Irreducible representations of finite groups

Abstract

The Flodmark-Blokker scheme for finding irreducible representations of finite groups is sufficiently general for treating all little groups of crystallographic space groups. This scheme is generalized in two respects: (a) The case when no element has a non-degenerate eigenvalue, but the group contains a set with commuting matrix representatives for each irreducible representation. (b) The case when any set of commuting irreducible matrix representatives has a common eigenspace with dimension higher than 1, degenerate with respect to the eigenvalues of the set. In case (b) we make linear combinations of group elements to find a complete set of commuting operators, the completeness being with respect to a degenerate subspace of an irreducible space of the group. We state a theorem to prove the generality of this method. The procedure is valid for arbitrary finite groups and it has been programmed for implementation by a computer.