Jones pairs

Abstract

Motivated by Jones' braid group representations constructed from spin models, we define a Jones pair to be a pair of n x n matrices (A, B) such that the endomorphisms XA and \B form a representation of a braid group. When A and B are type-II matrices, we call (A, B) an invertible Jones pair. We develop the theory of Jones pairs in this thesis.

Our aim is to study the connections among association schemes, spin models and four-weight spin models using the viewpoint of Jones pairs. We use Nomura's method to construct a pair of algebras from the matrices (A, B), which we call the Nomura algebras of (A, B). These algebras become the central tool in this thesis. We explore their properties in Chapters 2 and 3.

In Chapter 4, we introduce Jones pairs. We prove the equivalence of four-weight spin models and invertible Jones pairs. We extend some existing concepts for four-weight spin models to Jones pairs. In Chapter 5, we provide new proofs for some well-known results on the Bose-Mesner algebras associated with spin models.

We document the main results of the thesis in Chapter 6. We prove that every four-weight spin model comes from a symmetric spin model (up to odd-gauge equivalence). We present four Bose-Mesner algebras associated to each four-weight spin model. We study the relations among these algebras. In particular, we provide a strategy to search for four-weight spin models. This strategy is analogous to the method given by Bannai, Bannai and Jaeger for finding spin models.