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Home > Vol 6, No 1 (1997)

Kazumasa Nomura

Tokyo Ikashika University Kounodai, Ichikawa 272 Japan

To each symmetric n \times  n matrix W with non-zero complex entries, we associate a vector space N, consisting of certain symmetric n \times  n matrices. If W satisfies å _{x = 1} ⁿ \frac W _{a, x} W _{b, x} = n d _{a, b} ( a, b = 1,..., n), \sum\limits_{x = 1}^n {\frac{{W_{a,x} }}{{W_{b,x} }} = n{δ}_{a,b} } (a,b = 1,...,n), then N becomes a commutative algebra under both ordinary matrix product and Hadamard product (entry-wise product), so that N is the Bose-Mesner algebra of some association scheme. If W satisfies the star-triangle equation: \frac1 Ö n å _{x = 1} ⁿ \frac W _{a, x} W _{b, x} W _{c, x} = \frac W _{a, b} W _{a, c} W _{b, c} ( a, b, c = 1,..., n), \frac{1}{{\sqrt n }}\sum\limits_{x = 1}^n {\frac{{W_{a,x} W_{b,x} }}{{W_{c,x} }} = \frac{{W_{a,b} }}{{W_{a,c} W_{b,c} }}} (a,b,c = 1,...,n), then W belongs to N. This gives an algebraic proof of Jaeger's result which asserts that every spin model which defines a link invariant comes from some association scheme.

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