Leonard pairs, spin models, and distance-regular graphs

Abstract

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∈M⊆N(W), where M is the Bose-Mesner algebra of a distance-regular graph Γ; in this case we say that Γ affords W. If Γ affords a spin model, then each irreducible module for every Terwilliger algebra of Γ 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 Γ takes this form, then Γ affords a spin model. We explicitly construct this spin model when Γ has q-Racah type. The proof of our main result relies heavily on the theory of spin Leonard pairs.