To each symmetric n × n matrix W with non-zero complex entries, we associate a vector space N , consisting of certain symmetric n × n matrices. If W satisfies n ∑ x=1 Wa,x Wb,x = nδa,b (a, b = 1, . . . , n), then N becomes a commutative algebra under both ordinary matrix product and Hadamard product (entry-wise product), so that N is the Bose-Mesner algebra of some association scheme. If W sati...

Duality maps of finite abelian groups are classified. As a corollary, spin models on finite abelian groups which arise from the solutions of the modular invariance equations are determined as tensor products of indecomposable spin models. We also classify finite abelian groups whose Bose-Mesner algebra can be generated by a spin model.

A Leonard pair is an ordered of diagonalizable linear maps on a finite-dimensional vector space, that each act eigenbasis for the other one in irreducible tridiagonal fashion. In present paper we consider type pair, said to have spin. The notion spin model was introduced by V.F.R. Jones construct link invariants. symmetric matrix over C satisfies two conditions, called II and III conditions. It...

We inductively construct an explicit (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme. The key step is a constructive, linear algebraic interpretation of the GoldmanRota recurrence for the number of subspaces of a finite vector space. This interpretation shows that the up operator on subspaces has an explicitly given recursive structure. Using t...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi CHAPTER 1. A GENERAL OVERVIEW AND INTRODUCTION . . . . 1 CHAPTER 2. PRELIMINARIES: COMMUTATIVE ASSOCIATION SCHEMES AND THEIR TERWILLIGER ALGEBRAS . . . . . . . . . . . . . . . . . 4 2.1 The Bose-Mesner Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Dual Bose-Mesner Algebra . . . . . . ....