Analogs and Extensions of Selberg’s Integral

Abstract

This article will center on Selberg’s integral, some of its extensions, and an interesting connection with classical invariant theory. We present a direct approach to the evaluation of Selberg-type integrals of the Jack symmetric functions in the case n = 2. We show that in Selberg’s integral, the discriminant may be replaced by a large number of the classical invariants. Using results on hypergeometric functions of matrix argument, we extend some integrals of Mehta. Some analogs of Dyson’s integral, arising in Fourier series on compact groups, are evaluated. Using one of these latter integrals, we solve an old problem of H. Weyl by deriving a sharp upper bound on the number of irreducible components which can appear in the decomposition of tensor products of self-representations of the unitary group.Key wordsSelberg’s integralzonal polynomialhypergeometric function of matrix argumentrepresentation theory, invariant theoryspherical transformSchur functionmultipliers