An Algebra Associated with a Spin Model

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 n ∑ x=1 Wa,x Wb,x = nδa,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: 1 √ n n ∑ x=1 Wa,x Wb,x Wc,x = Wa,b Wa,cWb,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.