Ikuta, Takuya

Nomura, Kazumasa

Kyoto University Research Information Repository

Title General form of non-symmetric spin models (AlgebraicCombinatorics)Author(s)Ikuta, Takuya; Nomura, KazumasaCitation数理解析研究所講究録 (1999), 1109: 35-41Issue Date1999-08URL http://hdl.handle.net/2433/63313RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityGeneral Form of Non-Symmetric Spin ModelsTakuya IkutaKazumasa NomuraAbstract. A spin model (for link invariants) is a square matrix $W$ with non-zero complexentries which satisfies certain axioms. Recently [6] it was shown that ${}^{t}WW^{-1}$ is a permutationmatrix (the order of this permutation matrix is called the “index” of $W$), and a general formwas given for spin models of index 2. In the present paper, we generalize this general form toan arbitrary index $m$ . In particular, we give a simple form of $W$ when $m$ is a prime number.1 IntroductionSpin models were introduced by Vaughan Jones [7] to construct invariants of knots andlinks. A spin model is essentially a square matrix $W$ with nonzero entries which satis-fies two conditions (type II and type III conditions). In his definition of a spin model,Jones considered only symmetric matrices. It was generalized to non-symmetric case by$\mathrm{K}\mathrm{a}\mathrm{W}\mathrm{a}\mathrm{g}_{0}\mathrm{e}-\mathrm{M}\mathrm{u}\mathrm{n}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{a}- \mathrm{w}_{\mathrm{a}}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{i}[8]$ .Recently, Rangois Jaeger and the second author [6] introduced the notion of “index”of a spin model. For every spin model $W$ , the transpose ${}^{t}W$ is obtained from $W$ by apermutation of rows. Let $\sigma$ denote the corresponding permutation of $X=\{1, \ldots , n\}(n$is the size of $W$). Then the index $m$ is the order of a. In [6], it was shown that $X$ ispartitioned into $m$ subsets $X_{0},$ $X_{1},$ $\ldots,$ $X_{m-1}$ such that $W(x, y)=\eta^{i}-jW(y, x)$ holds forall $x\in X_{i},$ $y\in X_{j}$ . Moreover, the case of $m=2$ was deeply investigated, and a generalform of spin models of index 2 was given.In the present paper, we investigate the structure of spin models of an arbitrary index$m$ . In Section 4, we show that $W$ is decomposed into blocks $W_{ij}$ , and $W_{ij}$ splits intoKronecker product of two matrices $S_{ij}$ and $T_{ij}$ (Proposition 4.3). In Section 5, we giveconditions on $T_{ij}$ (Propositions 5.1 and 5.5). In Section 6, we apply this general form tosome special cases (Propositions 6.1 and 6.2). In particular, we give a simple form of $W$when the index $m$ is a prime number (Corollary 6.3).2 PreliminariesIn this section, we give some basic materials concerning spin models and associationschemes. For more details the reader can refer to [3, 7, 5, 6].Let $X$ be a finite non-empty set with $n$ elements. We denote by Mat$\mathrm{x}(\mathrm{C})$ the set ofsquare matrices with complex entries whose rows and columns are indexed byX-. For$W\in \mathrm{M}\mathrm{a}\mathrm{t}_{X}(\mathrm{c})$ and $x,$ $y\in X$ , the $(x, y)$-entry of $W$ is denoted by $W(x, y)$ .1109 1999 35-41 35A type II matrix on $X$ is a matrix $W\in \mathrm{M}\mathrm{a}\mathrm{t}_{X}(\mathrm{c})$ with nonzero entries which satisfiesthe type II condition:$\sum_{x\in X}\frac{W(a,x)}{W(b,x)}=n\delta_{a,b}$ (for all $a,$ $b\in X$ ).Let $W^{-}\in$ Mat$x(\mathrm{C})$ be defined by $W^{-}(x, y)=W(y, x)^{-1}$ . Then type II condition iswritten as $WW^{-}=nI$ (I denotes the identity matrix). Hence, if $W$ is a type II matrix,then $W$ is non-singulax with $W^{-1}=n^{-1}W^{-}$ It is clear that $W^{-1}$ and ${}^{t}W$ are also typeII matrices.A type II matrix $W$ is called a spin model on $X$ if $W$ satisfies type III condition:$\sum_{x\in X}\frac{W(a,x)W(b_{X)}}{W(c,x)},=D\frac{W(a,b)}{W(a,c)W(c,b)}$ (for all $a,$ $b,$ $c\in X$) (1)for some nonzero complex number $D$ . The number $D$ is called the loop variable of $W$ .Setting $b=c$ in (1), $\Sigma_{x\in X}W(a, X)=DW(b, b)^{-1}$ holds, so that the diagonal entries$W(b, b)$ is a constant, which is called the modulus of W.For a spin model $W$ with loop variable $D$ , any nonzero scalar multiple $\lambda W$ is a spinmodel with loop variable $\lambda^{2}D$ . Usually $W$ is normalized so that $D^{2}=n$ , but we allowany nonzero value of $D$ in this paper to simplify our arguments.Observe that, for any spin models $W_{i}$ on $X_{i}$ with loop variable $D_{i}(i=1,2)$ , their tensor(Kronecker) product $W_{1}\otimes W_{2}$ is a spin model with loop variable $D=D_{1}D_{2}$ . Conversely,it is not difficult to show that, if $W_{1}\otimes W_{2}$ and $W_{1}$ are spin models, then $W_{2}$ must be aspin model.A (class $d$) association scheme on $X$ is a partition of $X\cross X$ with nonempty relations$R_{0},$ $R_{1},$$\ldots,$$R_{d},$ where. $R_{0}=\{(x, x)|x\in X\}$ which satisfy the following conditions:(i) For every $i$ in $\{0,1, \ldots, d\}$ , there exists $i’$ in $\{0,1, \ldots, d\}$ such that$R_{i’}=\{(y, X)|(X, y)\in h\}$ .(ii) There exist integers $p_{ij}^{k}(i, j, k\in\{0,1, \ldots , d\})$ such that for every $(x, y)\in R_{k}$ , thereare precisely $p_{ij}^{k}$ elements $z$ such that $(x, z)\in R_{i}$ and $(z, y)\in R_{j}$ .(iii) $p_{i\mathrm{j}}^{k}=p_{ji}^{k’}$ for every $i,$ $j$ in $\{0,1, \ldots , d\}$ .Let $A_{i}$ denote the adjacency matrix of the relation $R_{i}$ , so $A_{i}\in \mathrm{M}\mathrm{a}\mathrm{t}_{x}(\mathrm{c})$ is a $\{0,1\}$-matrixwhose $(x, y)$-entry is equal to 1 if and only if $(x, y)\in R_{i}$ . Clearly $A_{0=}I,$ $A_{i}\circ A_{j}=\delta_{i,j}A_{i}$(entry-wise product), $\Sigma_{i=0^{A_{i}}}^{d}=J$ (all l’s matrix), and $A_{i}A_{j}=\Sigma_{k=0}^{d}p_{ij}A_{k}k$ hold. Thelinear span $A$ of $\{A_{0}, A_{1}, \ldots, A_{d}\}$ becomes a subalgebra of Mat$x(\mathrm{C})$ , called the Bose-Mesner algebra of the association scheme. Observe that $A$ is closed under entry-wiseproduct, $A$ is closed under transposition $Arightarrow {}^{t}A$ , and $A$ contains $I,$ $J$ .3 Associated PermutationLet $W$ be a spin model on $X$ . Then there exists an association scheme $R_{0},$ $\ldots,$ $R_{d}$ on $X$such that the corresponding Bose-Mesner algebra $A$ contains $W$ ([5] Theorem 11). In [6], it36was shown that ${}^{t}WW^{-1}=\mathrm{A}_{s}$ (the adjacency matrix of $R_{s}$ ) for some $s\in\{0,1, \ldots, d\}$ , andmoreover $A_{s}$ is a permutation matrix ([6] Proposition 2). Let $\sigma$ denote the correspondingpermutation on $X$ , so that $A_{s}(x, y)=1$ if $y=\sigma(x)$ and $A_{s}(x, y)=0$ otherwise. Theorder $m$ of $\sigma$ is called the index of $W$ .Observe that $m=1$ if and only if $W$ is symmetric. Also observe that, for two spinmodels $W_{i}$ of index $m_{i}(i=1,2)$ , the index of $W_{1}\otimes W_{2}$ is equal to the least commonmultiple of $m_{1}$ and $m_{2}$ . In particular, tensor product of a spin model of index $m$ with anysymmetric spin model has index $m$ .Lemma 3. 1 (i) $W(x, \sigma(x))=W(y, \sigma(y))$ $(x, y\in X)$ .(ii) $W(y, x)=W(\sigma(X), y)$ $(x_{l}y\in X)$ .(iii) Every orbit of a has length $m$ .Lemma 3. 2 There is a partition $X=x_{0}\cup\cdots\cup X_{m}-1^{\mathit{8}u}Ch$ that (for all $i_{J}j\in\{0,$ $\ldots m-$)$1\})$.$W(x, y)=\eta Wi-j(y)X)$ $($for all $x\in X_{i},$ $y\in X_{j})_{\}}$where $\eta$ denotes a primitive $m$ -root of unity. $M_{oreov}er_{J}$ for every $i_{f}\sigma(X_{i})=X_{j}$ holds forsome $j$ .We fix a primitive $m$-root of unity $\eta$ , and let $X_{0},$ $\ldots$ , $X_{m-1}$ be the partition of $X$ givenin Lemma 3. 2. We identify the index set $\{0,1, \ldots , m-1\}$ with $\mathrm{Z}_{m}=\mathrm{Z}/m$ Z. By Lemma3. 2, there is a permutation $\pi$ on $\mathrm{Z}_{m}$ such that $\sigma(X_{i})=x_{\pi}(i)(i\in \mathrm{Z}_{m}.)$ . Let $t$ denote theorder of $\pi$ , and set $k=m/t$.Lemna 3. 3 $\pi(i)-i=\pi(j)-j$ for all $i,$ $j\in \mathrm{Z}_{m}$ .Lemma 3. 4 There exists an automorphism $\varphi$ of the additive group $\mathrm{Z}_{m}$ such that$\pi(\varphi(i))=\varphi(i+k)$ for all $i\in \mathrm{Z}_{m}$ Moreover, $W(x, y)=(\eta^{\varphi(1)})^{i-j}W(y, x)$ for every$X\in x_{\varphi()_{Jy}}i\in X_{\varphi(}j)$ .Thus, by reordering the indices $\{0,1, \ldots , m-1\}$ by $\varphi$ , and by replacing $\eta$ with $\eta^{\varphi(1)}$ ,we may assume that$\pi(i)=i+k$ $(i\in \mathrm{Z}_{m})$ .4 General Form of $\mathrm{W}$We use the notation of the previous section. We also use the notation:$\gamma_{k}(\ell, i)=\eta-\ell i-(k/2)l(^{\ell 1)}-$ . (2)Proposition 4. 1 Let $i,$ $j\in \mathrm{Z}_{m}$ and $x\in X_{i_{\lambda}}y\in X_{j}$ . Then for $p_{J}\ell’\in \mathrm{Z}_{f}$$W(\sigma^{\ell_{(X}\ell’}),$$\sigma(y))=\gamma_{k}(p_{-}\ell’, i-j)W(x, y)$ . (3)Lemma 4. 2 If $m$ is $even_{f}$ then $ki\mathit{8}$ even.37For $i\in \mathrm{Z}_{m}$ , set$\Delta_{i}=\bigcup_{0h=}^{1}t-x_{i+hk}$ .Observe that $|\triangle_{i}|=t(n/m)=tn/(kt)=n/k$ , and that$X=\overline{\bigcup_{i=0}^{k1}}\Delta i_{)}$Since $\sigma(\triangle_{i})=\triangle_{i},$ $\triangle i$ is partitioned into a-orbits $Y_{\alpha}^{i}$ :$\Delta_{i}=\bigcup_{\alpha=1}^{r}Y_{\alpha}^{i}$ $(i=0, \ldots, k-1)$ ,where $r=|\Delta_{i}|/m=n/(mk)$ . Observe that $|Y_{\alpha}^{i}|=m$ and $|Y_{\alpha}^{i}\cap X_{i}|=k$ . We chooserepresentative elements$y_{\alpha}^{i}\in Y_{\alpha}^{i}\cap X_{i}$ $(i=0, \ldots, k-1, \alpha=1, \ldots, r)$ .Then$X=\{\sigma^{\ell}(y^{i}\alpha)|i=0, \ldots k-)1, \alpha=1, \ldots, r, P=0, , . . , m-1\}$,and$W(\sigma^{\ell}(y\alpha i), \sigma\ell’(y^{j}\beta))=\gamma k(^{p-\ell}J, i-j)W(y\alpha’ y_{\beta}^{j})i$for $l,$ $\ell’\in \mathrm{Z}_{m},$ $i,$ $j=0,$ $\ldots,$ $k-1$ and $\alpha,$ $\beta=1,$ $\ldots$ , $r$ .We define square matrices $T_{ij}$ of size $r$ and $S_{ij}$ of size $m$ $(i,j=0, . .. , k-1)$ by$T_{ij}(\alpha\beta\rangle)=W(y\alpha i, y^{j}\beta)$ $(\alpha,\beta=1, \ldots, r)$ ,$S_{ij}(\ell, p_{)}=\gamma_{k}(\ell-lli-)j)$ $(^{\ell,P=}/\mathrm{o}, \ldots, m-1)$ .For subsets $A,$ $B$ of $X$ , let $W|_{A\cross B}$ denote the restriction (submatrix) of $W$ on $A\cross B$ .For two matrices $S,$ $T$ , we denote the Kronecker product by $S\otimes T$ .Proposition 4. 3 For $i_{l}j=0:\ldots k-1$ ,$W|_{Y_{\dot{\alpha}}\mathrm{X}Y_{\beta}}j=^{\tau_{i}}j(\alpha,\beta)sij$$(\alpha,\beta=1, \ldots, r)$ ,and$W|_{\Delta_{\mathfrak{i}^{\mathrm{X}}}}\triangle_{j}=S_{ij}\otimes\tau_{i}j$ .Thus $W$ decomposes into blocks $W_{ij}=W|_{\triangle_{i^{\cross\triangle}j}}(i,j=0, \ldots k-)1)$ , and each blockhas the form $W_{ij}=S_{ij}\otimes T_{ij}(i,j=0, \ldots, k-1)$ .385 Type II and.Type III conditionsLet $m,$ $k,$ $t,$ $r$ be positive integers with $m=kt$ .Let $T_{ij}(i,j=0, \ldots, k-1)$ be any matrices of size $r$ with nonzero entries, and let $S_{ij}$$(i,j=0, \ldots , k-1)$ be the matrix of size $m$ defined by$S_{ij}(\ell,\ell J)=\gamma_{k}(\ell-^{p’},i-j)$ $(l,l’=0,\ldots,m-1)$ ,where $\gamma_{k}$ is defined by (2) for a prinitive $m$-root of unity $\eta$ . Now set$W_{ij}=s_{ij}\otimes\tau_{i}j$ $(i,j=0, \ldots, k-1)$ ,and let $W$ be the matrix of size $n=kmr$ whose $(i, j)$ block is $W_{ij}(i,j=0, \ldots , k-1)$ .We index the rows and the columns of $W$ by the set:$X=\{[i, \ell, \alpha]|0\leq i\leq k-1,0\leq\ell\leq m-1,1\leq\alpha\leq r\}$ ,so that$W([i,\ell,\alpha], [j,\ell’,\beta])=S_{i}j(\ell,P’)\tau_{i}j(\alpha,\beta)$ .Proposition 5. 1 $W$ is a type II $matr\dot{\eta}x$ if and only if $T_{ij}$ is a type II matrix for all $i$ ,$j\in\{0, \ldots, k-1\}$ .Lemma 5. 2 $A_{\mathit{8}Su}mek$ is even when $m$ is even. Then the matrix $W\mathit{8}atisfieS$ the typeIII condition (1) if and only if the following equation holds for all $i_{1_{J}}i_{2f}i_{3}\in\{0, \ldots , k-1\}$and for all $\alpha_{1_{J}}\alpha_{2:}\alpha_{3}\in\{1, \ldots , r\}$ :$\sum_{i=0}^{k1}-(_{\ell_{=}0}^{m-}\sum^{1}\eta^{-}\gamma k(\ell kl, i-i_{1}-i2+i_{3}))(_{\alpha}\sum_{=1}^{r}\frac{T_{i_{1)}i}(\alpha_{1},\alpha)\tau i_{2},i(\alpha_{2},\alpha)}{T_{i_{3},i}(\alpha_{3},\alpha)})$$=D \frac{T_{i_{1},i_{2}}(\alpha 1,\alpha_{2})}{T_{i_{1},i_{3}}(\alpha_{1},\alpha 3)Ti_{3},i_{2}(\alpha 3,\alpha 2)}$ .Lemma 5. 3 For all $u,$ $s(0\leq u\leq t-1,0\leq s\leq k-1)_{f}$$\gamma_{k}(u+St,j)=((-1)t-1\eta-tj)^{s}\gamma_{k}(u,j)$ .Lenma 5. 4 (i) If $t$ is odd, then$m- \ell 0\sum_{=}^{1}\eta-k\ell\gamma k(^{pj},)=\{$$k \sum_{=u0}^{t-1}\eta^{-uj}-ku(u+1)/2$ if $j\equiv 0$ (mod $k$),$0$ $\mathit{0}\grave{t}he7wise$ .39(ii) If $t$ and $k$ are even, then$m- \sum_{\ell_{=}0}^{1}\eta^{-k}\gamma\ell k(^{p},j)=\{$$k\Sigma_{u=}^{t-1-u}0\eta j-ku(u+1)/2$ if $j \equiv\frac{k}{2}$ (mod $k$ ),$0$ $otherwi\mathit{8}e$.Proposition 5. 5 $A_{\mathit{8}}sumek$ is even when $m$ is even. Then the matrix $W$ satisfiesthe type III condition (1) if and only if the following equation holds for all $i_{1},$ $i_{2y}i_{3}\in$$\{0, \ldots, k-1\}$ and for all $\alpha_{1},$ $\alpha_{2_{J}}\alpha_{3}\in\{1, \ldots, r\}$ :$(_{u=}^{t-} \sum^{1}\eta^{-}-i\hat{i})u(-ku(u+1)/2)\mathrm{o}(_{\alpha}\sum_{=1}^{r}\frac{T_{i_{1},i}(\alpha_{1},\alpha)T_{i_{2}},i(\alpha_{2},\alpha)}{T_{i_{3}i1}(\alpha_{3}1\alpha)})=(D/k)\frac{\tau_{i_{1},i_{2}}(\alpha_{12}\alpha)}{T_{i_{1},i_{3}}(\alpha_{1},\alpha 3)\tau i\mathrm{a}i_{2}(\alpha_{3,2}\alpha)},,$’where $\text{\^{i}}=i_{1}+i_{2}-i_{3}$ , and $i$ denotes the integer in $\{0, \ldots , k-1\}\mathit{8}uch$ that$i\equiv\{$\^i (mod $k$ ) if $t$ is odd,$\hat{i}+\frac{k}{2}$ (mod $k$ ) if $t$ is even.6 Some Special CasesWe use the notation in Section 4.Proposition 6. 1 Suppose $k=1$ . Then $m$ is $odd_{f}$ and$W=S\otimes^{\tau}$,where $S$ is a spin model of $\mathit{8}izem$ and index $m$ which is given by$S(p, p)=\eta^{-(1})/2)(^{\ell-}l’)(\ell-\ell’-1 (^{p\ell’=},0,1, \ldots m-)1)$ ,and $T$ is a $symmet_{7\dot{\eta}}c$ spin model of $\mathit{8}izen/m$ .Proposition 6. 2 Suppose $k=m$ . Then$W|_{x_{i^{\cross}}X_{j}}=S_{ij}\otimes\tau_{i}j$ $(i,j=0,1, \ldots, m-1))$and$S_{ij}(\ell, l/)=\eta)-(\ell-\mathit{1}^{J}(i-j) (p, p^{J}=0, \ldots)1m-)$.The matrices $T_{ij}$ are type II matrice8 of size $r=n/m^{2}$ . Moreover the following equationholds for all $i_{1},$ $i_{2},$ $i_{3}\in\{0, \ldots , m-1\}$ and for all $\alpha_{1_{2}}\alpha_{2f}\alpha_{3}\in\{1, \ldots, r\}$ :$\sum_{\alpha=1}^{r}\frac{T_{i_{1},i}(\alpha_{1},\alpha)T_{ii)}(2\alpha_{2},\alpha)}{T_{i_{3},i}(\alpha_{3},\alpha)}=(D/m)\frac{T_{i_{1},i_{2}}(\alpha 1\alpha 2)}{T_{i_{1},i_{3}}(\alpha_{1},\alpha \mathrm{s})T_{ii}(3,23,\alpha\alpha 2)},$,where $i$ denotes the integer in $\{0, \ldots, m-1\}\mathit{8}uch$ that $i\equiv i_{1}+i_{2}-i_{3}$ (mod $m$).40Corollary 6. 3 Let $W$ be a spin model on $X$ ofprime index $m$ . Then one of the following$holdS_{J}$ where $\eta$ denotes a $p_{\dot{\mathcal{H}}mit}ivem$-root of unity.(i) $W=S\otimes T$, where $S$ is a spin model of $\mathit{8}izem$ with$S(p,\ell/)=\eta-(1/2)\mathrm{t}\ell_{-\ell(-})\ell_{-}\ell’1)$ $(P, P’=0,1, \ldots, m-1)$ ,and $T$ is a symmetric spin model of size $|X|/m$ .(ii) $W$ decomposes into $m^{2}$ blocks $W_{ij}(i, j=0, \ldots , m-1)$ with $W_{ij}=S_{ij}\otimes T_{ij_{J}}$ where$S_{ij}$ are matrices of size $m$ defined by$S_{ij}(\ell, p’)=\eta^{-(l}-\ell \text{ })(i-\mathrm{j}\rangle$ $(^{\ell,\ell\prime}=0,1, \ldots, m-1)$ ,and $T_{ij}$ are type II matrice8 of $\mathit{8}i_{Zer}=n/m^{2}$ which satisfy the follouring equationfor all $i_{1_{2}}i_{2},$ $i_{3}\in\{0, \ldots , m-1\}$ and for all $\alpha_{1_{J}}\alpha_{2},$ $\alpha_{3}\in\{1, \ldots, r\}.\cdot$$\sum_{\alpha=1}^{r}\frac{T_{\dot{l}1},i(\alpha 1\alpha)T_{i_{2}},i(\alpha_{2},\alpha)}{T_{i_{3},i}(\alpha_{3},\alpha)},=(D/m)\frac{\tau_{i_{1},i_{2}}(\alpha_{12}\alpha)}{\tau_{i_{1},i_{3}}(\alpha_{1},\alpha_{3})T_{ii}(3,23,\alpha\alpha 2)},$ ,where $i$ denotes the integer in $\{0, \ldots, m-1\}$ such that $i\equiv i_{1}+i_{2}-i_{3}$ (mod $m$).References[1] E. Bannai, “Modular invariance property and spin models attached to cyclic group associa-tion schemes,” J. Stat. Plann. and Inference, 51 (1996), 107-124.[2] E. Bannai and Et. Bannai, “Spin models on finite cyclic groups,” J. Alg. Combin. 3 (1994),243-259.[3] E. Bannai and T. Ito, Algebraic Combinatorics I, $\mathrm{B}\mathrm{e}\mathrm{n}\mathrm{j}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{n}/\mathrm{C}\mathrm{u}\mathrm{m}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{g}_{\mathrm{S}}$, Menlo Park, 1984.[4] E. Bannai, Et. Bannai, and F. Jaeger, “On spin models, modular invariance, and duality,”J. Alg. Combin. 6 (1997), 203-228.[5] F. Jaeger, M. Matsumoto, and K. Nomura, “Bose-Mesner algebras related to type II matricesand spin models,” J. Alg. Combin. 8 (1998), 39-72.[6] F. Jaeger and K. Nomura, “Symmetric versus non-symmetric spin models for link invariants,”J. Alg. Combin., to appear.[7] V.F.R. Jones, “On knot invariants related to some statistical mechanical models,” Pac. J.Math. 137 (1989), 311-336.[8] K. Kawagoe, A. Munemasa, and Y. Watatani, “Generalized spin models,” J. of Knot Theoryand its Ramifications 3 (1994), 465-475.41

General form of non-symmetric spin models (Algebraic Combinatorics)

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General form of non-symmetric spin models (Algebraic Combinatorics)

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