Multiplicity, invariants, and tensor product decompositions of compact groups

Abstract

Decomposing tensor products of irreducible representations of compact groups almost always involves multiplicity, wherein some irreducible representations occur more than once in the direct sum decomposition. We show that the multiplicity can always be specified by polynomial group invariants. The setting is a Bargmann–Segal–Fock space in n×N complex variables, where n is the number of labels needed to specify the tensor product and N is the dimension of the fundamental representation of the compact group. Both the tensor product and direct sum bases are realized as polynomials in this space, and it is shown how Clebsch–Gordan and Racah coefficients can be computed by suitably differentiating these polynomials. The example of SU(N) is discussed in detail, and it is shown that the multiplicity can be computed as the solution of certain diophantine equations arising from powers of group invariants, namely minors of determinants.