An algebra associated with a spin model

Nomura, Kazumasa

Journal of Algebraic Combinatorics, Tome 6 (1997) no. 1, pp. 53-58.

Résumé

Summary: 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 $_{ a, b} ( a, b = 1,\dots , n)$, sumlimits_x = 1^n fracW_a,x W_b,x = $n{\delta }$_a,b (a,b = 1,$\dots ,n$), 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 $_{ x = 1} ^{ n} \frac W _{ a, x} W _{ b, x} W _{ c, x} = \frac W _{ a, b} W _{ a, c} W _{ b, c} ( a, b, c = 1,\dots , n)$, frac1$\sqrt n$ sumlimits_x = 1^n fracW_a,x W_b,x W_c,x = fracW_a,b W_a,c W_b,c (a,b,c = 1,$\dots ,n$), 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.

Détail

Keywords: spin model, association scheme, Bose-mesner algebra