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A Generalization of the Terwilliger Algebra

✍ Scribed by Eric S. Egge

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
226 KB
Volume
233
Category
Article
ISSN
0021-8693

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✦ Synopsis

bra generated by a given Bose Mesner algebra M and the associated dual Bose Mesner algebra M U . This algebra is now known as the Terwilliger algebra and is usually denoted by T. Terwilliger showed that each vanishing intersection number and Krein parameter of M gives rise to a relation on certain generators of T. These relations determine much of the structure of T, thought not all of it in general. To illuminate the role these relations play, we consider a certain generalization T T of T. To go from T to T T, we replace M and M U with a pair of dual character algebras C and C U . We define T T by generators and relations; intuitively T T is the associative ‫-ރ‬algebra with identity generated by C and C U subject to the analogues of Terwilliger's relations. T T is infinite dimensional and noncommutative in general. We construct an irreducible T T-module which we call the primary module; the dimension of this module is the same as that of C and C U . We find two bases of the primary module; one diagonalizes C and the other diagonalizes C U . We compute the action of the generators of T T on these bases. We show T T is a direct sum of two sided ideals T T and T T with T T isomorphic to a full matrix 0 1 0 algebra. We show that the irreducible module associated with T T is isomorphic to 0 the primary module. We compute the central primitive idempotent of T T associated with T T in terms of the generators of T T. ᮊ 2000 Academic Press 0 n Ž . complex entries, and let J g M ‫ރ‬ denote the matrix whose entries are all n 1. By a Bose Mesner algebra of order n we mean a commutative subalge-Ž . bra M of M ‫ރ‬ which contains J and which is closed under transposition n

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