Central intertwining numbers for representations of finite groups

In the case L = M, (R(P, P) forms a ring and we denote its center by C(R(P). We define the central intertwining number, Qd(L) of a representation L to be the vector space dimension of C(R(P). For finite groups Qe(L) gives the number of distinct irreducible components appearing in the decomposition of P into irreducible parts, where a component appearing with multiplicity « is counted but once. All results on central intertwining numbers should be interpreted from this point of view. Two representations, P and M are said to be disjoint (denoted P?M) if no subrepresentation of one is equivalent to any subrepresentation of the other. A representation is called a factor representation if it cannot be expressed as the direct sum of two disjoint representations. As is well known and easily proved, a representation P is a factor if and only if Qa(L) = 1. For finite dimensional representations, a factor is simply some integral multiple of an irreducible representation. For the representation theory of separable locally compact groups, factor representations form a natural building block. Heuristically, central intertwining numbers give the same type of information with respect to factor representations, as do intertwining numbers with respect to irreducible representations. In this paper we describe the situation only for finite groups, where the phenomenon is purely algebraic and free of measure theoretic difficulties. It is hoped that this will then serve as a useful prolegomena to the more general investigation.