Tensor Products of Representations

Abstract

If G is a reductive group in characteristic zero with two irreducible representations V1 and V2, we may form the tensor product representation V1 0 V2 where G acts diagonally. The problem is to decompose V1 0 V2 into its irreducible components with their multiplicities. Of course this may be done using Weyl's character formula but this is a very difficult calculation. This paper presents a general method for doing this calculation in a more efficient way. For the general linear group the method was explained in [2] with a minimum of group-theoretic language. For the other classical groups this paper will give explicit formulas. The method of this paper may be used for exceptional groups but for combinatorial reasons it does not seem worthwhile at this point to write an explicit formula. Perhaps it would be more useful to have a computer program for the calculation. As an after thought I have included an appendix on how to decompose an irreducible representation of G when restricted to a large reductive subgroup. The case of the general linear group is well-known.