A Simple Formula for the Moments of Unitarily Invariant Matrix Distributions

We present a new formula for computing arbitrary moments of unitarily invariant matrix distributions. The Schur-Weyl duality is used to decompose the expected value of tensor powers of the random matrices as a linear combination of projection operators onto unitary irreducible representations. The coefficients in this combination, which are labeled by Young diagrams, are expectations of products of determinants of the random matrices. It is demonstrated in a number of important cases, including matrix gamma and matrix beta distributions, that these coefficients can be simply computed from a knowledge of the normalization factors of the distributions.