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

Authors:	Brian Curtin Kazumasa Nomura

Abstract:	We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let $\mathcal{M}$ denote a Bose-Mesner algebra on a finite nonempty set X. Fix p X, and let $\mathcal{M}^ $ and $\mathcal{T}$ denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of $\mathcal{M}$ with respect to p. By a hyper-duality of $\mathcal{M}$ , we mean an automorphism of $\mathcal{T}$ such that $\psi (\mathcal{M}) = \mathcal{M}^ ,\psi ^2 (A) = ^t {\kern 1pt} A$ for all $A \in \mathcal{M}$ ; and $\left\| X \right\|\psi \rho$ is a duality of $\mathcal{M}$ . $\mathcal{M}$ is said to be hyper-self-dual whenever there exists a hyper-duality of $\mathcal{M}$ . We say that $\mathcal{M}$ is strongly hyper-self-dual whenever there exists a hyper-duality of $\mathcal{M}$ which can be expressed as conjugation by an invertible element of $\mathcal{T}$ . 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.

Keywords:	Bose-Mesner algebra Terwilliger algebra spin model
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