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

Curtin, Brian ; Nomura, Kazumasa

Journal of Algebraic Combinatorics, Tome 13 (2001) no. 2, pp. 173-186.

Résumé

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

Détail

Keywords: Bose-mesner algebra, Terwilliger algebra, spin model