Jones Pairs

Motivated by Jones' braid group representations constructed from spin models, we define {\sl a Jones pair} to be a pair of $\nbyn$ matrices $(A,B)$ such that the endomorphisms $X_A$ and $\D_B$ form a representation of a braid group. When $A$ and $B$ are type-II matrices, we call $(A,B)$ {\sl 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 \ref{Nomura} and \ref{IINom}. In Chapter \ref{JP}, 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 \ref{SpinModels}, 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 \ref{InvJP}. 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.