An Algebra Associated with a Spin Model

Abstract

To each symmetric n × n matrix W with non-zero complex entries, we associate a vector space N, consisting of certain symmetric n × n matrices. If W satisfies $$\sum\limits_{x = 1}^n {\frac{{W_{a,x} }}{{W_{b,x} }} = n{\delta }_{a,b} } (a,b = 1,...,n),$$ 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: $$\frac{1}{{\sqrt n }}\sum\limits_{x = 1}^n {\frac{{W_{a,x} W_{b,x} }}{{W_{c,x} }} = \frac{{W_{a,b} }}{{W_{a,c} W_{b,c} }}} (a,b,c = 1,...,n),$$ 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.