On resolving the multiplicity of arbitrary tensor products of the U(N) groups

Abstract

Representations of U(N) are realised as right translations on holomorphic Hilbert (Bargmann) spaces of n*N complex variables. r-fold tensor product spaces of irreducible representations of U(N) are shown to be isomorphic to subspaces of the holomorphic Hilbert spaces. Maps are exhibited which carry an irreducible representation of U(N) into these subspaces. The algebra of operators commuting with these maps is constructed and it is shown how eigenvalues of certain of these operators can be used to resolve the multiplicity. Several examples from U(3) are explicitly worked out.