Reduced characters for finite groups

Abstract

Reduced characters for a finite group G with respect to a subgroup G ₀, are defined by taking partial traces with respect to G ₀-invariant subspaces of the irreducible representations of G. The properties of reduced characters and their evaluation is discussed in detail and in particular a complete theory of reduced characters for simply reducible subgroup imbeddings is developed. In the general case the reduced characters of G contain all multiplicity-independent information about the imbedding G ⊇ G ₀. We apply our methods to the evaluation of all multiplicity-averaged G : G ₀ reduced Wigner coefficients which include, as a special case, all multiplicity-free reduced Wigner coefficients. It is shown that these coefficients are intimately connected with the algebra structure of G ₀-class functions on G.