Leonard pairs, spin models, and distance-regular graphs

A Leonard pair is an ordered pair of diagonalizable linear maps on a finite-dimensional vector space, that each act on an eigenbasis for the other one in an irreducible tridiagonal fashion. In the present paper we consider a type of Leonard pair, said to have spin. The notion of a spin model was introduced by V.F.R. Jones to construct link invariants. A spin model is a symmetric matrix over C that satisfies two conditions, called the type II and type III conditions. It is known that a spin model W is contained in a certain finite-dimensional algebra N(W), called the Nomura algebra. It often happens that a spin model W satisfies W is an element of M subset of N(W), where M is the Bose-Mesner algebra of a distance-regular graph Gamma; in this case we say that Gamma affords W. If Gamma affords a spin model, then each irreducible module for every Terwilliger algebra of Gamma takes a certain form, recently described by Caughman, Curtin, Nomura, and Wolff. In the present paper we show that the converse is true; if each irreducible module for every Terwilliger algebra of Gamma takes this form, then Gamma affords a spin model. We explicitly construct this spin model when Gamma has q-Racah type. The proof of our main result relies heavily on the theory of spin Leonard pairs. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

This paper explores Leonard pairs with spin, discussing the characteristics of spin models in finite-dimensional vector spaces and their application in algebras.