Generalized Clifford-Littlewood-Eckmann groups. II: Linear representations and applications (Q581665)

This paper presents applications of the decomposition of groups which can be presented as \[ G=<\omega,a_ 1,...,a_ r| \quad \omega^ n=1,\quad a^ n_ i=\omega^{e(i)}\forall i,\quad a_ ia_ j=\omega a_ ja_ i\quad \aleph i<j,\quad \omega a_ i=a_ i\omega \forall i>. \] The decomposition and classification of these groups was carried out in an earlier paper by the author [Part I, Pac. J. Math. (to appear)]. This paper begins by determining the irreducible complex representations for the ``building block groups'' of orders \(n^ 2\) and \(n^ 3\), and then showing how the representations for the general groups are constructed from them. This also determines a complete set of inequivalent matrix representations for the generalized Clifford algebras corresponding to these groups. The representation-theoretic results are then applied to determine the sizes of the maximal abelian subgroups of these groups, and to present a generalization of a result of Littlewood on maximal sets of anti-commuting matrices.