An algebra associated with a spin model. (English) Zbl 0865.05077

Nomura, Kazumasa

J. Algebr. Comb. 6, No. 1, 53-58 (1997).

Summary: To each symmetric \(n\times n\) matrix \(W\) with non-zero complex entries, we associate a vector space \(\mathcal N\), consisting of certain symmetric \(n\times n\) matrices. If \(W\) satisfies \[ \sum^n_{x=1} {W_{a,x}\over W_{b,x}}=n\delta_{a,b}\qquad (a,b=1,\dots,n), \] then \(\mathcal N\) becomes a commutative algebra under both ordinary matrix product and Hadamard product (entry-wise product), so that \(\mathcal N\) is the Bose-Mesner algebra of some association scheme. If \(W\) satisfies the star-triangle equation: \[ {1\over\sqrt n} \sum^n_{x=1} {W_{a,x}W_{b,x}\over W_{c,x}}= {W_{a,b}\over W_{a,c}W_{b,c}}\qquad (a,b,c=1,\dots,n), \] then \(W\) belongs to \(\mathcal N\). 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.

MSC:

05E30

Association schemes, strongly regular graphs

Keywords:

Bose-Mesner algebra; association scheme; spin model

References:

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