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Over forty years ago, Goethals and Seidel showed that if the adjacency algebra of a strongly regular graph $X$ contains a Hadamard matrix then $X$ is either of Latin square type or of negative Latin square type. We extend their result to complex Hadamard matrices and find only three additional families of parameters for which the strongly regular graphs have complex Hadamard matrices in their adjacency algebras. Moreover we show that there are only three distance regular covers of the complete graph that give complex Hadamard matrices.

Let A be an association scheme on q\geq 3 vertices. We show that the Bose-Mesner algebra of the generalized Hamming scheme H(n,A), for n\geq 2, is not the Nomura algebra of a type II matrix. This result gives examples of formally self-dual Bose-Mesner algebras that are not the Nomura algebras of type II matrices., Comment: 15 pages, minor revision

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., Comment: 155 pages, Thesis

Type-II matrices are a class of matrices used by Jones in his work on spin models. In this paper we show that type-II matrices arise naturally in connection with some interesting combinatorial and geometric structures.

In 1989, Vaughan Jones introduced spin models and showed that they could be used to form link invariants in two different ways--by constructing representations of the braid group, or by constructing partition functions. These spin models were subsequently generalized to so-called 4-weight spin models by Bannai and Bannai; these could be used to construct partition functions, but did not lead to braid group representations in any obvious way. Jaeger showed that spin models were intimately related to certain association schemes. Yamada gave a construction of a symmetric spin model on $4n$ vertices from each 4-weight spin model on $n$ vertices. In this paper we build on recent work with Munemasa to give a different proof to Yamada's result, and we analyse the structure of the association scheme attached to this spin model., Comment: 23 pages

We introduce and discuss Jones pairs. These provide a generalization and a new approach to the four-weight spin models of Bannai and Bannai. We show that each four-weight spin model determines a ``dual'' pair of association schemes.