On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups

Abstract

The present article deals with simply reducible groups which are finite or compact and with their unitary representations (more precisely: with their representations which are in unitary form) in general. A simply reducible group satisfies two criteria. First, all classes are ambivalent, i.e., contain with an element X also its reciprocal, X−1. Since the characters of all elements of a class are equal to each other in every representation, it follows that the characters of X and X−1 are equal. If the representation is in unitary form, it is at once evident that the characters of reciprocals are conjugate complex. It therefore follows from the ambivalent nature of all classes that all characters are real in every representation of a simply reducible group. As a result, every representation is equivalent to the conjugate complex representation.