This paper presents
applications of the decomposition of generalized Clifford-Littlewood-Eckmann groups,
or CLE-groups, which are given by presentations of the type
We
begin by studying the irreducible complex representations of the “building block
groups” of orders n2 and n3, 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 ain= 1, but allow aiaj= ωkajai for fixed k dividing n, where possibly
k > 1. The irreducible complex representations of these groups are then
calculated.
