数理解析研究所講究録
第 846 巻 1993 年 90-100

90

Hadamard Matrices and Generalized Spin Models
九州大 (理)山田美枝子(Mieko Yamada)

Abstract
The concept of spin models was introduced by V.F. Jones in 1989. K.Kawagoe,
A.Munemasa and Y.Watatani generalized it by removing the condition of symmetry. Recently E.Bannai and E.Bannai further generalized the concept of spinmodels which is called 4-weight spin models or generalized generalized spin models. On the otherhand, A.A. Ivanov and I.V. Chuvaeva showed that symmetric
amorphous association schemes of class 4 obtained from Hadamard matrices. An
infinite family of Hadamard matrices and of complex Hadamard matrices can beconstructed by fusing the relations of these amorphous association schemes.
We show the necessary and sufficient condition that these Hadamard matrices
give generalized spin models of symmetric Hadamard type and of pseudo-Jones
type. A special class of Hadamard matrices satisfies this necessary and sufficientcondition. Furthermore Hadamard matrices constructed by fusing amorphous association schemes are also contained in the special class if Hadamard matrices giving
these amorphous assoication schemes are contained in the special class. It meansthat there exist infinite families of generalized spin models of symmetric Hadamard
type and of pseudo-Jones type.

Introduction

1

The concept of spin models was introduced by V.F. Jones [5] in 1989 to give the
Link invariant. K.Kawagoe, A.Munemasa and Y.Watatani [6] generalized it by removing
the condition of symmetry. Recently E.Bannai and E.Bannai [1] further gneralized theconcept of spin models which is called 4-weight spin models or generalized generalized
spin models.
Definition 1 [E.Bannai-E.Bannai, [1]] Let $X$ be a finite set and $w;,$ $(i=1,2,3,4)$ be$functionsonX\cross XtoC$ . $Then(X, w_{1}, w_{2},w_{3}, w_{4})is4$ -weight spin model of loop variable
Difthe following conditions are satisfied for any
:
and
(1)
,
$\alpha,$$\beta$$\gamma\in X$$w_{1}(\alpha, \beta)w_{3}(\beta,\alpha)=1,$ $w_{2}(\alpha,\beta)w_{4}(\beta, \alpha)=1$(2)$\Sigma_{x\in X}w_{1}(\alpha, x)w_{3}(x,\beta)=n\delta_{\alpha,\beta},$$\Sigma_{x\in X}w_{2}(\alpha,x)w_{4}(x,\beta)=n\delta_{\alpha,\beta_{J}}$,$(3a)\Sigma_{x\in X}w_{1}(\alpha, x)w_{1}(x, \beta)w_{4}(\gamma,x)=Dw_{1}(\alpha, \beta)w_{4}(\gamma,\alpha)w_{4}(\gamma, \beta)$

91$(3b)\Sigma_{x\in X}w_{1}(x, \alpha)w_{1}(\beta, x)w_{4}(x,\gamma)=Dw_{1}(\beta, \alpha)w_{4}(\alpha,\gamma)w_{4}(\beta,\gamma)$,

where $D^{2}=n=|X|$ .
Let be a diagram of an oriented link. We color the regions of in black and white
so that the unbounded region is colored white and adjacent regions have different colors.
We construct an oriented graph assigning a black region to a vertex and a crossing to anedge. We get exactly four kinds of crossings according to the colors of the regions andthe orientations of links. Then we attach four weights, 1,2,3,4 to four kinds of edges,
namely to four kinds of crossings, respectively. Then we get an oriented graph with fourkinds of weights.
Denote the weight for an edge
. Let $X$ be a finite set withbybe complex valued functions defined on $X\cross X$ .
$|X|=n=D^{2}$ . Letandis defined by
Under these assumptions, the partition function
$L$

$L$

$n(\alphaarrow\beta)$$\alphaarrow\beta$$n$$w_{1},$ $w_{2},$ $w_{3}$$w_{4}$$Z_{L}$$Z_{L}=D^{-v(L)} \sum_{\alpha\sigma}\prod_{arrow\beta}w_{n(\alphaarrow\beta)}(\sigma(\alpha),\sigma(\beta))$where a state is a map from the vertices of the graph to $X$ and $v(L)$ is a number ofvertices of the graph.
If (X,) is a 4-weight spin models with loop variable , then the partition
functionis invariant under the Reidemeister moves of types II and III. See the details
in [1].
We consider the special case of 4-weight spin models. Let
for
$\{+,
-\}$
$i=1,2,3,4$ . Let andbe from. A 4-weghit spin models withandis called a generalized spin model of Jones type. A 4-weight spinmodels withandis called a generalized spinmodel of pseudo-Jones type. Further, a 4-weight spin models withandis called a generalized spin model of Hadamard type.
K. Nomura [7] constructed a family of symmetric spin models of Jones type of loop
variablefrom Hadamard matrices of order $4n$ . M. Wakimoto [8] showed that spinmodels of Jones type and 4-weight spin models can be constructed by using Lie algebra.
We treat here generalized spin models of pseudo-Jones type and of symmetric Hadamard
type. First we give the definitions.
$\sigma$$D$$w_{1},$ $w_{2},$ $w_{3},$ $w_{4}$$Z_{L}$$W_{i}=(w_{i}(\alpha, \beta))_{\alpha,\beta\in X}$$\epsilon’$$W_{1},$ $W_{2}\in\{.W_{\epsilon}, W_{\epsilon}^{t}\}$$\epsilon$$W_{3},$ $W_{4}\in\{W_{\epsilon’}, W_{\epsilon}^{t},\}$$W_{1},$ $W_{4}\in\{W_{\epsilon}, W_{\epsilon}^{t}\}$$W_{2},$ $W_{3}\in\{W_{\epsilon’}, W_{\epsilon}^{t},\}$$W_{1},$ $W_{3}\in\{W_{\epsilon}, W_{\epsilon}^{t}\}$$W_{2},$ $W_{4}\in\{W_{\epsilon’}, W_{\epsilon}^{t},\}$$4\sqrt{n}$Definition 2 [E.Bannai-E.Bannai, [1]] (X,) is a generalized spin model of pseudoJones type if the following conditions are satisfied for any.
and$w_{+},$ $w_{-}$$\alpha,$(0) $w_{+}(\alpha,\beta)=w_{+}(\beta,\alpha),$(1J)$w_{+}(\alpha,\beta)w_{-}(\alpha, \beta)=1$$w_{-}(\alpha, \beta)=w_{-}(\beta, \alpha)$,,$\beta$$\gamma\in X$

92,$(2J)\Sigma_{x\in X}w_{+}(\alpha, x)w_{-}(x, \beta)=n\delta_{\alpha,\beta}$$(3P)\Sigma_{x\in X}w_{+}(\alpha, x)w_{+}(x, \beta)w_{+}(\gamma, x)=Dw_{+}(\alpha,\beta)w_{+}(\gamma, \alpha)w_{+}(\gamma,\beta)$.

) is a generalized spin model of symDefinition 3 [E.Bannai-E.Bannai, [1]] (X,:andmetric Hadamard type if the following conditions are satisfied for any$w_{+},$ $w_{-}$$\alpha,$(0)$w_{+}(\alpha,\beta)=w_{+}(\beta, \alpha),$ $w_{-}(\alpha, \beta)=w_{-}(\beta, \alpha)$$\gamma\in X$$\beta$,(1H) $W_{+}oW+=J,$ $W_{-}oW_{-}=J$,$(2H)W_{+}^{2}=nI,$ $W_{-}^{2}=nI$,,$( 3H)\sum_{x\in X}w_{\epsilon’}(\alpha, x)w_{\epsilon’}(x, \beta)w_{\epsilon}(\gamma, x)=Dw_{\epsilon’}(\alpha, \beta)w_{\epsilon}(\gamma, \alpha)w_{\epsilon}(\gamma, \beta)$

where is an Hadamard product,
unit matrix.
$0$

2

$J$

is the matrix whose entries are all 1 and$I$is theAmorphous association schemes and Hadamard
matrices

Theorem 1 [A.A. Ivanov-I.V. Chuvaeva, [3]] Let $H=(h_{i,j})$ be an Hadamard matrix of$(0\leq i\leq 4)$. The subsets
order $4n$ and $\Omega=\{0,1,2, \ldots, 4n-1\}$ . Putof$X\cross X$ aredefined by$X=\Omega\cross\Omega$$R_{4}=\{(x, x)|x\in X\}$,

$R_{1}=\{((x_{1}, x_{2}), (y_{1}, y_{2}))|x_{1}=y_{1}\}$,$R_{2}=\{((x_{1}, x_{2}), (y_{1}, y_{2}))|x_{2}=y_{2}\}$,$R_{3}=\{((x_{1}, x_{2}), (y_{1}, y_{2}))|h_{x_{1}x_{2}}h_{y_{1}y_{2}}h_{x_{1}y2}h_{y_{1}x_{2}}=1\}$,$R_{4}=\{((x_{1}, x_{2}), (y_{1}, y_{2}))|h_{x_{1}x_{2}}h_{y_{1}y_{2}}h_{x_{1}y_{2}}h_{y_{1}x_{2}}=-1\}$

Then (X,$R_{i},$$R_{0},$ $R_{1},$ $R_{2},$ $R_{3},$ $R_{4}$.

) is an amorphous association scheme of class 4.

Let $Y=(X, \{R_{i}\}_{0\leq i\leq d})$ be a commutative association scheme. A partition$(1 \leq i\leq e)$ andof the index set is said to be admissible if$R_{\alpha’}=\{(x, y)|(y, x)\in R_{\alpha}\}$ . Let
for some $j,$ $(1\leq i,j\leq e)$ where. If (X,) becomes an association scheme for every admissble
partition, then is defined to be amorphous.
$\Lambda_{0},$$\Lambda_{1},$$\Lambda_{e}$$\ldots,$$\Lambda_{0}=\{0\},$ $\Lambda_{i}\neq\phi$$\Lambda_{i}’=\{\alpha’|\alpha\in\Lambda_{i}\},$$R_{\Lambda_{i}}= \bigcup_{\alpha\in\Lambda_{i}}R_{\alpha}$$Y$$\{R_{\Lambda_{i}}\}_{0\leq i\leq e}$$\Lambda_{1}’=\Lambda_{j}$93Corollary 1 [2] The valencies and intersection numbers of an amorphous association
scheme mentioned in Theorem 1 are given as follows:(1) $k_{1}=k_{2}=4n-1,$ $k_{3}=(2n-1)(4n-1),$ $k_{4}=2n(4n-1)$(2) $p!_{i}=g^{2}:-3g_{i}+4n,p_{||}j=g_{i}(g_{i}-1),p_{j}^{i}=_{k}k\lrcorner_{1}g_{i}(g_{i}-1),p_{*j}^{l}=g;g_{j}$for $i\neq j\neq l,$ $0\leq i,j,$ $l\leq 4$ , where $g_{1}=g_{2}=1,$ $g_{3}=2n-1,g_{4}=2n$ .

We have the following theorem by using these amorphous association schemes.
$(0\leq i\leq 4)$ be adjacency matrices of an amorphous association
Theorem 2 Let
scheme obtained from an Hadanard matrix of order $4n$ by Theorem 1. Then
$A_{i},$(1) $M_{1}=A_{0}+A_{1}+A_{2}+A_{3}-A_{4},$ $M_{2}=A_{0}+A_{1}-A_{2}-A_{3}+A_{4},$ $M_{3}=A_{0}-A_{1}+A_{2}-A_{3}+A_{4}$are regular symmetric Hadamard matrices of order $(4n)^{2}$ ,(2) $L_{1}=A_{0}+A_{1}+A_{2}i+A_{3}i-A_{4}i$ and $L_{2}=A_{0}+A_{1}i+A_{2}+A_{3}i-A_{4}i$ are regularsymmetric complex Hadamard matrices of $(4n)^{2}$ ,
where

$i$

is a primitive$4^{th}$root of unity.

By using Theorems 1 and 2 repeatedly and by using the tensor products of matrices,
we have,
Corollary 2 (1) There exists an infinite family of regular symmetaric Hadamard matrices of order $(4n)^{2l},$ : a positive integer,$l$(2) there exists an infinite family of regular symmetaric complex Hadamard matricesof order $(4n)^{2l},$ : a positive integer.
$l$3Classes of Hadamard matrices

Definition 4 Two Hadamard matrices are said to be equivalent if one can be obtained
from the other by a sequence of the following operations:(1) Permute rows(or columns),(2) multiply any row(or column) by-l.

94
Let $H=(h_{i,j})$ be an Hadamard matrix. Let $I=(i_{1},i_{2}, i_{3}, i_{4})$ be a 4-subset of theindex set $\Omega=\{0,1, \ldots,4n-1\}$ . We define.

$N_{I}=N_{\langle i_{1},i_{2},i_{3},i_{4})}= \sum_{j=0}^{4n-1}hhhh$

Then$N_{I}$

is invariant under Hadamard transformation for columns. If we define$S_{k}=\#\{(i_{1}, i_{2}, i_{3}, i_{4})|N_{\langle i_{1},i_{2},i_{3},i_{4})}=k\}$$C_{k}=S_{k}+S_{-k}$

then
If

,

,

is invariant under Hadamard transformation (1) and (2) in the above definition.
of two Hadamard matrices are different, they are inequivalent.

$C_{k}$$C_{k}’ s$Lemma 1 $N_{I}\equiv 0(mod 4)$ .
$i=$andbe equivalent Hadamard matmces. Let
andCorollary 3 Let
respectively. Then there exists
and
0,1,2,3,4 be adjacency mat ni ces obtained froma permutation matrix such that
$A_{i}$$H_{2}$$H_{1}$$H_{1}$$A_{i}’,$$H_{2}$$P$$A_{i}’=PA_{i}P$for

4$0\leq i\leq 4$.

Generalized spin models of symmeteric Hadamardtype

We give a necessary and sufficient condition that Hadamard matrices
in Theorem 2 give generalized spin models of symmetric Hadamard type.

$M_{1},$ $M_{2}$Theorem 3 Let $H$ be a normalized Hadamard matrix of order $4n$ andbe adjacency mat nces obtained from $H$ .

$A_{1},$

and$M_{3}$$(0\leq i\leq 4)$(1) $W+=W_{-}=M_{1}=A_{0}+A_{1}+A_{2}+A_{3}-A_{4}$ gives a generalized spin model ofsymmetric Hadamard type if and only if the following conditions $(a),(b)$ are satisfied
and $\gamma_{2}\in\Omega^{*}=\{1,2, \ldots,4n-1\}$ :for any$\beta_{1},$

$(a)$

when

$\beta_{2},\gamma_{1}$,$(h_{\beta_{1}}\rho_{2}, h_{\gamma_{1}\gamma_{2}}, h_{\beta_{1}\gamma_{2}}, h_{\gamma_{1}\beta_{2}})=(1,1,1,1)_{f}(1,1,- 1,- 1)_{f}(1,- 1,- 1,1)$$(1,- 1,1,- 1),(- 1,- 1,- 1,- 1)$ ,$\sum_{l=-n}^{n}\theta_{l}l=n$

,

95
$(b)$

when

$(h_{\beta_{1}\beta_{2}}, h_{\gamma_{1}\gamma_{2}}, h_{\beta_{1}\gamma_{2}}, h_{\gamma_{1}\beta_{2}})=(1,1,- 1,),$$(1_{J}- 1,- 1,- 1)$,$\sum_{l=-n}^{n}\theta_{l}l=0$

where$\theta_{l}=\#\{x_{1}|h_{x_{1}\beta_{2}}h_{x_{1}\gamma_{2}}=1, N_{(0,\beta_{1},\gamma_{1},x_{1})}=4l\}$.

(2) $W+=W_{-}=M_{2}=A_{0}+A_{1}-A_{2}-A_{3}+A_{4}$ gives a generalized spin modelof symmetric Hadamard type if and only if the above conditions $(a)$ and $(b)$ areand.
satisfied for any$\beta_{1},$$\beta_{2},$$\gamma_{2}\in\Omega^{*}$$\gamma_{1}$(3) $W+=W_{-}=M_{3}=A_{0}-A_{1}+A_{2}-A_{3}+A_{4}$ gives a generalized spin model ofsymmetric Hadamard type if and only if the transpose matrix
satisfies the above
conditions $(a)$ and $(b)$ for any
and.
$H^{t}$$\beta_{1},$$\beta_{2},$$\gamma_{2}\in\Omega^{*}$$\gamma_{1}$To prove the Theorem 3, the following lemma is useful.
Lemma 2 Let $H$ be a normalized Hadamard matrex of order $4n$ . Choose three rows
and . Then$\alpha_{1}=0,$$\beta_{1}$$\gamma_{1}$$\sum_{l=-n}^{n}\xi_{l}l=n$where$\xi_{l}=\#\{x_{1}|N_{(\alpha_{1},\beta_{1},\gamma_{1},x_{1})}=4l\}$. It is also true for columns.

$0\leq i\leq 3$ , are regular symmetProof of Theorem 3. (1) Since Hadamard matrices
ric, the conditions (0),(1H),and (2H) hold. Therefore we get a necessary and sufficientcondition by verifying the condition (3H).
, we may assume one of them, say ,When we choose three rows
and ofis equal to $0=(0,0)$ . We obtain only one inequivalent normalized Hadamard matrix onwhichever entry we normalize an Hadamard matrix. Assume that$(0,0)$ . The row
and the column
can be transformed into the row and the column
with all 1 entries by multiplying some rows and columns by-l. Denote this Hadamard
matrix by $H’$ . Then we get the normalized Hadamard matrix $H$ by permuting rows andcolumms of $H’$ ;$H=QH’R$ ,$M_{i},$$\alpha,$$\beta$$\gamma$$M_{1}$$\alpha$$\alpha.=(\alpha_{1}, \alpha_{2})\neq 0=$$\alpha_{1}$$\alpha_{2}$where and are permutation matrices. Hence if the permutations
rows and columns ofsimultaneously, we obtain the same matrix
exists a permutation matrix such that
$Q$

$R$

$M_{1}$$P$.

$M_{1}=PM_{1}P$$Q$

and act on the. Namely there$M_{1}$$R$

96
Put $M_{1}=(m(\alpha, \beta))$ where
The left-hand side of the star triangle relation (3H)$m(\alpha, \beta)=h_{\alpha_{1}\alpha_{2}}h_{\beta_{1}\beta_{2}}h_{\alpha_{1}\beta_{2}}h_{\beta_{1}\alpha_{2}},$$\alpha=(\alpha_{1}, \alpha_{2}),\beta=(\beta_{1}, \beta_{2})$.

$S( \alpha, \beta,\gamma)=\sum_{x\in X}m(\alpha, x)m(\beta, x)m(\gamma,x)$

is invariant under the column permutation of$\alpha=0$$M_{1}$.

When
regularity of

or
or
or
. We may assume that$\alpha=\beta=\gamma$$M_{1}$$\alpha=\beta$$\beta=\gamma$$\gamma=\alpha$. Since and$\beta$run over $X$ , we may put$\gamma$, the condition (3H) is satisfied from the. $i^{Fromh_{00}}=h_{0x_{2}}=h_{x_{1}0}=0$ ,$\alpha\neq\beta\neq\gamma$$S(0, \beta,\gamma)=\sum_{x\in X}m(0, x)m(\beta,x)m(\gamma, x)=h_{\beta_{1}\beta_{2}}h_{\gamma_{1}\gamma_{2}}\sum_{x_{1}}h_{x_{1}\beta_{2}}h_{x_{1}\gamma_{2}}\sum_{x_{2}}h_{x_{1}x_{2}}h_{\beta_{1}x_{2}}h_{\gamma_{1}x_{2}}$Put$N_{x_{1}}=N_{t^{0,\beta_{1},\gamma_{1},x_{1})}}=\Sigma_{x_{2}}h_{x_{1}x_{2}}h_{\beta_{1}x_{2}}h_{\gamma_{1}x_{2}}$. Since$N_{0}=N_{\beta_{1}}=N_{\gamma_{1}}=0$$S(0, \beta,\gamma)=h_{\beta_{1}\beta_{2}}h_{\gamma_{1}\gamma_{2}}\sum_{x_{1}\neq 0,\beta_{1},\gamma_{1}}h_{x_{1}\beta_{2}}h_{x_{1}\gamma_{2}}N_{x_{1}}$We defineThen$\theta_{l}=\#\{x_{1}|N_{x_{1}}=4l, h_{x_{1}\beta_{2}}h_{x_{1}\gamma_{2}}=1\}$

and

,

.
.

$\eta\iota=\#\{x_{1}|N_{x_{1}}=4l, h_{x_{1}}\rho_{2}h_{x_{1}\gamma_{2}}=-1\}$.

$S(0, \beta,\gamma)=4h_{\beta_{1}\beta_{2}}h_{\gamma_{1}\gamma_{2}}\sum_{l=-n}^{n}(\theta_{l}-\eta_{l})l$

Next we consider the right-hand side of the star triangle relation (3H).
$4n\cdot m(\alpha,\beta)m(\beta,\gamma)m(\gamma, \alpha)=4nh_{\beta_{1}\gamma_{2}}h_{\gamma_{1}\beta_{2}}$.

Hence we have$\sum_{l=-n}^{n}(\theta_{l}-\eta_{l})l=n\cdot m(\beta,\gamma)$.

Since from Lemma 2,$\sum_{l=-n}^{n}\xi_{l}l=\sum_{l=-n}^{n}(\theta_{l}+\eta_{l})l=n$,

the necessary and sufficient condition is given by$\sum_{l=-n}^{n}\theta_{l}l=n(m(\beta,\gamma)+1)/2$.

to eachNotice that we may exchange the rows
and , and the columns
andother. Hence the values
reduce to the 7 cases in Theorem 3.
(2) We can prove in the same way as the following.
(3) Similarly to the case (1), we verify the condition (3H) for. We may put$\alpha=0$ . Let $W+=W_{-}=(w(x, y))_{x,y\in X}$ and $x=(x_{1}, x_{2}),$ $y=(y_{1}, y_{2})$ . Then$\beta_{1}$$\beta_{2}$$\gamma_{1}$$(h_{\beta_{1}\beta_{2}}, h_{\gamma_{1}\gamma 2}, h_{\beta_{1}\gamma 2}, h_{\gamma_{1}\beta_{2}})$$\alpha\neq\beta\neq\gamma$$w(x,y)=\{\begin{array}{l}1(x,y)\in R_{0}orR_{2}-h_{x1x2}h_{y_{1}y_{2}}hx_{1}y_{2}h_{y_{1}x_{2}}(x,y)\in R_{1},R_{3}orR_{4}\end{array}$$\gamma_{2}$97
where $R;,$ $0\leq i\leq 4$ , are defined in Theorem 1.
, it is easy to prove that the condition (3H) holds.
When
We distinguish 2 cases.
Case 1. Exactly one ofis contained in .
, and
. We get the right, namelyFirst we supposeofhand side of the condition(3H) is. Now we verify the left-hand side
the condition.
$(\alpha, \beta),$ $(\beta,\gamma),$ $(\alpha, \gamma)\in R_{2}$$(\alpha, \beta),$$(\beta,\gamma),$$(\alpha, \beta)\in R_{2}$$R_{2}$$(\alpha, \gamma)$$\alpha_{2}=\beta_{2}$$(\beta, \gamma),$$(\alpha,\gamma)\not\in R_{2}$$4nh_{\beta_{1}\gamma_{2}}$$S(0, \beta,\gamma)$$S(O, \beta,\gamma)$$=$$\sum_{(\alpha,x),(\beta,x)\in R_{2},(\gamma,x)\not\in R_{2}}w(\alpha, x)w(\beta, x)w(\gamma, x)$$+ \sum_{\langle\alpha,x),\langle\beta,x)\not\in R_{2},\langle\gamma,x)\in R_{2}}w(\alpha, x)w(\beta, x)w(\gamma, x)$$+ \sum_{(\alpha,x),(\beta,x)\not\in R_{2},(\gamma,x)\not\in R_{2}}w(\alpha, x)w(\beta, x)w(\gamma, x)$$=$$- \sum_{x_{1}}h_{\gamma_{1}\gamma_{1}}h_{x_{1}\alpha_{2}}h_{\gamma_{1}\alpha_{2}}h_{x_{1}\gamma_{2}}+\sum_{x_{1}}h_{x_{1}\gamma_{2}}h_{\beta_{1}\beta_{2}}h_{x_{1}\gamma_{2}}h_{\beta_{1}\gamma_{2}}h_{x_{1}\beta_{2}}$$-h_{\beta_{1}\alpha_{2}}h_{\gamma_{1}\gamma_{2}} \sum_{x_{2}}h_{\beta_{1}x_{2}}h_{\gamma_{1}x_{2}}\sum_{x_{1}}h_{x_{1}x_{2}}h_{x_{1}\alpha_{2}}h_{x_{1}\gamma_{2}}$$=$$h_{\beta_{1}\beta_{2}}h_{\beta_{1}\gamma_{2}} \sum_{x_{1}}h_{x_{1}\alpha_{2}}$$=$$4nh_{\beta_{1}\gamma_{2}}$Hence the condition (3H) holds. We can prove that the condition (3H) holds for the casesin the same way.
and.
Case 2.
.
The right-hand side of (3H) is$(\beta, \gamma)\in R_{2},$ $(\alpha, \beta)_{s}(\alpha, \gamma)\not\in R_{2}$$(\alpha,\gamma)\in R_{2},$ $(\alpha, \beta),$ $(\beta, \gamma)\not\in R_{2}$$(\alpha,\beta),$ $(\beta, \gamma),$ $(\alpha,\gamma)\not\in R_{2}$$-4nh_{\beta_{1}\gamma_{2}}h_{\gamma_{1}\beta_{2}}$$S(0, \beta,\gamma)$$=$$\sum_{(\alpha,x)\in R_{2},(\beta,x),\langle\gamma,x)\not\in R_{2}}w(\alpha, x)w(\beta, x)w(\gamma, x)$$+ \sum_{(\beta,x)\in R_{2},(\alpha,x),(\gamma,x)\not\in R_{2}}w(\alpha,x)w(\beta, x)w(\gamma, x)$$+ \sum_{\langle\gamma,x)\in R_{2},(\alpha,x),(\beta,x)\not\in R_{2}}w(\alpha,x)w(\beta, x)w(\gamma, x)$$+ \sum_{\langle\alpha,x),(\beta,x),(\gamma,x)\not\in R_{2}}w(\alpha, x)w(\beta, x)w(\gamma, x)$$=$$h_{\beta_{1}\beta_{2}}h_{\gamma_{1}\gamma 2} \sum_{x_{1}}h_{x_{1}}\ h_{x_{1}\gamma_{2}}+h_{\gamma_{1}\gamma_{2}}h_{\gamma_{1}\beta_{2}} \sum_{x_{1}}h_{x_{1}\gamma_{2}}+h_{\beta_{1}\beta_{2}}h_{\beta_{1}\gamma_{2}}\sum_{x_{1}}h_{x_{1}\beta_{2}}$$-h_{\beta_{1}\beta_{2}}h_{\gamma\}\gamma_{2}} \sum_{x_{2}}h_{\beta_{1}x_{2}}h_{\gamma_{1}x_{2}}\sum_{x_{1}}h_{x_{1}x_{2}}h_{x_{1}\beta_{2}}h_{x_{1}\gamma_{2}}$$=$$-h_{\beta_{1}\beta_{2}}h_{\gamma_{1}\gamma_{2}} \sum_{x_{2}}h_{\beta_{1}x_{2}}h_{\gamma_{1}x_{2}}\sum_{x_{1}}h_{x_{1}x_{2}}h_{x_{1}\beta_{2}}h_{x_{1’l2}}$

98
Putting$\tilde{N}_{x_{2}}=\tilde{N}_{(0,/\approx,\gamma_{2},x_{2})}=\Sigma_{x_{1}}h_{x_{1}x_{2}}h_{x_{1}\beta_{2}}h_{x_{1}\gamma_{2}}$. Since$\tilde{N}_{0}=\tilde{N}_{\beta_{2}}=\tilde{N}_{\gamma_{2}}=0$,.

$S(0, \beta,\gamma)=-h_{\beta_{1}\beta_{2}}h_{\gamma_{1}\gamma_{2}}\sum_{x_{2}\neq 0,\beta_{2},\gamma_{2}}h_{\beta_{1}x_{2}}h_{\gamma_{1}x_{2}}\tilde{N}_{x_{2}}$DefineThen$\tilde{\theta}_{l}=\#\{x_{1}|\tilde{N}_{x_{1}}=4l, h_{\beta_{1}x_{2}}h_{\gamma_{1}x_{2}}=1\}$

and, $\tilde{\eta}_{l}=\#\{x_{1}|\tilde{N}_{x_{1}}=4l, h_{\beta_{1}x_{2}}h_{\gamma_{1}x_{2}}=-1\}$.

.

$S(0, \beta,\gamma)=-h_{\beta_{1}C_{2}}h_{\gamma_{1}\gamma_{2}}\sum_{l=-n}^{n}(\tilde{\theta}_{l}-\tilde{\eta}\iota)l$

From the Lemma 2, it follows that$\sum_{l=-n}^{n}\tilde{\theta}_{l}l=n(m(\beta,\gamma)+1)/2$

is a necessary and sufficient condition. It means that the transpose matrix
the c\’onditions (a) and (b).

5$H^{t}$

satisfies$\square$Generalized spin models of pseudo-Jones typeWe give a necessary and sufficient condition that complex Hadamard matrices
andin Theorem 2 give generalized spin models of pseudo-Jones type.

$L_{1},\overline{L_{1}}$$L_{2},\overline{L_{2}}$Theorem 4 Let be a primitiveroot of unity and $H$ be a normalized Hadamard$(0\leq i\leq 4)$ be adjacency matrices obtained
matrix of order $4n$ . Letfrom $H$.
$4^{th}$$i$$A_{i},$(1) $W+=L_{1}=A_{0}+A_{1}+A_{2}i+A_{3}i-A_{4}i,$gives a generalized spin model$(a)$and $(b)$ in Theorem 3 areof pseudo-Jones type if and only if the conditions.
andsatisfied for any$W_{-}=\overline{L_{1}}$$\beta_{1},$$\beta_{2},\gamma_{1}$$\gamma_{2}\in\Omega^{*}$(2) $W+=L_{2}=A_{0}+A_{1}i+A_{2}+A_{3}i-A_{A}i,$gives a genemlized spin modelof pseudo-Jones type if and only if the transpose mat’rix satisfies the conditions$(a)$ and $(b)$and.
for any$W_{-}=\overline{L_{2}}$$H^{t}$$\beta_{1},$$\beta_{2},\gamma_{1}$$\gamma_{2}\in\Omega^{*}$

Proof. (1) Let $W+=(w(x, y))_{x,y\in X}$ and $x=(x_{1}, x_{2}),$ $y=(y_{1}, y_{2})$ . Then the entry$w(x, y)$ is given by$w(x, y)=\{\begin{array}{l}1(x,y)\in R_{0}orR_{1}-i\cdot h_{x_{1}x_{2}}h_{y_{1}y_{2}}h_{x_{1}y_{2}}h_{y_{1}x_{2}}(x,y)\in R_{2},R_{3}orR_{4}\end{array}$

We can prove the theorem similarly to the Theorem 3.

$\square$99
Remark.

The following (1) $-(3)$ are equivalent and (4) $-(5)$ are equivalent:(1) $W+=W_{-}=M_{1}$ gives a generalized spin model of symmetric Hadamard type.

(2) $W+=W_{-}=M_{2}$ gives a generalized spin model of symmetric Hadamard type.

(3) $W+=L_{1},$$W_{-}=\overline{L_{1}}$gives a generalized spin model of pseudo-Jones type.

(4) $W+=W_{-}=M_{3}$ gives a generalized spin model of symmetric Hadamard type.
(5) $W+=L_{2},$6$W_{-}=\overline{L_{2}}$gives a generalized spin model of pseudo-Jones type.

A special class of Hadamard matrices and generalized spin models

Theorem 5 Assume that an Hadamard matrix of order $4n$ satisfies$(c)C_{4n}= \frac{1}{4}(\begin{array}{l}4nn\end{array}),$$C_{0}= (\begin{array}{l}4n4\end{array})-\frac{1}{4}(\begin{array}{l}4n3\end{array}),$$C_{l}=0(l\neq 0,4n)$.

Then the normalized matrix of $H$ satisfies the necessary and sufficient conditions $(a),(b)$in Theorem 3. It implies thatobtained from $H$ give spin models of symmetricgives a generalized spin model of pseudo-Jones type.
Hadamard type and$M_{1},$ $M_{2}$$L_{1},$$\overline{L_{1}}$Proof. It turns out that the condition (c) means there exists only one row such
thath
for fixed three rows
and . Assume that $H$ satisfies
the condition (c). Denote the normalized Hadamard matrix $\cdot byH’$ . Assume that thesatisfies $N_{x_{1}}=N(O, \beta_{1}, \gamma_{1}, x_{1})=4n$ (the case $N_{x_{1}}=-4n$ does not occur). Ifrow, which is equivalent to $m(\beta,\gamma)=1$ , thenandfor $l\neq 0,$ .
, which is equivalent to $m(\beta, \gamma)=-1$ , thenIf
for $l\neq 0$ . It follows
that $H’$ satisfies the necessary and sufficient conditions (a) and (b).
$x$$N_{\langle\alpha,\beta,\gamma,x)}=N_{x}=\pm 4n$$\alpha,$$\beta$$\gamma$$x_{1}$$\dot{\theta}_{l}=0$$\theta_{n}=1$$h_{x_{1}\beta_{2}}h_{x_{1}\gamma_{2}}=1$$n$$\theta_{l}=0$$h_{x_{1}\beta_{2}}h_{x_{1}\gamma_{2}}=-1$$\square$

It is obvious that if the transpose matrix H satisfies the above condition $(c),$ $thenthe$
normalized matrix ofsatisfies the necessary and sufficient condition (3) in Theorem
3.
$H^{t}$

Corollary 4 Assume that both $H$ andsatisfy the condition $(c)$ in Theorem 5. $H$ is
not necessarily equivalent to.
$(0\leq i\leq 4)$ be adjacency matrices obtainedLetfrom $H,$ respectively.
$H^{t}$$H^{t}$$A_{i},$ $A_{i}’,$$H^{t}$

100
Then $M_{1}=A_{0}+A_{1}+A_{2}+A_{3}-A_{4}$ and $M_{1}’=A_{1}’+A_{2}’+A_{3}’-A_{4}’$ also satisfy the condition$(c)$ . Namely,
infinite families constructed from $H$ and mentioned in Corollay 2 satisfy
the condition $(c)$ .
Hence there exist infinite families of genemlized spin models of pseudo-Jones type and, l:positive integer.
of symmetric Hadamard type with loop variable$H^{t}$$(4n)^{2l}$There is only 1 inequivalent class of Hadamard matrices of orders 4 and 8. They satisfy
the condition (c). There are 5 inequivalent classes of order 16. Class I according to theclassification by M.Hall Jr. satisfies the condition (c) but other classes not.

参考文献
[1] E.Bannai and E.Bannai, Generalized generalized spin models, preprint.
[2] T.Ito, A.Munemasa
M. Yamada, Amorphous association schemes over characteristic 4, Europ. J. Combinatorics, 12 (1991), 513-526.
$ai_{1}d$[3] A.A. Ivanov and I.V. Chuvaeva, Action of the groupon Hadamard matrices, Investigations in Algebraic Theory of Combinatorial Objects, VNIISI, Moscow, Institutefor System Studies, 1985, 159-169(in Russian).
$M_{12}$[4] F.Jaeger, Storongly regular graphs and spin models for the Kauffman polynomial,Geom. Dedicata, 44 (1992), 23-52.
[5] V.F.R. Jones, On knot invariants related to some statistical mechanical models, $Pac$ .
J. Math. 137 (1989), 311-334.
[6] K.Kawagoe, A. Munemasa and Y. Watatani, Generalized spin models, preprint.
[7] K. Nomura, Spin models constructed from Hadamard matrices, J. Combinatorial
Theory Ser. A (to appear).
[8] M. Wakimoto, Private communication.