Bose-Mesner algebras related to type II matrices and spin models

A type II matrix is a square matrix W with non-zero complex entries such that the entrywise quotient of any two distinct rows of W sums to zero. Hadamard matrices and character tables of abelian groups are easy examples, and other examples called spin models and satisfying an additional condition can be used as basic data to construct invariants of links in 3-space. Our main result is the construction, for every type II matrix W, of a Bose-Mesner algebra N(W), which is a commutative algebra of matrices containing the identity I, the all-one matrix J, closed under transposition and under Hadamard (i.e., entrywise) product. Moreover, if W is a spin model, it belongs to N(W). The transposition of matrices W corresponds to a classical notion of duality for the corresponding Bose-Mesner algebras N(W). Every Bose-Mesner algebra encodes a highly regular combinatorial structure called an association scheme, and we give an explicit construction of this structure. This allows us to compute N(W) for a number of examples.