Spin models and strongly hyper-self-dual Bose-Mesner algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let M denote a Bose-Mesner algebra on a finite nonempty set X. Fix p is an element of X, and let M* and T denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of M with respect to p. By a hyper-duality of M, we mean an automorphism psi of T such that psi (M) = M*,psi (M*) = M; psi (2)(A) = (t)A for all A is an element of M; and |X|psi rho is a duality of M. M is said to be hyper-self-dual whenever there exists a hyper-duality of M. We say that M is strongly hyper-self-dual whenever there exists a hyper-duality of M which can be expressed as conjugation by an invertible element of T. We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.