Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

Abstract

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let {\cal M} denote a Bose-Mesner algebra on a finite nonempty set i>X. Fix p \in X, and let {\cal M}^* and {\cal T} denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of {\cal M} with respect to i>p. By a hyper-duality of {\cal M}, we mean an automorphism ψ of {\cal T} such that \psi({\cal M}) = {\cal M}^*,\psi({\cal M}^*) = {\cal M}; \psi^{2}(A) ={}^{\rm t}\kern-1.3pt{A} for all A \in {\cal M}s and |X\,{|}\,\psi \rho is a duality of {\cal M}. {\cal M} is said to be hyper-self-dual whenever there exists a hyper-duality of {\cal M}. We say that {\cal M} is strongly hyper-self-dual whenever there exists a hyper-duality of {\cal M} which can be expressed as conjugation by an invertible element of {\cal 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.