Invariant theory, generalized Casimir operators, and tensor product decompositions ofU(N)

Abstract

One of the central problems in the representation theory of compact groups concerns multiplicity, wherein an irreducible representation occurs more than once in the decomposition of the n-fold tensor product of irreducible representations. The problem is that there are no operators arising from the group itself whose eigenvalues can be used to label the equivalent representations occurring in the decomposition. In this paper we use invariant theory along with so-called generalized Casimir operators to show how to resolve the multiplicity problem for the U(N) groups. The starting point is to augment the n-fold tensor product space with the contragredient representation of interest and construct a subspace of U(N) invariants. The setting for this construction is a polynomial space embedded in a Fock space of complex variables which carries all the irreducible representations of U(N) (or GLN ). The dimension of the invariant subspace is equal to the multiplicity occurring in the tensor product decomposition. Generalized Casimir operators are operators from the universal enveloping algebra of outer product U(N) groups that commute with the diagonal U(N) action and whose eigenvalues can be used to label the multiplicity. Using the notion of dual representations we show how to rewrite the generalized Casimir operators and prove that they act invariantly on the invariant subspace. A complete set of commuting generalized Casimir operators can therefore be used to construct eigenvectors that form an orthonormal basis in the invariant subspace. Different sets of generalized commuting Casimir operators generate different orthonormal bases in the invariant subspace; the overlaps between the eigenvectors of different commuting sets of generalized Casimir operators are called invariant coefficients. We show that Racah coefficients are special cases of invariant coefficients in which the generalized Casimir operators have been chosen with respect to a definite coupling scheme in the tensor product. The paper concludes with an example of the threefold tensor product of the eight-dimensional irreducible representation of U(3) in which the multiplicity of the chosen irreducible representation is 6. Eigenvectors in the six-dimensional invariant subspace are computed for different sets of generalized Casimir operators and invariant coefficients, including Racah coefficients.