Algebraic Graph Theory Seminar - Paul Terwilliger

Title: 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 this talk 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 the complex numbers that satisfies two conditions, called type II and type III. It is known that a spin model W is contained in a certain finite-dimensional algebra N(W), called the Nomura algebra of W.

It often happens that a spin model W is contained in the Bose-Mesner algebra M of a distance-regular graph G, and moreover M is contained in N(W). In this case we say that G affords W.

If G affords a spin model, then each irreducible module for every Terwilliger algebra of G takes a certain form, recently described by Caughman, Curtin, Nomura, and Wolff. In the talk we show that the converse is true; if each irreducible module for every Terwilliger algebra of G takes this form, then G affords a spin model. We explicitly construct this spin model when G has q-Racah type.

The proof of our main result relies heavily on the theory of spin Leonard pairs; the first half of the talk is about this theory.

This talk is based on joint work with Kazumasa Nomura.