Generalized Clifford-Littlewood-Eckmann groups II: Linear representations and applications

Abstract

We begin by studying the irreducible complex representations of the building block of orders n 2 and n 3 , and how the representations for the composite groups are constructed from them. This of course also gives a complete set of inequivalent irreducible matrix representations for the generalized Clifford algebras corresponding to these groups. We apply these representation-theoretic results to determine the size of the maximal abelian subgroups of these groups, and to present a generalization of a result of Littlewood on maximal sets of anticommuting matrices. In the final section we consider an alternative generalization of the CLE-groups, in which we require a n = 1, but allow aiaj = ω k ajai for fixed k dividing n, where possibly k > 1. The irreducible complex representations of these groups are then calculated. These representations have been studied from the standpoint of projective representations of ( /n ) r in [S-I]. However we feel that the presentation given here is somewhat clearer. The results again are of interest to physicists in a number of applications (see [S-I], [Kw]).Throughout this paper the notation and conventions will follow those of [Sm1], [Sm2]. The results of [Sm 2] concerning the explicit decomposition of the groups will be used extensively here. The corresponding generalized Clifford algebras are studied in [Sm3].