The Multiplicities of a Dual-thin Q-polynomial Association Scheme

Abstract

. Let Y = (X; fR i g 0iD ) denote a symmetric association scheme, and assume that Y is Q-polynomial with respect to an ordering E 0 ; :::; ED of the primitive idempotents. In [1, p.205], Bannai and Ito conjectured that the associated sequence of multiplicities m i (0 i D) of Y is unimodal. Stanton [7] made the related conjecture that m i m i+1 and m i mD\Gammai for i ! D=2. We prove that if Y is dual-thin in the sense of Terwilliger, then the Stanton conjecture is true. 1 Introduction For a general introduction to association schemes, we refer to [1], [2], [5], or [8]. Our notation follows that found in [3]. Throughout this article, Y = (X; fR i g 0iD ) will denote a symmetric, D- class association scheme. Our point of departure is the following well-known result of Taylor and Levingston. 1.1 Theorem. [6] If Y is P -polynomial with respect to an ordering R 0 ; :::; RD of the associate classes, then the corresponding sequence of valencies k 0 ; k 1 ; : : : ; kD is unimodal. Fur..