moment_injection}, and eq. \\ref{eq_red_press_energy_increase} can be directly cast into the AMR formalism. Additionally, because of our approximate treatment of IR-trapping (see App. \\ref{app_rad_press}), we force energy conservation: $V_{cell} \\Delta e_k \\leq \\Delta t\\,(L_{\\rm ion} + L_{\\rm uv})$, i.e. the deposited energy must not exceed the radiative input energy. Finally, we recall here that non-thermal energy is dissipated with a time scale $t_{\\rm diss}$, as described in the beginning of Sec. \\ref{sec_numerical}.\n\n\\section{Results}\\label{sec_result}\n\nAt $z=6$ ($t \\simeq 920\\, \\rm Myr$), the simulated zoom-in region contains a group of 15 DM haloes that host galaxies. We target the most massive halo ($M_{\\rm h} = 1.8\\times 10^{11}{\\rm M}_{\\odot}$) that hosts \\quotes{\\emph{Dahlia}}, which is a galaxy characterized by a stellar mass of $M_{\\star}=1.6\\times 10^{10}{\\rm M}_{\\odot}$, therefore representative of a typical LBG galaxy at that epoch. {Dahlia} has 14 satellites located within $\\simeq 100 \\,{\\rm kpc}$ from its centre. The six largest ones have a DM mass in the range $M_{\\rm h} = 2.5\\times 10^{9}{\\rm M}_{\\odot} - 1.2\\times 10^{10}{\\rm M}_{\\odot}$, and they host stars with total mass $M_{\\star}\\lsim 10^{9}{\\rm M}_{\\odot}$. Additionally, there are eight smaller satellites ($M_{\\rm h} \\simeq 10^{7}{\\rm M}_{\\odot}$), with $M_{\\star}\\simeq 10^{5}{\\rm M}_{\\odot}$.\n\n\\subsection{Overview}\\label{sec_res_barions}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.99\\textwidth,height=0.33\\textwidth]{plots_pdf\/maps\/landscape_completo_type3_zoomed_rtdensmap_27.pdf}\n\\vspace{.3pt}\n\n\\includegraphics[width=0.99\\textwidth,height=0.33\\textwidth]{plots_pdf\/maps\/landscape_completo_type3_zoomed_rttemp_27.pdf}\n\\vspace{.3pt}\n\n\\includegraphics[width=0.99\\textwidth,height=0.33\\textwidth]{plots_pdf\/maps\/landscape_completo_type3_zoomed_rtpress_27.pdf}\n\\vspace{.3pt}\n\n\\includegraphics[width=0.99\\textwidth,height=0.33\\textwidth]{plots_pdf\/maps\/landscape_completo_type3_zoomed_rtmetal_27.pdf}\n\\caption{\n(Caption next page.) %\n\\label{fig_mappe_hydro}\n}\n\\end{figure*}\n\\addtocounter{figure}{-1}\n\\begin{figure*}\n\\caption{(Previous page.) %\nMaps of the simulated galaxy {Dahlia} $z=6$. From left to right we plot subsequent zooms on the galaxy. From top to bottom we plot the density ($n$), temperature ($T$), pressure ($P$) and metallicity ($Z$). Each map is obtained\\textsuperscript{\\ref{footnote_pymses}} by mass-averaging the physical quantity along the line of sight piercing the field of view and centred on {Dahlia}. In all panels the physical scale is indicated as an inset. Movies of {Dahlia} can be found at \\url{https:\/\/www.researchgate.net\/profile\/Andrea_Pallottini}.\n}\n\\end{figure*}\n\nWe start by looking at the overall properties of {Dahlia} on decreasing scales. In the following we refer to Fig. \\ref{fig_mappe_hydro}, which shows the simulated density ($n$), temperature ($T$), total (thermal+kinetic) pressure ($P$), and metallicity ($Z$) maps\\footnote{Most of the maps of this paper are obtained with a customized version of \\textlcsc{pymses} \\citep{labadens:2012aspc}, a visualization software that implements optimized techniques for the AMR grid of \\textlcsc{ramses}.\\label{footnote_pymses}}\nat $z=6$.\n\n\\subsubsection{Environment (scale $\\simeq 160$~kpc)}\n\n{Dahlia} sits at the centre of a cosmic web knot and accretes mass from the intergalactic medium (IGM) mainly via 3 filaments of length $\\simeq 100\\,{\\rm kpc}$, confirming previous findings \\citep[][]{dekel:2009nat}. These overdense filaments ($n\\simeq 10^{-2}{\\rm cm}^{-3}$) are slightly colder ($T\\simeq10^{3.5}{\\rm K}$) than the IGM ($\\langle T \\rangle\\simeq10^{4.5}{\\rm K}$) as a consequence of their shorter radiative cooling time ($t_{\\rm cool}\\propto n^{-1}$). Along these cold streams, pockets of shock-heated ($T\\gsim10^{4.5}{\\rm K}$) gas produced by both structure formation and feedback (SN and winds) are visible.\n\nThe galaxy locations can be pinpointed from the metallicity map, showing a dozen of metal-polluted regions. The size of the metal bubbles ranges from $\\simeq20$ kpc in the case of {Dahlia} to a few ${\\rm kpc}$ for the satellites. Bubble sizes increase with the total stellar mass (see \\citetalias{pallottini:2014_sim}, in particular Fig. 13), and age of the galaxy stellar population. \n\nOn these scales, the pressure is dominated by the thermal component ($P\\simeq P_{\\rm th}\\sim 10^4{\\rm K}\\,{\\rm cm}^{-3}$); higher values of pressure, associated to non-thermal feedback effects (e.g. gas bulk motion), are confined around star forming regions, again traced by the metallicity distribution.\n\n\\subsubsection{Circumgalactic medium (scale $\\simeq 50$~kpc)}\\label{sec_CGM}\n\nTo investigate the circumgalactic medium (CGM), we zoom in a region within $\\sim 3\\, r_{\\rm vir} = 47.5$ kpc from {Dahlia}'s centre. On these scales, we can appreciate the presence of several {Dahlia}'s satellites, i.e. extended (few ${\\rm kpc}$) structures that are $\\sim 100$ times denser than the filament in which they reside. Two of these density structures are particularly noticeable. These are located at a distance of $\\sim10~{\\rm kpc}$ from the centre in the upper left and lower left part of the map, respectively. By looking at the metallicity distribution, we find that both satellites reside within their own metal bubble, which is separated from Dahlia's one. This clearly indicates an in-situ star formation activity.\n\nAdditionally, the density map shows about $20$ smaller ($\\sim 10-100\\,{\\rm pc}$) overdense clumps ($n\\gsim 10\\,{\\rm cm}^{-3}$). The ones within {Dahlia}'s metal bubble are enriched to $Z\\simeq {\\rm Z}_{\\odot}$. This high $Z$ value is indicative of in-situ self-pollution, which possibly follows an initial pre-enrichment phase from {Dahlia}. Clumps outside {Dahlia} metal bubble have on average an higher density ($n\\sim 10^{2}{\\rm cm}^{-3}$). Since these clumps are unpolluted, they have not yet formed stars, as the effective density threshold for star formation is $\\sim 25\/(Z\/{\\rm Z}_{\\odot}){\\rm cm}^{-3}$ (see eq. \\ref{eq_critical_density} and Sec. \\ref{sec_model_sf}). Such clumps represent molecular cloud complexes caught in the act of condensing as the gas streams through the CGM \\citep{ceverino:2016MNRAS}. Such clumps have gas mass in the range $10^5 - 10^6 {\\rm M}_{\\odot}$, and are not DM-confined, as the DM density field is flat on their location.\n\nStar forming regions are surrounded by an envelope of hot ($T\\simeq 10^{5.5}{\\rm K}$), diffuse ($n\\gsim 10^{-2}\\,{\\rm cm}^{-3}$) and mildly enriched ($Z\\sim 10^{-2}{\\rm Z}_{\\odot}$) gas produced by SN explosions and winds. In the centre of star forming regions, instead, the gas can cool very rapidly due to the high densities\/metallicities. Nevertheless, these regions are highly pressurized due to bulk motions mostly driven by radiation pressure (see Fig. \\ref{fig_feedback_vs_time}).\n\n\\subsubsection{ISM (scale $ \\simeq 10$~kpc)}\\label{sec_small_scale}\n\nThe structure of Dahlia's ISM emerges once we zoom in a region $\\sim 0.5\\, r_{\\rm vir}$ from its centre. In the inner region ($\\simeq 2\\,{\\rm kpc}$), a counterclockwise disk spiral pattern is visible, since the field of view is perpendicular to the rotation plane of the galaxy (see \\citealt{gallerani:2016outflow} for the analysis of the velocity field of {Dahlia}). The presence of disks in these early systems has already been suggested by other studies. For example, \\citet{feng:2015apj} show that already at $z \\sim 8$ nearly $70\\%$ of galaxies with $M_{\\star}\\simeq 10^{10}{\\rm M}_{\\odot}$ have disks (see also Sec. \\ref{sec_final_results}).\n\nThe spiral central region and the spiral arms are dense ($n\\simeq 10^{2}{\\rm cm}^{-3}$) and cold ($T\\simeq 10^3{\\rm K}$), and the active SF produces a large in-situ enrichment ($Z\\simeq {\\rm Z}_{\\odot}$). Winds and shocks from SN have no effect in the inner part of the galaxy, because of the high density and short cooling time of the gas; this implies that metals remain confined within $\\sim 2\\,{\\rm kpc}$.\n Within spiral arms radiation pressure induced bulk motions largely dominate the total pressure, which reaches values as high as $P\\gsim 10^{6.5}{\\rm K}\\,{\\rm cm}^{-3}$. The imprint of SN shocks is evident in the temperature map in regions with $T\\gsim 10^5{\\rm K}$. Shock driven outflows originated in spiral arms travel outward in the CGM, eventually reaching the IGM if outflow velocities exceed the escape velocity ($\\sim 100\\,{\\rm km}\\,{\\rm s}^{-1}$, see Fig. 4 in \\citealt{gallerani:2016outflow}).\n\nOutflows are either preferentially aligned with the galaxy rotation axis, or they start at the edge of the disk. However, when spherically averaged, infall and outflow rates are nearly equal ($\\sim 30\\,{\\rm M}_{\\odot}\/{\\rm yr}$ at $z\\sim6$, \\citealt{gallerani:2016outflow}), and the system seems to self-regulate \\citep[see also][]{dekel:2014}.\n\nOutside the disk, clumps with density $n\\simeq 10^{2}{\\rm cm}^{-3}$ are also present and are actively producing stars. These isolated star forming MCs are located at a distance $\\gsim 2\\,{\\rm kpc}$ from the centre, and show up as spots of high pressure ($P\\gsim 10^7{\\rm K}\\,{\\rm cm}^{-3}$); some of this MCs are completely disrupted by internal feedback and they can be recognized by the low metallicity ($Z\\sim 10^{-3}{\\rm Z}_{\\odot}$): this is consistent with the outcome of numerical simulations of multiple SN explosions in single MC \\citep[e.g.][]{kortgen:2016}.\n\n\\subsubsection{Radial profiles}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.485\\textwidth]{plots_pdf\/eos\/profile_sph_profile_dahlia_27.pdf}\n\\caption{\nDensity ($n$, blue), metallicity ($Z$, red) and molecular hydrogen density ($n_{\\rm H2}$, yellow) radial profile ($r$) with respect to {Dahlia} centre. The profiles are spherically averaged, as indicated by the $\\langle\\,\\rangle_{V}$ operator, and the upper axis shows the radial distance $r$ as a function of the virial radius of {Dahlia} ($r_{\\rm vir}$).\n\\label{fig_sph_profile}\n}\n\\end{figure}\n\nFig. \\ref{fig_sph_profile} shows spherically averaged density, metallicity, and ${\\rm {H_2}}$~density profiles for the gas. The density profile rapidly decreases from $n\\sim 30\\,{\\rm cm}^{-3}$ at $r\\sim 0$ to $n\\sim 0.1\\,{\\rm cm}^{-3}$ at $r\\sim 6\\,{\\rm kpc} (\\sim 0.5\\,r_{\\rm vir})$, and then flattens at larger distances. Such profile is consistent with the average profile of $z=4$ galaxies presented in \\citetalias{pallottini:2014_sim}. There we claimed that the density profile is universal once rescaled to the halo virial radius (see also \\citealt{liang:2016}). Superposed to the mean density profile, local peaks are clearly visible: they result from individual clumps\/satellites, as discussed above. \n\nThe central metallicity is close to the solar value, but by $r\\sim12\\,{\\rm kpc}\\sim r_{\\rm vir}$ it has already dropped to $Z=Z_{floor}$. Within $0\\lsim r\/{\\rm kpc}\\lsim6$, the metallicity gradient closely tracks the density profile, while for $6\\lsim r\/{\\rm kpc}\\lsim15$ the decrease is steeper. \\citet{pallottini:2014cgmh} find that the metallicity profile is not universal, however it usually extend up to few virial radii, as for {Dahlia}; further insights can be obtained by analyzing the $n$-$Z$ relation (Sec. \\ref{sec_eos}).\n\nIn Fig. \\ref{fig_sph_profile} we note that the $Z$ gradient found in {Dahlia} at $z = 6$ is slightly steeper than the one inferred from observations of $z\\sim 3$ galaxies: i.e. we find $\\Delta Z\/r \\sim -0.1\\, {\\rm dex}\/{\\rm kpc}$ while the observed ones are $\\sim 0\\, {\\rm dex}\/{\\rm kpc}$ \\citep{wuyts:2016} and $\\sim +0.1\\, {\\rm dex}\/{\\rm kpc}$ \\citep{troncoso:2013arxiv1311}. This suggests that the metallicity profile evolve with cosmic time and that the flattening is likely caused by stellar feedback, which in our Dahlia may occur in the following Gyr of the evolution. However, to prove such claim we should evolve the simulation to $z\\sim3$.\n\nThe ${\\rm {H_2}}$~profile is spiky, and each peak marks the presence of a distinct SF region\\footnote{We remind that the profiles are volume-weighted, thus the plotted $n_{\\rm H2}$ accounts for the fact that ${\\rm {H_2}}$~is present only in a fraction of the gas at a given radius.}. In {Dahlia} ${\\rm {H_2}}$~is mainly concentrated within $r\\lsim 0.5\\,{\\rm kpc}$ and it is distributed in the disk-like structure seen in Fig. \\ref{fig_mappe_hydro} (see Sec. \\ref{sec_final_results}). The location of the other peaks correspond to the satellites, which are mostly co-located with metallicity peaks. With increasing metallicity, in fact, lower densities are needed to form ${\\rm {H_2}}$~(eq. \\ref{eq_critical_density}).\n\n\\subsection{Star formation and feedback history}\\label{sec_sfr_result}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/sameplot_dahlia_27.pdf}\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/sfr_dahlia_7.pdf}\n\\caption{\n{\\bf Left}: {Dahlia} and satellites cumulative stellar masses ($M_{\\star}$, upper left panel) and star formation rates ($SFR$, lower left panel) as a function of cosmic time ($t$). For each galaxy, individual $M_{\\star}$ and $SFR$ are plotted with a solid line, coloured accordingly to the total dark matter mass ($M_{\\rm h}$) of the host halo at $z=6$. For both $M_{\\star}$ and $SFR$, {Dahlia}'s tracks are plotted with a blue line, and the totals ({Dahlia}+satellites) are in black. {\\bf Right:} $SFR$ as a function of cosmic time, with individual galaxies defined by the merger history up to $z\\simeq8.5$. Note the different $M_{\\rm h}$ colourbar scale with respect to the left panel. \n\\label{fig_sfr_smf_energy}\n}\n\\end{figure*}\n\nWe analyze the SF history of {Dahlia} and its major satellites by plotting in Fig. \\ref{fig_sfr_smf_energy} the cumulative stellar mass ($M_{\\star}$) and star formation rate ($SFR$) vs. time\\footnote{The $SFR$ is averaged in steps of $\\simeq 3\\,\\rm Myr$. We have checked that smaller steps do not alter the following analysis.}.\n\nFor the whole galaxy sample, the time averaged ($\\pm$ r.m.s.) specific star formation is $\\langle{\\rm sSFR}\\rangle= (16.6 \\pm 32.8)\\,{\\rm Gyr}^{-1}$. This mean value is comparable to that obtained by previous simulations of high-$z$ galaxies \\citep{wise:2012radpres} and broadly in agreement with $z\\sim 7$ observations \\citep{Stark:2013ApJ}. At early times the $sSFR$ reaches a maximum of $\\sim 100\\,{\\rm Gyr}^{-1}$, while a minimum of $3.0\\,{\\rm Gyr}^{-1}$ is found during the late time evolution. Both the large ${\\rm sSFR}$ range and maximum at early times are consistent with simulations by \\citet{shen:2014}. At late times, the $sSFR$ is in agreement with analytical calculation \\citep{behroozi:2013apj}, and with $z=7$ observations \\citep{gonzalez:2010}, although we note {Dahlia} has a larger stellar mass with respect to the galaxies in the sample ($M_{\\star}\\simeq 5\\times 10^9{\\rm M}_{\\odot}$).\n\nAt all times, {Dahlia} dominates both the stellar mass and star formation rate, whose mean value is $\\langle SFR\\rangle \\simeq (35.3 \\pm 32.7)\\,{\\rm M}_{\\odot}\/{\\rm yr}$. Its stellar mass grows rapidly, and it reaches $M_{\\star}\\sim 10^{9}{\\rm M}_{\\odot}$ by $t\\simeq 400\\,{\\rm Myr}$ ($z=11$), i.e. after $\\simeq 120\\,{\\rm Myr}$ from the first star formation event. Such rapid mass build-up is due to merger-induced SF, that plays a major role at high-$z$ \\citep{poole:2016MNRAS,behroozi:2013apj,salvadori2010MNRAS}. The $SFR$ is roughly constant from $z\\sim 11$ to $z\\sim 8.5$ and reaches a maximum of $\\simeq 130\\, {\\rm M}_{\\odot}\/{\\rm yr}$ at $z\\sim 6.7$. With respect to observations of $z\\sim6$ LBG galaxies \\citep[e.g.][]{stanway:2003MNRAS,stark:2009apj} the $SFR$ and $M_{\\star}$ of {Dahlia} are above the mean values, but still consistent within one sigma. Additionally, the combination of $SFR$, $M_{\\star}$, and $Z_{\\star}$ for {Dahlia} are compatible with the fundamental mass metallicity relation observed in local galaxies \\citep{mannucci:2010mnras}.\n\nThe total stellar mass in satellites is $M_{\\star}\\sim 10^{9}{\\rm M}_{\\odot}$. Typically, SF starts with a burst, generating $\\sim 10^{7.5} {\\rm M}_{\\odot}$ of stars during the first $\\simeq 20\\,\\rm Myr$. Then the $SFR$ exponentially declines and becomes intermittent with a bursty duty cycle of $\\sim100\\,\\rm Myr$. This process can be explained as follows. As an halo forms, at its centre the density of the gas slowly rises. When the density is higher than the critical density of ${\\rm {H_2}}$~formation (eq. \\ref{eq_critical_density}), the gas in the inner region is converted into stars in few free-fall times. Then feedback, and in particularly coherent SN explosions ($t_{\\star}\\gsim 10\\,\\rm Myr$, see Fig. \\ref{fig_gamete_tables}), quenches $SFR$, and the star formation activity becomes self-regulated. As mergers supply fresh gas, the $SFR$ suddenly goes out of equilibrium and becomes bursty again. Note that self-regulation is possible only for major satellites, since smaller ones ($M_{\\rm h}\\lsim 10^8{\\rm M}_{\\odot}$) cannot retain a large fraction of their gas following feedback events due to their shallow potential wells (see \\citetalias{pallottini:2014_sim}).\n\nNote that the duty cycle and the amplitude of the burst are fairly in agreement with observations of $M_{\\star}\\sim10^8-10^{10}{\\rm M}_{\\odot}$ galaxies at $z\\lsim0.3$ \\citep{kauffmann:2014mnras}. Furthermore, in our satellites we find that the typical behavior of the burst phases -- starburst - quiescent - post-starburst -- is qualitatively similar to what found by \\citet{read:2016mnras}, that simulate the evolution of a $M_{\\star}\\simeq 10^9{\\rm M}_{\\odot}$ galaxy for $\\simeq 1\\, {\\rm Gyr}$ (see also \\citealt[][]{teyssier:2013mnras,read:2016mnras_b} for further specific studies on the bursty nature of this kind of galaxies).\n\nSince individual galaxies are defined as group of star particles in the same DM halo at $z=6$, the SF history accounts for the sum of all the stars that formed in different progenitors of the considered halo. For comparison, in the right panel of Fig. \\ref{fig_sfr_smf_energy} we plot the $SFR$ of individual halos defined by their merger history at $z=8.7$. Galaxies with active SF at $300-550$ Myr merge into {Dahlia} at a later time, thus they do not appear individually in the left panel of Fig. \\ref{fig_sfr_smf_energy}.\n\nSuperimposed to the global trend, the SF history of {Dahlia} and its satellites fluctuates on time scales of $\\sim 10\\,\\rm Myr$, corresponding to the time scale of energy deposition by feedback \\citep[see e.g.][]{torrey:2016arxiv}.\n\n\\subsubsection{Star formation efficiency}\\label{sez_sfr_efficiency}\n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/eos\/std_eos_ele_semenov_dahlia_27.pdf}\n\\caption{Effective star formation efficiency ($\\zeta_{\\rm sf}\\,f_{\\rm H2}$) vs density ($n$) at $z=6$. The distribution is ${\\rm {H_2}}$~mass weighted; we consider gas within $3\\, r_{\\rm vir} = 47.5$ kpc from {Dahlia} centre.\n\\label{fig_cfr_semenov}}\n\\end{figure}\n\n$\\zeta_{\\rm sf}\\,f_{\\rm H2}$ represents the quantity of gas converted in stars within a free-fall time (see eq. \\ref{eq_sfr_tot}). In Fig. \\ref{fig_cfr_semenov} we plot the effective star formation efficiency ($\\zeta_{\\rm sf}\\,f_{\\rm H2}$) as a function of gas density, weighted by the ${\\rm {H_2}}$~mass fraction at $z=6$. Most of the ${\\rm {H_2}}$~is contained in the range $n=10-100 {\\rm cm}^{-3}$, and the effective efficiency $\\zeta_{\\rm sf}\\,f_{\\rm H2}$ varies from $10^{-3}$ to $10^{-1}$. Since $\\zeta_{\\rm sf}= \\mathrm{const.} =0.1$, the spread is purely due to the dependence of $f_{\\rm H2}$ on density and metallicity (see Fig. \\ref{fig_kmt_test}). Note that by construction $\\zeta_{\\rm sf}\\,f_{\\rm H2}\\leq0.1$, and the plot does not show values very close to such limit, since gas with higher effective efficiency is converted into stars within a few free-fall times (eq. \\ref{eq_sfr2}).\n\nInterestingly, our ${\\rm {H_2}}$-based star formation criterion is reminiscent of a density threshold one, as below $n \\simeq 3\\, {\\rm cm}^{-3}$ the efficiency drops abruptly (eqs. \\ref{eqs_sfr_equivalence}). However, an important difference remains, i.e. in the present model at any given density the efficiency varies considerably as a result of the metallicity dependence. The relation between efficiency and density is also similar to that found by \\citet{semenov:2015} (\\citetalias{semenov:2015}). This is striking because these authors use a star formation efficiency that depends on the turbulent velocity dispersion of the gas, with no notion of the local metallicity. This comparison is discussed further in Sec. \\ref{sec_conclusioni}.\n\n\\subsubsection{Feedback energy deposition}\\label{sec_feedback_res}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/otf_ist_dahlia.pdf}\n\\caption{Rate of energy deposition in the gas, ${\\rm d}E\/{\\rm d}t$, by feedback processes as a function of cosmic time. Different contributions (SN, wind and radiation) are plotted with a different colour, and we additionally distinguish between the kinetic (thick lines) and thermal (thin lines) energy variation. By definition, radiation pressure has no thermal contribution. Note the jump at $t\\simeq500\\,\\rm Myr$ due to the onset of radiation pressure by AGB stars. The upper axis indicates the corresponding redshift.\n\\label{fig_feedback_vs_time}\n}\n\\end{figure}\n\nAs discussed in Sec. \\ref{sezione_blast}, only a small fraction of the available energy produced by stars can couple to the gas. During the simulation, we find that the time average efficiency of the conversion is $f \\sim 0.1\\%$, regardless of the feedback type. These low efficiencies imply that energy is mostly dissipated within MCs where the stars reside and produce it. For SN and winds, such small efficiency is a consequence of the short cooling times in MCs (see also App. \\ref{app_blastwave}). For radiation pressure the efficiency is limited by the relatively small dust optical depths (see also App. \\ref{app_rad_press}).\n\nNote that, typically in simulations \\citep[e.g.][]{wise:2012radpres,agertz:2012arxiv}, energy from stars is directly deposited in the gas, and then dissipation (mostly by radiative losses) occurs during the hydrodynamical time step. Within our scheme, instead, the deposited energy is already dissipated within high density cells, where cooling is important. Nevertheless, this does not appear to determine major differences in, e.g., $SFR$ history and ISM thermodynamics, as discussed in Sec. \\ref{sec_eos}.\n\nIn Fig. \\ref{fig_feedback_vs_time} we plot the energy deposition rate in the gas by various feedback processes as a function of time. Most evidently, \\emph{radiation dominates the energy budget at all times}: $\\dot E_{rad} \\simeq 10^{2} \\dot E_{SN}\\simeq 10^3 \\dot E_{w}$. The ratios of these energy rates somewhat reflect the stellar inputs shown in Fig. \\ref{eqs_stellar_inputs}, although this is not a trivial finding, given that the interplay among different feedback types is a highly non-linear process.\n\nAs expected, the energy deposition rate behaves as $\\dot E \\propto SRF^q$, with $q \\simgt 1$, apart from fluctuations and jumps as the one at $t\\simeq 500\\,\\rm Myr$. The scaling can be understood by simple dimensional arguments. Assume that most of the energy is deposited by radiation pressure. In the optically thick limit, we can combine eqs. \\ref{eq_red_press_energy_increase} and \\ref{eq_rad_moment_injection} to write $\\dot E_{rad} \\Delta t \\simeq (L_{\\rm uv} \\Delta t)^2 \/ (M_{g}\\,c^2)$, where $M_{g}$ is the gas mass accelerated by radiation, and we neglect ionizing radiation. Then, using \\ref{eqs_def_rad_energy}, we can write $\\dot E_{rad} \\propto SFR\\,(M_{\\star}\/M_{g})$. Initially, $M_{g}\\simeq M_{\\star}$, thus $\\dot E_{rad} \\propto SFR$. Once the gas mass is expelled from the star forming region or converted into stars, $M_{g}\\ll M_{\\star}$. Thus the deposition rate increases faster than the $SFR$ and it is very sensitive to the amount of gas mass around the sources.\n\nThe previous argument holds until the gas remains optically thick. This is warranted by AGB metal\/dust production which becomes important after for stellar ages $t_{\\star}\\sim 100\\,\\rm Myr$ (see Fig. \\ref{fig_gamete_tables}). When combined with the parallel increase of UV photons by the same sources, it is easy to interpret the rapid increase of the radiative feedback efficiency at $t\\simeq500\\,\\rm Myr$, i.e. after $\\simeq 200 \\,\\rm Myr$ from the first star formation events in {Dahlia}. We checked this interpretation by looking at the IR-trapping recorded on the fly during the simulation. We find that on average $f_{\\rm ir}\\simeq 10^{-2}$ for $t\\lsim 500\\, \\rm Myr$, and $f_{\\rm ir}\\simeq 0.1$ at later times, thus confirming our hypothesis.\n\nThe energy deposition rates for different feedback types are highly correlated in time (Pearson coefficients $\\gsim 0.7$). This is partially due to the fact that the same stellar population inputs wind, radiation and supernova energy in the gas. Additionally, as we have just seen for the case of AGB star, different types of feedback are mutually dependent. For example, radiation pressure is more effective when the gas is metal and dust enriched by SN and AGB stars; winds and SN can more efficiently couple with low density gas (longer cooling time).\n\nNote that short and intense peaks in energy deposition rate correspond to the complete disruption of multiple MCs. This occurs following strong SF events in small satellites ($M_{\\rm h} \\sim 10^{7}{\\rm M}_{\\odot}$) that cannot retain the gas and sustain a continuous star formation activity.\n\nFinally, we remind that, when compared with observational\/analytical constraints, the $SFR$ and $M_{\\star}$ of {Dahlia} are higher then the mean, but still consistent within one sigma. We caution that this might imply a somewhat weak feedback prescription.\n\n\\subsubsection{Feedback effects on ISM thermodynamics}\\label{sec_eos}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.485\\textwidth]{plots_pdf\/eos\/eos_dahlia_27.pdf}\n\\includegraphics[width=0.485\\textwidth]{plots_pdf\/eos\/eos_pressure_dahlia_27.pdf}\n\n\\includegraphics[width=0.485\\textwidth]{plots_pdf\/eos\/eos_metal_dahlia_27.pdf}\n\\includegraphics[width=0.485\\textwidth]{plots_pdf\/eos\/metal_vs_density_dahlia_27.pdf}\n\\caption{\nEquation of State of the gas within $\\simeq47.5\\,{\\rm kpc}$ $(3\\, r_{\\rm vir})$ from {Dahlia} centre at $z=6$. Each EOS consists in a mass- or metal-weighted probability distribution function (PDF) as specified by the colourbar. We plot the PDF in the $n$-$T$ plane ({\\bf upper left panel}), in the $n$-$P$ plane ({\\bf upper right panel}), the metal-mass weighted PDF in the $n$-$T$ plane ({\\bf lower left panel}), and mass-weighted relation between gas $n$ and $Z$ ({\\bf lower left right}). Mean relations and r.m.s. dispersions are overplotted with solid black and dashed lines, respectively. In the upper horizontal axis of each panel, we indicate the overdensity ($\\Delta$) corresponding to $n$. The density range of rarefied, diffuse and dense phases used in the text are indicated. For the panels on the left, the rarefied gas is additionally divided in \\emph{photo-ionized} ($T<10^{4.5}{\\rm K}$) and \\emph{shock-heated} ($T\\geq10^{4.5}{\\rm K}$). See Tab. \\ref{tagella_eos_riassunto} for a summary of the total values.\n\\label{fig_eos_1}\n}\n\\end{figure*}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lcccc}\n\\hline\\hline\n& mass & rarefied & diffuse & dense \\\\\n\\hline\nGas & $1.3 \\times 10^{10}{\\rm M}_{\\odot}$ & $44\\%$ & $34\\%$ & $22\\%$ \\\\\nMetals & $ 4.2\\times 10^5{\\rm M}_{\\odot}$ & $5\\%$ & $25\\%$ & $70\\%$\\\\\n${\\rm {H_2}}$ & $ 3.6\\times 10^8{\\rm M}_{\\odot}$ & $0\\%$ & $1\\%$ & $99\\%$\\\\\n\\hbox{C~$\\scriptstyle\\rm II $} & $ 2.2\\times 10^5{\\rm M}_{\\odot}$ & $4\\%$ & $22\\%$ & $74\\%$\\\\\n\\end{tabular}\n\\caption{\nSummary of the gas masses for total, metal, \\hbox{C~$\\scriptstyle\\rm II $}, and ${\\rm {H_2}}$~within $\\sim 47.5\\,{\\rm kpc}$ $(3\\, r_{\\rm vir})$ from {Dahlia} center. In the table, we report also the fraction that is contained in different gas phases\\textsuperscript{\\ref{footnote_phases}}: \\emph{rarefied} ($\\log(n\/{\\rm cm}^3)\\leq -1$), \\emph{diffuse} ($-1<\\log(n\/{\\rm cm}^3)\\leq 1$) and \\emph{dense} ($\\log(n\/{\\rm cm}^3)> 1$). Discussion about gas and metal mass is found in Sec. \\ref{sec_eos}; analysis of ${\\rm {H_2}}$~and \\hbox{C~$\\scriptstyle\\rm II $}~is in Sec. \\ref{sec_final_results} (see also App. \\ref{sez_cloudy_model} for \\hbox{C~$\\scriptstyle\\rm II $}~calculation).\n\\label{tagella_eos_riassunto}}\n\\end{table}\n\nFeedback leaves clear imprints in the ISM thermodynamics. For convenience, we classify ISM phases according to their density: we define the gas to be in the \\emph{rarefied}, \\emph{diffuse}, and \\emph{dense} phase if $n \\leq 0.1\\,{\\rm cm}^{-3}$, $0.1 \\leq n\/{\\rm cm}^{-3}\\leq 10$, $n > 10\\,{\\rm cm}^{-3}$, respectively\\footnote{Compared to the definitions used in \\citet{klessen:2014review}, the rarefied corresponds to the warm and hot ionized medium, the diffuse phase to the cold and warm neutral medium and the dense phase to the molecular gas.\\label{footnote_phases}}.\n\nWe focus at $z=6$ and consider the gas in a region within $\\simeq 47.5\\,{\\rm kpc}$ $(3\\, r_{\\rm vir})$ from {Dahlia}'s centre, essentially the scale of the CGM described in Sec. \\ref{sec_CGM}. This region contains a total gas mass of $1.3\\times 10^{10} {\\rm M}_{\\odot}$, and metal mass of $4.2\\times 10^5{\\rm M}_{\\odot}$ (additional data in Tab. \\ref{tagella_eos_riassunto}).\n\nFig. \\ref{fig_eos_1} shows the Equation of State (EOS, or phase diagram) of the gas. The fraction of gas in the rarefied, diffuse and dense phases is $44\\%$, $34\\%$ and $22\\%$; these phases contain $5\\%$, $25\\%$ and $70\\%$ of the metals, respectively. Thus, while the gas mass is preferentially located in the lower density phases, metals are mostly found in dense gas, i.e. star forming regions\/MC. Additionally only $\\sim 30\\%$ of the considered volume shows $Z>10^{-3}{\\rm Z}_{\\odot}=Z_{\\rm floor}$, i.e. it has been polluted by stars in the simulation. We note that the EOS in the $n$-$T$ plane is fairly consistent with the one found in other high-$z$ galaxy simulations \\citep[e.g. see Fig. 5 in][]{wise:2012radpres}. Comparison between the EOS in the $n$-$T$ and $n$-$P$ plane highlights the relative importance of different feedback types.\n\nThe \\emph{rarefied} gas is characterized by long cooling times. Thus, once engulfed by shocks, such phase becomes mildly enriched ($\\langle Z\\rangle \\sim10^{-2}{\\rm Z}_{\\odot}$) and remains hot ($T\\sim10^{6}{\\rm K}$). The enriched rarefied gas preferentially populates the $n\\simeq 10^{-3}{\\rm cm}^{-3}$ and $T\\simeq10^{6.5}{\\rm K}$ region of the phase diagram. However, part of the rarefied gas has $T\\simeq10^{4}{\\rm K}$. This gas component has a temperature set by the equilibrium between adiabatic cooling and the photo-heating by the UV background; it feeds the accretion onto Dahlia, but it is not affected by stellar feedback. As such it is not central in the present analysis.\n\nThe \\emph{dense} gas is mostly unaffected by shocks and it is concentrated in the disk. Typically, such gas has $n\\sim 10^2 {\\rm cm}^{-3}$ and $T\\sim10^{2}{\\rm K}$, thus a thermal pressure $P_{\\rm th}\/k \\sim 10^4 {\\rm cm}^{-3}\\,{\\rm K}$ is expected. However, the total gas pressure is $P\/k \\sim 10^7 \\,{\\rm cm}^{-3}$ K (see the $P$-$n$ EOS). The extra contribution is provided in kinetic form by radiation pressure, thanks to the strong coupling with the gas allowed by the high optical depth of this phase. This leads to the important implication that the central structure of {Dahlia} is radiation-supported (see also Sec. \\ref{sec_final_results}).\n\nThe \\emph{diffuse} gas acts as an interface between the dense disk gas and the rarefied gas envelope. Diffuse gas is found both in hot ($T\\sim10^{5}{\\rm K}$) and cold ($T\\sim10^{3}{\\rm K}$) states. The cold part has a sufficiently high mean metallicity, $Z\\sim 0.1{\\rm Z}_{\\odot}$, to allow an efficient cooling of the gas. This is highlighted by the metal-weighted EOS, where we can see that most of the metals present in the diffuse phase are cold.\n\nNote that the phase diagram also shows evidence for the classical 2-phase medium shape for pressures around $P\/k \\sim 10^3 {\\rm cm}^{-3}\\,{\\rm K}$, while at higher (and lower) pressures only one stable phase is allowed; nevertheless, at any given pressure a range of densities can be supported. Such situation, though, is highly dynamic and does not correspond to a true thermal equilibrium.\n\nA final remark is that by $z=6$ a $n-Z$ correlation is already in place, although considerable scatter is present. The relation gets steeper at large densities, and at the same time the scatter decreases. Such relation arises from the superposition of the analogous relation for metal bubbles of individual galaxies ({Dahlia} and satellites). The scatter instead results from the fact that the slope of the $n-Z$ relation depends on the $SFR$ history (for an in-depth analysis see \\citetalias{pallottini:2014_sim}). The average $n-Z$ relation found is consistent with the results from $z\\simeq3$ galaxies \\citep{shen:2014}.\n\n\\subsection{Additional ISM properties}\\label{sec_final_results}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.32\\textwidth]{plots_pdf\/maps\/landscape_c_rt_surf_dens_kmt_H2_27.pdf}\n\\includegraphics[width=0.32\\textwidth]{plots_pdf\/maps\/landscape_c_rt_surf_dens_CII_27.pdf}\n\\includegraphics[width=0.32\\textwidth]{plots_pdf\/maps\/landscape_c_rt_emission_CII_map_27.pdf}\n\\vspace{.5pt}\n\n\\includegraphics[width=0.32\\textwidth]{plots_pdf\/maps\/landscape_c_edgeon_rt_surf_dens_kmt_H2_27.pdf}\n\\includegraphics[width=0.32\\textwidth]{plots_pdf\/maps\/landscape_c_edgeon_rt_surf_dens_CII_27.pdf}\n\\includegraphics[width=0.32\\textwidth]{plots_pdf\/maps\/landscape_c_edgeon_rt_emission_CII_map_27.pdf}\n\\caption{\nFace-on ({\\bf upper panels}) and edge-on ({\\bf lower panels}) $z=6$ {Dahlia} surface maps for ${\\rm {H_2}}$~density ($\\Sigma_{\\rm H2}\/(\\msun\\,{\\rm kpc}^{-2})$ {\\bf left panels}), \\hbox{C~$\\scriptstyle\\rm II $}~density ($\\Sigma_{\\rm CII}\/(\\msun\\,{\\rm kpc}^{-2})$ {\\bf middle panels}), and \\hbox{[C~$\\scriptstyle\\rm II $]}~brightness ($S_{\\rm [CII]}\/(\\lsun\\,{\\rm kpc}^{-2})$ {\\bf left panels}). The scale is $10$~kpc, as in the right-most panels of Fig. \\ref{fig_mappe_hydro} (Sec. \\ref{sec_small_scale}). Note that lower limits for the maps are drawn for visualization purposes ($\\log(\\Sigma_{\\rm H2}\/(\\msun\\,{\\rm kpc}^{-2}))\\simeq \\log(S_{\\rm [CII]}\/(\\lsun\\,{\\rm kpc}^{-2})) \\simeq 5$, $\\log(\\Sigma_{\\rm CII}\/(\\msun\\,{\\rm kpc}^{-2}))\\simeq 2$). Additionally, an average of the maps is plotted in Fig. \\ref{fig_mappe_results_profili}.\n\\label{fig_mappe_tutte}\n}\n\\end{figure*}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/dahlia_profiles.pdf}\n\\caption{\nRadially-averaged profiles for face-on (full colour) and edge-on (transparent and hatched) views of Dahlia at $z=6$. {\\bf Upper left:} \n${\\rm {H_2}}$~surface density; {\\bf upper right:} stellar surface density; {\\bf lower left:} \\hbox{C~$\\scriptstyle\\rm II $}~surface density; {\\bf lower right:} \\hbox{[C~$\\scriptstyle\\rm II $]}~surface brightness.\n\\label{fig_mappe_results_profili}\n}\n\\end{figure}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lccc}\n\\hline\n~ & \\multicolumn{2}{c}{$r_{1\/2}\/{\\rm kpc}$} & approximate\\\\\n~ & face-on & edge-on & value at $r_{1\/2}$\\\\\n\\hline\t\t\t\n${\\rm {H_2}}$\t& 0.59 & 0.36 &\t$10^{8.23}{\\rm M}_{\\odot}$\\\\\n\\hbox{C~$\\scriptstyle\\rm II $}\t& 0.64 & 0.38 &\t$10^{5.14}{\\rm M}_{\\odot}$\\\\\nstars\t& 0.37 & 0.23 &\t$10^{9.89}{\\rm M}_{\\odot}$\\\\\n\\hbox{[C~$\\scriptstyle\\rm II $]}\t& 0.60 & 0.36 &\t$10^{7.25}{\\rm L}_{\\odot}$\\\\\n\\end{tabular}\n\\caption{\nSummary of the effective radius ($r_{1\/2}$) for the ${\\rm {H_2}}$, \\hbox{[C~$\\scriptstyle\\rm II $]}, \\hbox{C~$\\scriptstyle\\rm II $}~and stellar component in {Dahlia} at $z=6$ for the face-on and edge-on case and corresponding values for the mass\/luminosity. For each entry, $r_{1\/2}$ is defined as the radius including half of the mass\/luminosity. The quoted radii have a reference error of $\\pm 0.015\\,{\\rm kpc}$. Note that the approximate values of masses\/luminosity at $r_{1\/2}$ are insensitive to the orientation of the projection (face-on\/edge-on). The full profiles are shown in Fig. \\ref{fig_mappe_results_profili}.\n\\label{tagella_halflightradius}}\n\\end{table}\n\nWe conclude our analysis by inspecting the distribution of two key ISM species, molecular hydrogen and \\hbox{C~$\\scriptstyle\\rm II $}, along with the expected surface brightness of the corresponding $158\\mu$m \\hbox{[C~$\\scriptstyle\\rm II $]}~line. The surface maps of these quantities in Dahlia ($z=6$) are shown in Fig. \\ref{fig_mappe_tutte} for the face-on and edge-on view cases. For reference, in Fig. \\ref{fig_mappe_results_profili} we additionally plot the radially-averaged profiles of the same quantities, and in Tab. \\ref{tagella_halflightradius} we give their typical radial scales.\n\n\\subsubsection{Molecular Hydrogen}\n\n{Dahlia} has a total ${\\rm {H_2}}$~mass $M_{\\rm H2}\\simeq 3.6\\times 10^8{\\rm M}_{\\odot}$, that is mainly concentrated in a disk-like structure of radius $\\simeq 0.6\\,{\\rm kpc}$ and scale height $\\simeq 200\\,{\\rm pc}$, with a sharp cut off beyond these scales\\footnote{Such scales are calculated by using the principal component analysis of the ${\\rm {H_2}}$~distribution around the galaxy.}. The disk has mean surface density $\\langle\\Sigma_{\\rm H2}\\rangle \\simeq 10^{7.5}\\msun\\,{\\rm kpc}^{-2}$, that is approximately constant with radius and presents perturbed spiral arms along which the density is enhanced by a factor $\\simeq 3$. The spiral arms are less pronounced than in a more massive, MW-like galaxy (see \\citetalias{semenov:2015} and \\citealt{ceverino:2015}). This trend with mass has already been pointed out by \\citet{ceverino:2010MNRAS}.\n\nThe disk is composed by dense ($n\\gsim 25\\,{\\rm cm}^{-3}$), enriched ($Z\\simeq 0.5\\,{\\rm Z}_{\\odot}$), radiation-pressure supported gas, as already discussed. It is fed by frequent mergers driving fresh gas to the centre, and supports a star formation rate per unit area of $\\simeq 15\\,{\\rm M}_{\\odot}\\,{\\rm yr}^{-1}\\,{\\rm kpc}^{-2}$, i.e. more than 1000 times the Milky Way value. Fragmentation of the disk is relatively weak \\citep[cfr.][]{mayer:2016}, as indicated by smooth surface density map, and also paralleling the flat metallicity profile in the inner $\\simeq\\,{\\rm kpc}$.\nFor the fragmentation of the ${\\rm {H_2}}$~component, we caution that this result has been obtained assuming a uniform UV interstellar field; stronger fragmentation in the ${\\rm {H_2}}$~distribution may occur when accounting for local radiation sources: Lyman-Werner photons from these sources might in fact locally dissociate the ${\\rm {H_2}}$~by generating pockets of \\HI~in the distribution.\n\nWhile most of the ${\\rm {H_2}}$~gas resides in the disk, we can clearly distinguish 3 clumps of molecular gas both in the face-on and edge-on maps. These clumps are located few kpc away from the centre, and are characterized by sizes of $\\sim 150\\,{\\rm pc}$ and $M_{\\rm H2} \\sim 5\\times10^6 {\\rm M}_{\\odot}$. Such clumps are Jeans-unstable and form stars as they infall and stream through the CGM, as it can be appreciated by comparing ${\\rm {H_2}}$~and stellar mass profiles (Fig. \\ref{fig_mappe_results_profili}). \nThe stellar mass profiles also highlight the presence of 3 stellar clumps at $r\\sim 1\\,{\\rm kpc}$ with no associated ${\\rm {H_2}}$. These \\quotes{older} clumps share the same nature of the previous ones, but the ${\\rm {H_2}}$~has been already consumed and\/or dispersed by the star formation activity that produced the stars present at $z=6$.\n\n\\subsubsection{Singly ionized Carbon}\n\nThe \\hbox{C~$\\scriptstyle\\rm II $}~abundance is calculated by post-processing the simulation outputs with the photoionization code \\textlcsc{cloudy} (\\citealt{cloudy:2013}, and see App. \\ref{sez_cloudy_model}). The result is shown in Fig. \\ref{fig_mappe_tutte}. {Dahlia} contains a \\hbox{C~$\\scriptstyle\\rm II $}~mass of $M_{\\rm CII} = 2.2\\times 10^5 {\\rm M}_{\\odot}$, accounting for $\\sim 50\\%$ of the total metals produced. About 74\\% of the \\hbox{C~$\\scriptstyle\\rm II $}~mass is located in the dense phase, $22\\%$ in the diffuse phase, $4\\%$ in the rarefied phase. Note that the \\hbox{C~$\\scriptstyle\\rm II $}~mass phase distribution differs only for $\\lsim 10\\%$ from the $Z$ distribution (see Tab. \\ref{tagella_eos_riassunto}). The difference arises because shock-heated gas can be collisionally excited to higher ionization states. Thus, to first order, we expect the \\hbox{C~$\\scriptstyle\\rm II $}~spatial distribution to follow the metallicity one.\n\nThe face-on \\hbox{C~$\\scriptstyle\\rm II $}~surface density has a central maximum ($\\Sigma_{\\rm CII} \\sim 10^5\\msun\\,{\\rm kpc}^{-2}$), it gradually decreases to up to $\\simeq 1.2\\,{\\rm kpc}$, and drastically drops to $\\Sigma_{\\rm CII} \\lsim 10^2\\msun\\,{\\rm kpc}^{-2}$ beyond that radius (see also Fig. \\ref{fig_mappe_results_profili}). Thus, most of the \\hbox{C~$\\scriptstyle\\rm II $}~is located into the disk, but a more extended envelope containing a sizable fraction of mass exists. On top of this smooth distribution, there are \\hbox{C~$\\scriptstyle\\rm II $}~enhancements corresponding to the ${\\rm {H_2}}$~clumps described above. \n\nThe \\hbox{C~$\\scriptstyle\\rm II $}~profile is similar for edge-on and face-on case. However, the edge-on has a higher \\hbox{C~$\\scriptstyle\\rm II $}~central density and a steeper slope. While the higher central value is obviously due to the larger column density encountered along the disk, the sharp drop is related to metal transport. As most of the star formation activity is located in the disk, metals above it can be only brought by outflows which become progressively weaker with distance. Metal outflows originating from the centre are preferentially aligned with the rotation axis, and the pollution region starting from the edge is stretched by the disk rotation and by tidal interaction with satellites.\n\n\\subsubsection{Emission from singly ionized carbon}\n\nWe finally compute the expected \\hbox{[C~$\\scriptstyle\\rm II $]}~line emission using the same prescriptions of \\citet{Vallini:2013MNRAS,vallini:2015}, as detailed in App. \\ref{sez_cloudy_model}. Note that for the present work we assume uniform UV interstellar radiation. This approximation is valid in the MW, where variations around the mean field value are limited to a factor of 3. The results are plotted in Fig. \\ref{fig_mappe_tutte}.\n\nWithin $1\\,{\\rm kpc}$ from the centre the \\hbox{[C~$\\scriptstyle\\rm II $]}~emission structure closely follows the \\hbox{C~$\\scriptstyle\\rm II $}~distribution, and we find $S_{\\rm [CII]}\/{\\rm L}_{\\odot} \\simeq 200\\, \\Sigma_{\\rm CII}\/{\\rm M}_{\\odot}$. At larger radii the \\hbox{[C~$\\scriptstyle\\rm II $]}~surface brightness suddenly drops, although the peaks associated with ${\\rm {H_2}}$~clumps are preserved. This result holds both for the face-on and edge-on cases. \n\nSuch behavior can be understood as follows. Take a typical MC with $n = 10^2{\\rm cm}^{-3}$, $Z={\\rm Z}_{\\odot}$, and total mass $M$. Its \\hbox{[C~$\\scriptstyle\\rm II $]}~\nluminosity is $L_{[\\rm CII]}\/{\\rm L}_{\\odot} \\simeq 0.1 (M\/{\\rm M}_{\\odot})$ \\citep[]{vallini:2016a,goicoechea:2015apj}. Also, the \\hbox{[C~$\\scriptstyle\\rm II $]}~emission is $\\propto Z\\,n$ for $n\\lsim 10^3$, i.e. the critical density for \\hbox{C~$\\scriptstyle\\rm II $}~collisional excitation by H atoms \\citep{Vallini:2013MNRAS}. Then, \n\\be\\label{eq_stima_luminosita}\nL_{[\\rm CII]} \\simeq 0.1\\, \\left({n\\over 100\\, {\\rm cm}^{-3}}\\right) \\left({Z\\over {\\rm Z}_{\\odot}}\\right) \\left({M\\over {\\rm M}_{\\odot}}\\right)\\,{\\rm L}_{\\odot}\\,.\n\\ee\nIn the central kpc, where $n \\simeq 10^2{\\rm cm}^{-3}$ and $Z\\simeq {\\rm Z}_{\\odot}$, the luminosity depends only on the molecular mass contained in the disk, and the same holds even for ${\\rm {H_2}}$~clumps outside the disk. The envelope is instead more diffuse ($n\\lsim 10{\\rm cm}^{-3}$) and only mildly enriched ($Z\\lsim10^{-1}{\\rm Z}_{\\odot}$). As a result, its \\hbox{[C~$\\scriptstyle\\rm II $]}~luminosity per unit mass is lower. \n\nThe emission from this diffuse component is further suppressed by the CMB \\citep[][]{dacunha:2013apj,pallottini:2015_cmb,vallini:2015}. Namely, for gas with $n\\lsim 0.1\\,{\\rm cm}^{-3}$, the upper levels of the \\hbox{[C~$\\scriptstyle\\rm II $]}~transition cannot be efficiently populated through collisions, thus the spin temperature of the transition approaches the CMB one, and to a first order the gas cannot be observed in emission.\n\nIn summary, $\\simeq 95\\%$ of {Dahlia} \\hbox{[C~$\\scriptstyle\\rm II $]}~emission comes from dense gas located in the ${\\rm {H_2}}$~disk. Indeed, the \\hbox{[C~$\\scriptstyle\\rm II $]}~half light radius coincides with the ${\\rm {H_2}}$~half mass radius, i.e. $0.59\\,{\\rm kpc}$ ($0.36\\,{\\rm kpc}$) in the face-on (edge-on) case (see also Tab. \\ref{tagella_halflightradius}). Within such radius, the molecular gas has a mass $M_{\\rm H_{2}}\\simeq 1.69\\times10^8{\\rm M}_{\\odot}$ and the luminosity is $L_{\\rm CII}\\simeq 1.78\\times10^7{\\rm M}_{\\odot}$, i.e. with a \\hbox{[C~$\\scriptstyle\\rm II $]}-${\\rm {H_2}}$~scaling ratio consistent within $15\\%$ from the simple estimate in eq. \\ref{eq_stima_luminosita}.\n\nDahlia has a total \\hbox{[C~$\\scriptstyle\\rm II $]}~luminosity $L_{\\rm CII} \\simeq 3.5\\times 10^{7}{\\rm L}_{\\odot}$; this is fainter than expected on the basis of the local \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$ relation \\citep[$L_{\\rm CII}\\sim 10^8 - 10^9 {\\rm L}_{\\odot}$, i.e.][]{delooze:2014aa}. However, at high-$z$, such relation seems to hold only for a small subset of the observed galaxies \\citet[i.e.][]{capak:2015arxiv,Willott:2015arXiv15}. The majority of the observed galaxies show a strong \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$~deficit, when considering both detections \\citep[e.g. BDF3299, A383-5.1][]{maiolino:2015arxiv,knudsen:2016arxiv} and upper limits \\citep[e.g. Himiko, IOK1, MS0451-H][]{ouchi2013,ota:2014apj,knudsen:2016arxiv}.\n\nFor {Dahlia}, the \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$~deficit depends on multiple factors. The main contribution from \\hbox{[C~$\\scriptstyle\\rm II $]}~emission is in the ${\\rm {H_2}}$~disk, that on average has $\\langle Z\\rangle\\simeq 0.5 {\\rm Z}_{\\odot}$, i.e. slightly lower then solar. Additionally, the gas in the disk is efficiently converted in stars ($SFR\\simeq100{\\rm M}_{\\odot}\/{\\rm yr}$) and has $\\langle n\\rangle\\simeq 25\\,{\\rm cm}^{-3}$, thus the \\hbox{[C~$\\scriptstyle\\rm II $]}~emission is hindered (eq. \\ref{eq_stima_luminosita}). Finally, there is a marginal contribution to \\hbox{[C~$\\scriptstyle\\rm II $]}~from the diffuse and rarefied phase: $\\simeq30\\%$ of \\hbox{C~$\\scriptstyle\\rm II $}~is locked in a the low density and metallicity gas that gives a negligible contribution to \\hbox{[C~$\\scriptstyle\\rm II $]}~emission, particularly because of CMB suppression.\n\n\\section{Summary and discussion}\\label{sec_conclusioni}\n\nWith the aim of characterizing the internal properties of high-$z$ galaxies, we have performed an AMR zoom-in simulation of \\quotes{Dahlia}, a $z\\simeq6$ galaxy with a stellar mass of $M_{\\star}=1.6\\times10^{10}{\\rm M}_{\\odot}$, therefore representative of LBGs at that epoch. We follow the zoom-in region with a gas mass resolution of $10^{4}{\\rm M}_{\\odot}$ and a spatial resolution of $30\\,{\\rm pc}$.\n\nThe simulation contains a rich set of physical processes. We use a star formation prescription based on a ${\\rm {H_2}}$~dependent Schmidt-Kennicutt relation. The ${\\rm {H_2}}$~abundance is computed from the \\citetalias{krumholz:2009apj} model (Fig. \\ref{fig_kmt_test}). Using stellar evolutionary models \\citep{padova:1994,starburst99:1999}, we include chemical, radiative and mechanical energy inputs, accounting for their time evolution and metallicity dependence on the stellar population properties (Fig. \\ref{fig_gamete_tables}). We include feedback from SN, winds and radiation pressure with a novel, physically motivated coupling scheme between gas and stars. We also compute \\hbox{C~$\\scriptstyle\\rm II $}~abundance and the $158\\mu$m \\hbox{[C~$\\scriptstyle\\rm II $]}~emission, by post-processing the outputs with \\textlcsc{cloudy} \\citep{cloudy:2013}, and a FIR~emission model drawn from radiative transfer numerical simulations \\citep{Vallini:2013MNRAS,vallini:2015}.\n\nThe main results can be summarized as follows:\n\n\\begin{itemize}\n\n\\item[\\bf 1.] {Dahlia} sits at the centre of a cosmic web knot, and accretes mass from the intergalactic medium mainly via 3 filaments of length $\\simeq 100\\,{\\rm kpc}$ (Fig. \\ref{fig_mappe_hydro}). Dahlia has $\\sim 6$ major satellites ($M_{\\star}\\lsim 10^{9}{\\rm M}_{\\odot}$) and is surrounded by $\\sim 10$ minor ones ($M_{\\star}\\sim 10^{5}{\\rm M}_{\\odot}$). The latter represent molecular cloud (MC) complexes caught in the act of condensing as the gas streams through the circumgalactic medium (Fig. \\ref{fig_sph_profile}). {Dahlia} dominates both the stellar mass ($M_{\\star}\\sim 10^{10}{\\rm M}_{\\odot}$) and the SFR of the galaxy ensemble ($SFR\\simeq 100\\,{\\rm M}_{\\odot}\\,{\\rm yr}^{-1}$, Fig. \\ref{fig_sfr_smf_energy}).\n\n\\item[\\bf 2.] Only a small fraction of the available energy produced by stars couples to the gas, as energy is mostly dissipated within MCs where the stars reside. Radiation dominates the feedback energy budget by a factor $> 100$ (Fig. \\ref{fig_feedback_vs_time}). \n\n\\item[\\bf 3.] By $z=6$ {Dahlia} forms a ${\\rm {H_2}}$~disk of mass of $M_{\\rm H2}= 3.6\\times 10^{8}{\\rm M}_{\\odot}$, effective radius $0.6\\,{\\rm kpc}$, and scale height $200\\,{\\rm pc}$ (Fig. \\ref{fig_mappe_tutte}). The disk is dense ($n\\gsim 25\\,{\\rm cm}^{-3}$), enriched ($Z\\simeq 0.5\\,{\\rm Z}_{\\odot}$), and it is fed by frequent mergers driving fresh gas to the centre, and supports a star formation rate per unit area of $\\simeq 15\\,{\\rm M}_{\\odot}\\,{\\rm yr}^{-1}\\,{\\rm kpc}^{-2}$. \n\n\\item[\\bf 4.] The disk is mostly unaffected by SN shocks, and it is pressure-supported by radiation. SN\/winds drive hot metal outflows (Fig. \\ref{fig_eos_1}), that are either preferentially aligned with the galaxy rotation axis, or start at the edge of the disk.\n\n\\item[\\bf 5.] The total \\hbox{[C~$\\scriptstyle\\rm II $]}~luminosity of {Dahlia} is $10^{7.55}{\\rm L}_{\\odot}$, and $\\simeq 95\\%$ of the emission is co-located with the ${\\rm {H_2}}$~disk (Fig. \\ref{fig_mappe_results_profili}). The diffuse, enriched material surrounding {Dahlia} contains $30\\%$ of the \\hbox{C~$\\scriptstyle\\rm II $}~mass, but it negligibly contributes to the \\hbox{[C~$\\scriptstyle\\rm II $]}~emission (Fig. \\ref{fig_mappe_tutte}) due to its low density ($n\\simeq 10\\,{\\rm cm}^{-3}$) and metallicity ($Z\\simeq10^{-1}{\\rm Z}_{\\odot}$). {Dahlia} is under-luminous with respect to the local \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$ relation; however, its luminosity is consistent with upper limits derived for most $z\\sim6$ galaxies. \n\\end{itemize}\n\nWe find clear indications that the SF subgrid prescription might considerably affect the \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$ relation and the ISM structure, as noted also by \\citep{hopkins:2013arxiv}. This is because stars form in gas of different densities depending on the chosen prescription. \nIn our simulation gas is converted into stars with an efficiency $\\zeta_{\\rm sf}\\,f_{\\rm H2}$, where the ${\\rm {H_2}}$~fraction is computed from the \\citetalias{krumholz:2009apj} model and we set $\\zeta_{\\rm sf}=0.1$. In \\citetalias{semenov:2015} the SF follows a \\textit{total} (i.e. not molecular) density Schmidt-Kennicutt relation. Further the SF efficiency depends on the free-fall time and the turbulent eddy turnover time. The SF relation is derived from an empirical fit to MC simulations \\citep{padoan:2012}, with no notion of the local metallicity. \n\nInterestingly, although the approaches are considerably different, the resulting efficiencies are compatible: in \\citetalias{semenov:2015} the bulk of the star forming gas has $n\\sim 10^{1.5}{\\rm cm}^{-3}$, as in {Dahlia} (Fig. \\ref{fig_cfr_semenov}). However, with respect to \\citetalias{semenov:2015}, Dahlia misses part of the very dense, star forming gas, and its corresponding contribution to \\hbox{[C~$\\scriptstyle\\rm II $]}~from $Z\\sim{\\rm Z}_{\\odot}$ MCs with $n\\sim10^{3}{\\rm cm}^{-3}$. These MC are expected to have high \\hbox{[C~$\\scriptstyle\\rm II $]}~fluxes (see eq. \\ref{eq_stima_luminosita}), but their abundance might be low \\citep{padoan:2012}.\nFurther investigation is needed before we draw any solid conclusion. To this aim, we plan to upgrade our simulations to a more sophisticated non-equilibrium ${\\rm {H_2}}$~evolution model. This is because the chemical equilibrium assumed in \\citetalias{krumholz:2009apj} does not hold in low-metallicity regimes. \n\nAnother important caveat is that we have assumed a uniform UV background. Instead, discrete sources (stellar clusters) might have a strong impact on star formation. For example, Lyman-Werner photons might locally dissociate the ${\\rm {H_2}}$~by generating pockets of \\HI~in the gas distribution. Thus, unshielded (low dust column density) gas in the disk would contribute only marginally to the SFR.\n\nFurthermore, a uniform UVB assumption likely leads to inaccurate computation of the ISM thermodynamic state. We find that $Z\\simeq 10^{-3}{\\rm Z}_{\\odot}$ gas with $n\\gsim 10^{2}\\,{\\rm cm}^{-3}$ has $T\\simeq 10^{4}$ (Fig. \\ref{fig_eos_1}), with the temperature been set by the UVB heating. However, such gas should be likely able to self-shield from the impinging UVB, whereas internal radiation sources could still play a role \\citep[e.g.][]{gnedin:2010}. \n\nFinally, local FUV flux variations can change the \\hbox{[C~$\\scriptstyle\\rm II $]}~emission from individual regions of the galaxy. Also, very high FUV fluxes can photoevaporate MC on short time scales ($\\lsim t_{\\rm ff}$ for gas with $Z\\sim 10^{-2}{\\rm Z}_{\\odot}$, \\citealt[][]{vallini:2016a}). This effect are particularly important, as it might be responsible for the displacement between the \\hbox{[C~$\\scriptstyle\\rm II $]}~and the UV emitting region observed in BDF3299 \\citep{maiolino:2015arxiv}, and in some of the \\citet{capak:2015arxiv} galaxies. To solve these problems, a multi-frequency radiative transfer computation must be coupled to the present simulations. This work is ongoing and will be presented elsewhere. \n\n\n\n\\section*{Acknowledgments}\nWe are grateful to the participants of \\emph{The Cold Universe} program held in 2016 at the KITP, UCSB, for discussions during the workshop. \nWe acknowledge the {\\tt AGORA} project members and the {\\tt DAVID} group for stimulating discussion. \nWe thank the authors and the community of \\textlcsc{pymses} for their work. \nWe thank B. Smith for support in implementing \\textlcsc{grackle}. \nThis research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915.\nS.S. was supported by the European Research Council through a Marie-Skodolowska-Curie Fellowship, project PRIMORDIAL-700907.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\n\n    Support Vector Machines (SVMs) are one of the most widely used algorithms for classification problems. Originally proposed in the works of Boser et al~\\cite{boser1992training} and Cortes and Vapnik~\\cite{cortes1995support}, they can be defined as learning machines which construct an $n$-dimensional decision boundary surface (also called a hyperplane) that optimally separates data into positive and negative classes by maximizing the margin of separation between them. \n    \n    Contrary to artificial neural networks, which provide a local minimum solution to the optimization problem, SVMs provide a unique globally optimal solution for the margin separation problem, which is addressed through the application of a kernel-based learning method. In this context, a kernel is understood as a similarity function that is applied to each data point to map the original non-linear observations into a higher dimensional space where the observations may become linearly separable. A wide range of different kernels have been proposed in the literature, targeting specific classification problems~\\cite{Haykin08}. The Gaussian kernel (also referred to as the Radial Basis Function kernel (RBF)), is probably the most widely used kernel demonstrating 'state of the art' performance in a variety of  classification problems~\\cite{Bhattacharyya11}.  \\\\\n\\indent More recently, new mappings between non-linear separable observations and higher dimensional feature spaces have been proposed with the purpose of extending the capabilities of SVMs towards theoretically feasible quantum-inspired machine learning algorithms~\\cite{schuld2019quantum,schuld2019machine,havlivcek2019supervised,mehta2019high,killoran2019strawberry,killoran2018continuous,srinivasan2018learning,adhikary2019supervised,bartkiewicz2019experimental}.  For instance, under a quantum theoretical perspective, by mapping data into coherent states which are a superposition of eigen-functions of a quantum harmonic oscillator with minimum Heisenberg  uncertainty, the RBF kernel can be understood as the inner product of two coherent states  \\cite{kubler2019quantum}. The coherent state of this harmonic oscillator comprises the following properties:\n\\begin{enumerate*}[label=({\\it \\roman*})]\n  \\item It is obtained by the displacement operators on the ground state;\n  \\item It is an eigenfunction of the annihilation operator;\n  \\item It satisfies the minimum uncertainty relation, i.e., $\\Delta(\\mathbf{x})=\\Delta(\\mathbf{p})=\\sigma\/\\sqrt{2}$, in which $\\Delta(\\mathbf{x})$ and $\\Delta(\\mathbf{p})$ are respectively the variance of the position and momentum of the harmonic oscillator;\n  \\item It is over-complete.\n\\end{enumerate*}\n\\\\\n\\indent \nThe over-completeness property implies an arbitrary function can be expressible as a linear combination of kernel functions in a \"reproducing Hilbert space\" \\cite{combescure2012coherent}. Any of the first three above-mentioned properties lead to a definition of generalized coherent states, although property ({\\it iv}) is necessary for the definition of coherent states. For example, while a Gazeau-Klauder coherent state is defined by property ({\\it ii}) and fulfils property ({\\it iii}), displacement-type coherent states are obtained by displacement operators on reference states  \\cite{ali2000coherent}.\n\\indent \nRecently, Schuld and Killoran proposed to map data from the original space to a feature space by using squeezed coherent states \\cite{schuld2019quantum}. Squeezed  states are coherent states that saturate the Heisenberg uncertainty principle in such a way that the variance of position and momentum depend on a so-called squeezed parameter. Therefore, the reduced uncertainty is one of its quadrature components, while increased uncertainty is the latter, i.e., $\\Delta(\\mathbf{x})=\\exp{\\zeta}\/\\sqrt{2}$ and $\\Delta(\\mathbf{p})=\\exp{-\\zeta}\/\\sqrt{2}$, where $\\zeta$ is the  squeezing parameter. The squeezing parameter controls uncertainty via a quadrature component, while the third  property of coherent states are preserved.\\\\\n\\indent \nGiven the large number of kernel functions currently being proposed in the literature, the question naturally arises as to which kernel function to apply. In SVM-based classification problems, the appropriate choice of a kernel is fundamental, however, the current 'trial-and-error' nature of selecting the best kernel poses significant challenges, especially when one considers kernels that can support both classical and quantum-inspired machine learning algorithms, which renders the kernel choice problem an open research question \\cite{Ali06}.\\\\\n\\indent To address this problem (and taking as basis the work of Schuld and Killoran on squeezed coherent states~\\cite{schuld2019quantum}), we propose a generalised meta-kernel from which the RBF kernel (and other kernels) can be derived by using a deformed Weyl-Heisenberg (dW-H) algebra, dependent on a parameter $\\alpha \\in \\mathbb{R}$. By applying the associated displacement operator on the reference state, the non-linear coherent state is generated by considering the specific value of the parameter, i.e. $\\alpha=2$, $\\alpha=0$, and $\\alpha=-2$, $SU(2)$, W-H, and $SU(1,1)$-coherent state are respectively recovered. \\cite{dehdashti2013coherent,dehdashti2015realization,castillo2019polynomial}.  \nThe choice of $\\alpha$ allows a specific kernel function to be defined. Therefore, the theory of coherent states can be seen as providing a meta-kernel from which kernel functions can be derived. To the best of our knowledge, no such theory of meta-kernels presently exists.\n \n By means of a feature mapping, data is mapped into the feature space, represented by the deformed coherent space. Schematically, this is illustrated in Figure \\ref{fig1}. Geometrically, the feature space constructed by the dW-H coherent state is a surface of revolution with constant curvature, i.e., the surfaces associated with $\\alpha-SU(2)$ and $\\alpha-SU(1,1)$ are respectively a positive compact surface, and negative surface, while $\\alpha=0$ produces a flat surface. Therefore, a kernel function defined in any one of the configurations is an inner product of two elements on the related surface. Through this process, our dW-H algebra acts like a meta-theory from which  a new class of two parameter non-linear kernel functions can be derived.\n\\begin{figure}[t]\n  \\centering\n   \n  \\includegraphics[scale=.55]{fig1.pdf}\n  \\caption{Schematic representation of the SVM method based on the Weyl-Heisenberg algebra, showing the mapping of data into the feature space, represented by the dW-H coherent state.}\n  \\label{fig1}\n\\end{figure}\n\n\nThe paper will proceed as follows. In Section \\ref{coherent_states}, we provide a brief introduction to dW-H coherent states which are then expressed in kernel functions. We also describe the geometric properties of the feature spaces in which these kernel functions are defined. In Section \\ref{test_design}, a test design is formulated for an illustrative evaluation of the introduced kernel functions, from the standpoint of enrichment of Gaussian strategies in SVM classification. \nAccompanying empirical results are presented, along with visualisations that aid descriptions of relevant observations.  Section \\ref{discussion} discusses the benefits of the algebra within SVM classification based on the results of the empirical evaluation.\n\\section{Deriving kernel functions from deformed Coherent States}\n\\label{coherent_states}\nA supervised machine learning (ML) classification problem can be formalised in the following way. Given a set of $N$ training examples $\\{ (x_{1}, y_{1}), (x_{2}, y_{2}), \\cdots, (x_{N}, y_{N})\\}$, where $x_{i}$ corresponds to the $ith$ training example, each training example is represented by a set of input features, and $y_{i}$ corresponds to the 'ground truth' label of the training example $x_{i}$, the objective of ML is to learn a model, $h(x)$, that represents the training set. Ideally, the outcome is to generate the model that is most capable of correctly predicting the class labels of unseen instances. \n\n\nOne way of predicting unseen examples is through the application of a similarity function, a \\textit{kernel}, between the unseen input instance ${x^{\\prime}}$ and each of the training inputs, ${x_i}$, learned during the training phase.\n\nKernel methods, $K(x, x^{\\prime})$, use the inner product between any two inputs $x, x^{\\prime} \\in \\mathcal{X}$, as distance measures in order to construct models that capture the properties of a data distribution. These distance measures can be defined in a feature space $\\mathcal{X}$, depending on whether the data is linear, or non-linearly separable.\n\nThe left side of Figure \\ref{fig1} schematically represents this process. One can define a complex Hilbert space as the feature space, where a feature mapping $\\phi: \\mathcal{X}\\rightarrow \\mathcal{F}$, in which $\\mathcal{F}$ is a complex Hilbert state,   $\\phi: x \\rightarrow |\\phi(x) \\rangle $, implies a kernel function can be defined as $K(x, x^{\\prime} ) =\n \\langle\\phi(x), \\phi(x^{\\prime}) \\rangle$. By operating a dW-H algebra on the reference state, the feature space is constructed with deformed coherent states. These coherent states depend on the attributed sign of the parameter $\\alpha$, meaning `positivity' or `negativity' defines an $\\alpha-SU(2)$ coherent state or $\\alpha-SU(1,1)$ coherent state respectively; while $\\alpha=0$ stands for a harmonic oscillator coherent state ( see Figure \\ref{fig1}). For the sake of simplicity, we define dW-H coherent states for positive and negative values, separately. The first is titled the $\\alpha-SU(2)$ coherent state, defined as follows:\n \\begin{eqnarray}\\label{eq3}\n|x;z,\\alpha\\rangle\n&=&\\left[1+\\tan^{2}\\left(z\\sqrt{\\frac{\\alpha}{2}}\\right)\\right]^{-k}\\sum_{m=0}^{2k} (-1)^{m} e^{-imx}\\nonumber\\\\\n&\\times&\\tan^{m}\\left(z\\sqrt{\\frac{\\alpha}{2}}\\right) \\sqrt{\\frac{(2k)!}{m!(2k-m)!}}|k,m\\rangle,\n\\end{eqnarray} \n in which $\\alpha \\geq 0$, and  $k \\in N$. The second is named the $\\alpha-SU(1,1)$ coherent state:\n \\begin{eqnarray}\\label{eq2}\n|x;z,\\alpha\\rangle&=&\\left[1-\\tanh^{2}\\left(z\\sqrt{\\frac{|\\alpha|}{2}}\\right)\\right]^{k}\\sum_{m=0}^{\\infty} (-1)^{m} e^{-imx}\\nonumber\\\\\n&\\times&\\tanh^{m}\\left(z\\sqrt{\\frac{|\\alpha|}{2}}\\right) \\sqrt{\\frac{\\Gamma[2k+m]}{m!\\Gamma[2k]}}|k,k+m\\rangle,\n\\end{eqnarray} \nwhere $\\alpha\\leq 0$, and $k\\in N$. \n Note that in the case of $\\alpha = \\pm 2$, coherent states (\\ref{eq3}) and (\\ref{eq2}), respectively reduce $SU(2)$ and $SU(1,1)$ coherent states. It was also shown that if $\\alpha $ approaches zero, both coherent states (\\ref{eq3}) and (\\ref{eq2}) reduce to the harmonic oscillator coherent states  \\cite{dehdashti2013coherent} , i.e.,\n \\begin{eqnarray}\n |z\\rangle=e^{-|z|^{2}}\\sum_{m=0}^{\\infty} \\frac{z^{m} }{\\sqrt{m!}}|m\n \\rangle.\n \\end{eqnarray}\n \\indent Hence,  by considering a multi-dimensional input set in a data set of vectors $\\mathbf{x} = (x_{1},\\cdots,x_{N})^{T} \\in \\mathbb{R}^{N}$, one can define the joint state of $N$ deformed coherent states,\n \\begin{align*}\n & \\phi: (x_{1},\\cdots,x_{N}) \\rightarrow \\\\ & |x_{1},z,\\alpha\\rangle\\otimes |x_{2},z,\\alpha\\rangle \\otimes \\cdots \\otimes |x_{N},z,\\alpha\\rangle. \\numberthis\n \\end{align*}{}\n Therefore, the kernel  is defined as the following:\n \\begin{eqnarray}\n K(\\mathbf{x},\\mathbf{x^{\\prime}})= \\prod_{i=1}^{N}\\langle x_{i};z,\\alpha | \n x_{i}^{\\prime};z,\\alpha \\rangle.\n \\end{eqnarray}\nIn the case of $\\alpha-SU(2)$ feature space, the kernel is obtained as follows:\\\\\n\\begin{tcolorbox}[boxsep=2pt,left=2pt,right=2pt,top=4pt,bottom=4pt]\n\\bf{$\\bf{\\alpha-SU(2)}$ Kernel Function}\n\\begin{eqnarray}\nK(\\mathbf{x},\\mathbf{x^{\\prime}})= \\prod_{i=1}^{N} \\left[\\frac{1+\\tan^{2}\\left(z\\sqrt{\\frac{|\\alpha|}{2}}\\right)e^{i(x_{i}-x^{\\prime}_{i})}}{1+\\tan^{2}\\left(z\\sqrt{\\frac{|\\alpha|}{2}}\\right)}\\right]^{2k},\n\\end{eqnarray}\n\\end{tcolorbox}\nMoreover, in the case of $\\alpha-SU(1,1)$ feature space, the kernel is given as follows:\n\\begin{tcolorbox}[boxsep=2pt,left=2pt,right=2pt,top=4pt,bottom=4pt]\n\\bf{$\\bf{\\alpha-SU(1,1)}$ Kernel Function}\n\\begin{eqnarray}\nK(\\mathbf{x},\\mathbf{x^{\\prime}})= \\prod_{i=1}^{N} \\left[\\frac{1-\\tanh^{2}\\left(z\\sqrt{\\frac{|\\alpha|}{2}}\\right)}{1-\\tanh^{2}\\left(z\\sqrt{\\frac{|\\alpha|}{2}}\\right)e^{i(x_{i}-x^{\\prime}_{i})}}\\right]^{2k}.\n\\end{eqnarray}\n\\end{tcolorbox}\n\\indent For understanding the role of $\\alpha$ and $k$, we study the geometrical properties of the above-mentioned feature spaces.  We can define the line element of the feature space, by using the Fubini\u2013Study metric \\cite{bengtsson2017geometry}, that is,\n\\begin{eqnarray}\nds^{2}=\\| d|x;z,\\alpha\\rangle\\|^{2}-|\\langle x;z,\\alpha|d|x;z,\\alpha\\rangle|^{2},\n\\end{eqnarray}\n By using the above definition, the metric of $\\alpha-SU(2)$ feature space is obtained by\n\\begin{eqnarray}\\label{eq11}\nds^{2}=k\\alpha dz^{2}+\\frac{k}{2} \\sin^{2} \\left(z\\sqrt{2\\alpha}\\right)dx^{2},\n\\end{eqnarray}\nwhich describes a positive constant curvature with the scalar Ricci $R=4\/k$. This is in fact a surface of revolution conforming with a sphere \\cite{carinena2005central}.\nBy using the same method, the metric of the $\\alpha-SU(1,1)$ feature space is given by\n\\begin{eqnarray}\\label{eq10}\nds^{2}=k|\\alpha| dz^{2}+\\frac{k}{2} \\sinh^{2} \\left(z\\sqrt{2|\\alpha|}\\right)dx^{2}.\n\\end{eqnarray}\nwhich describes a negative constant curvature with the scalar Ricci $R=-4\/k$, conformal with pseudo-spheres \\cite{carinena2005central}. Figure \\ref{parametric_feature_spaces} shows topological categories of feature spaces associated with $\\alpha-SU(2)$ and $\\alpha-SU(1,1)$ coherent states.\n\\begin{figure}[t]\n  \\centering\n   \n  \\includegraphics[scale=0.6]{fig2.pdf}\n  \\caption{Spatial definition of feature spaces for kernel functions $\\alpha-SU(2)$ \\textbf{(A.)} And $\\alpha-SU(1,1)$ \\textbf{(B.)} Under parametric configuration $\\alpha = 2$ And $k = 1$}\n  \\label{parametric_feature_spaces}\n\\end{figure}\nAs seen above, two differing categories of the deformed coherent states are defined with different topologies: one ($\\alpha-SU(2)$) is constructed on a truncated Hilbert space, forming a compact constant curvature feature space, conforming with a sphere. The other ($\\alpha-SU(1,1)$) is built on an infinite Hilbert space that leads to a feature space conforming with a pseudo-sphere, with negative constant curvature.\n\n\nIn the next section, we present a illustrate the empirical effectiveness of the meta-theoretically derived kernel functions in different experimental settings.\n\n\\section{Empirical evaluation of the kernel functions}\\label{test_design}\n\nIn order to assess the effectiveness of the proposed kernel functions, we conducted a series of experiments using different well known synthetic datasets of the literature, namely 1) Python's \\textit{scikit-learn} library: the circles, moons, and 2) the iris dataset. \nThe circles and the moons dataset involves binary features in a binary classification problem. The iris dataset is a multiclass classification task with three classes and four features.\n\n\nSince the goal of SVMs is to find the maximum-margin hyperplane, a set of parameters are needed to control the error between these margins (Figure~\\ref{fig:svm_hyper} shows this optimization in the high-dimensional feature space). For the RBF kernel, two parameters play a major role in this optimization process:\n\\begin{itemize}\n\\item Hyperparameter $C$: is a regularization parameter that controls the trade-off between the decision boundary and mis-classification term. It basically controls how much mis-classifications are tolerable during the optimization problem.\n\\item $\\gamma$, which controls the non-linearity of the decision boundary. It defines how far influences the calculation of plausible line of separation. A low $\\gamma$ takes into consideration far away points to influence the decision boundary; a high $\\gamma$ considers only points that are close to the decision boundary.\n\\end{itemize}\n\\begin{figure}[t]\n   \n   \n    \\centering\n    \\includegraphics[scale=.6]{fig3.pdf}\n    \\caption{Kernel Trick. A kernel is applied to each data point to map the original non-linear observations into a higher dimensional space where the observations may become linearly separable through a hyperplane.}\n    \\label{fig:svm_hyper}\n\\end{figure}\nWhen considering the proposed meta-kernel, we have a set of new parameters that extend the current RBF kernel with a set of new non-linear functions that are based on the  dW-H algebra. These extra parameters also enable the construction of a feature space over a surface of revolution with constant curvature. Theoretically, this could lead to significant improvements when the dataset is distributed along revolution surfaces. The parameters for both $SU(1,1)$ and $SU(2)$ kernels are the following:\n\\begin{itemize}\n    \\item Parameter $k$ is related to the curvature of the feature surface and  controls the non-linearity of the decision boundary in such way that high values of the parameter consider points that are near to the decision boundary whilst low values cause points further away to influence the decision boundary. Figures \\ref{vis_su11} and \\ref{fig:vis_su2} illustrate the rule of the parameter $k$.   \n    \\item By considering a fixed curvature, i.e., $k=const.$, the product of parameters $z$ and $\\sqrt{\\alpha}$, as an extra parameter   $z\\sqrt{\\alpha}$, controls the decision boundary as well. Figure   \\ref{vis_su11} and \\ref{fig:vis_su2} indicates a schematic behaviour of parameters $\\alpha$ and $k$, for $\\alpha-SU(1,1)$ and $\\alpha-SU(2)$ respectively.  \n\\end{itemize}\n\\begin{figure}[t]\n \n   \n    \\centering\n  \\includegraphics[scale=0.55]{fig4.pdf}\n  \\caption{Shape of the  kernel function obtained by $\\alpha-SU(1,1)$-coherent state  for different strength hyperparameters $\\alpha$ and $k$, while $z=1$. The input $x$ is fixed at $(0, 0)$ and $x^{\\prime}$ is varied.}\n  \\label{vis_su11}\n\\end{figure}\n\\begin{figure}[t]\n  \n  \n\\centering\n  \\includegraphics[scale=.55]{fig5.pdf}\n  \\caption{Shape of the  kernel function obtained by $\\alpha-SU(2)$-coherent state  for different strength hyperparameters $\\alpha$ and $k$, while $z=1$. The input $x$ is fixed at $(0, 0)$ and $x^{\\prime}$ is varied.}\n  \\label{fig:vis_su2}\n\\end{figure}\n\nFigure~\\ref{fig:svc_classification} illustrates different decision boundaries that can be computed using the different kernels. One can see the different non-linearity properties of the $\\alpha-SU(2)$ and $\\alpha-SU(1,1)$ kernels compared to the standard RBF kernel. \n\n\n\\begin{figure}[t]\n   \n\\centering\n  \\includegraphics[scale=.7]{fig6.pdf}\n  \\caption{Decision boundaries computed using different kernels for the synthetic datasets \\textit{Moons} and \\textit{Circles}.}\n  \\label{fig:svc_classification}\n\\end{figure}\n\nFor evaluation purposes, the parameters that provided the best results in terms of precision were found using a grid search approach (for more details on the evaluation code and experimental setup. For the synthetic datasets, Moons and Circles, 1000 samples were generated, where 70\\% were used for the training process and 30\\% were used for the evaluation task. To make the classification task more challenging, we applied noise factors of 0.3 and 0.1 to these datasets, respectively. Learning curves were analysed to ensure unbiased results and no overfitting.  Table~\\ref{tab:results} summarises the results.\n\n\\begin{figure}[]\n   \n\\centering\n  \\includegraphics[scale=.7]{table.pdf}\n  \\caption{Results obtained using the proposed kernels $\\alpha$-$SU(1,1)$ and $\\alpha$-$SU(2)$ for different datasets. Note that for the Moons and Circles synthetic dataset, we generated 1000 samples of which $70\\%$ were used for training and the remaining $30\\%$ for test. These datasets were generated using a noise factor of 0.3 and 0.1, respectively.}\n\\label{tab:results}\n\\end{figure}\n\n\n The proposed meta-kernels $\\alpha$-SU(1,1) and $\\alpha$-SU(2) exhibit state of the art performance when compared with the RBF kernel. \n These kernels provide a significant advantage for data points distributed over curved surfaces. Given that it is hard to find benchmark datasets with those characteristics, the results presented in Table~\\ref{tab:results} suggest an cautiously encouraging first step.\n\n\n\n\\section{Discussion} \\label{discussion}\n\n\nIn SVM-based classification problems, the appropriate choice of a kernel is fundamental to achieve high classification performance. \nHowever, the current 'trial-and-error' nature of selecting the best kernel poses significant challenges, especially when one considers kernels that can support both classical and quantum-inspired machine learning algorithms~\\cite{Ali06}. \nIn this section, a visual analysis is provided of how different kernels are derived from the $\\alpha$ and $k$ parameters of both $\\alpha-SU(1,1)$ and $\\alpha-SU(2)$ kernels in order to promote a clear discrimination between kernel functions. \n\nThe $\\alpha$ parameter controls allows a specific kernel function is derived from  deformed Weyl-Heisenberg (dW-H) algebra. \nA value of $\\alpha=0$ is a flat surface. When $\\alpha$ is small, it contributes to an almost flat surface, when maintaining $k$ low. On the other hand, when $\\alpha$ is high, it contributes to squeeze the function towards its center. Figure~\\ref{vis_su11} shows the impact of parameter $\\alpha$ in in the $\\alpha-SU(1,1)$ kernel.\nRegarding the $\\alpha-SU(2)$ kernels, the parameter $\\alpha$ generates a kernel that maps the non-linear observations into a higher dimensional space that `folds' the data in the feature space. A high value in $\\alpha$ and $k$ squeeze  these folds towards the center of the distribution of the geodesic distances between the data points as visualized in Figure~\\ref{fig:vis_su2}.\n\n\n\n\n\n\nIn terms of the empirical evaluation of the kernels, Table~\\ref{tab:results} indicates the the best results were obtained with low values of $\\alpha = 0.1$, $1.0 \\leq k \\leq 2.0$, with the $z$ parameters in the range: $ 0.2 \\leq z \\leq 4.6$. \n\n\n\\section{Conclusion}\nIn this paper, \nby using the theory of non-linear coherent states, we put forward a meta-kernel approach for deriving kernel functions for use in ML. \nMore specifically, data is mapped into a feature space which is defined as a deformed coherent state as defined by a deformed Weyl-Heisenberg algebra.\nThis algebra unifies the well-known $SU(2)$, Weyl-Heisenberg, and $SU(1,1)$ groups, through a common  parameter $\\alpha$.\nIn addition, by studying tgeometrical properties of feature space constructed on the dW-H coherent state, we showed that the meta-kernel function applies associated  surfaces of revolution as feature spaces identified with non-linear coherent states.  \nAn empirical investigation compares the $\\alpha-SU(2)$ and $\\alpha-SU(1,1)$ kernels derived from the meta-kernel which shows performance similar to the Radial Basis kernel.\n\nKernel functions drive developments in the field of machine learning and the meta-kernel function presented in this paper opens new theoretical avenues for the definition and exploration of kernel functions.\\\\\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgement}\nThis research was supported by the Asian Office of Aerospace Research and Development (AOARD) grant: FA2386-17-1-4016.\n\n\\appendices\n\\section{Geometrical Properties of Feature surfaces}\nThe  Christoffel symbols of the second kind according to definition are give by\n\\begin{eqnarray}\n\\Gamma_{ij}^{k}=\\frac{1}{2}g^{kl}\\left[\\partial_{i}g_{jl}+\\partial_{j}g_{il}-\\partial_{l}g_{ij}\\right]\n\\end{eqnarray}\nin which $g_{ij}$ is the $(i,j)$th component of the metric, $g^{ij}=g_{ij}^{-1}$ and $\\partial_{i}$ is an abbreviation of $\\frac{\\partial}{\\partial x_{i}}$. Also, according to standard notation, the Einstein summation convention is applied, i.e.,  summation over a set of indexed terms in a formula, e.g. $g^{ij}g_{jk}=g^{i1}g_{1k}+g^{i2}g_{2k}$. By using the metric (\\ref{eq11}), non-zero components of \nthe  Christoffel symbols of the second kind are respectively given by:\n\\begin{eqnarray}\n\\Gamma_{xz}^{x}&=&\\sqrt{2\\alpha}\\cot (\\sqrt{2\\alpha}\\ z)\\nonumber\\\\\n\\Gamma_{xx}^{z}&=&-\\frac{1}{2\\sqrt{2\\alpha}}\\sin (2\\sqrt{2\\alpha}\\ z)\n\\end{eqnarray}\nAlso, according to definition, the Ricci tensor is given by\n\\begin{eqnarray}\nR_{ij}=\\partial_{k}\\Gamma_{ij}^{k}-\\partial_{i}\\Gamma_{kj}^{k}+\n\\Gamma_{ij}^{k}\\Gamma_{kl}^{l}-\n\\Gamma_{ik}^{l}\\Gamma_{lj}^{k}.\n\\end{eqnarray}{}\nHence,\nthe non-zero Ricci tensors are given by:\n\\begin{eqnarray}\nR_{xx}= \\sin^{2} \\left[\\sqrt{2\\alpha} z\\right], R_{zz}=2\\alpha. \n\\end{eqnarray}\nThe Ricci scalar, which gives the curvature, is obtained by\n\\begin{eqnarray}\nR=\\frac{4}{k}\n\\end{eqnarray}\n\\indent In the case of metric (\\ref{eq10}), non-zero components are given by\n\\begin{eqnarray}\n\\Gamma_{xz}^{x}&=&\\sqrt{2\\alpha}\\coth (\\sqrt{2\\alpha}\\ z)\\nonumber\\\\\n\\Gamma_{xx}^{z}&=&\\frac{-1}{2\\sqrt{2\\alpha}}\\sinh (2\\sqrt{2\\alpha}\\ z)\n\\end{eqnarray}\nand the Ricci tensor are given by\n\\begin{eqnarray}\nR_{xx}=\\sinh^{2}(z\\sqrt{2\\alpha}), R_{zz}=-2\\alpha \n\\end{eqnarray}\nThe Ricci scalar, which gives the curvature, is obtained by\n\\begin{eqnarray}\nR=-\\frac{4}{k}\n\\end{eqnarray}\n\n\n\n\n\n\n\\bibliographystyle{unsrt} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}