Representations of Sn*U(N) in repeated tensor products of the unitary groups

Abstract

The n-fold tensor product space generated by a given irreducible representation of the unitary group U(N) is a representation space for the symmetric group S_n as well as for U(N). Using ideas from the theory of dual pairs, such tensor product spaces are decomposed into irreducible representations of U(N) times reducible representations of S_n. Computationally effective formulae for the multiplicity of irreducible representations of S_n are given. Generating sets of invariant polynomials from the enveloping algebra of U(N) that commute with the S_n action are exhibited; it is shown that the eigenvalues of such operators can be used to break the multiplicity occurring in the dual U(N)xS_n action.