The complex Wishart distribution and the symmetric group

Abstract

Let V be the space of (r, r) Hermitian matrices and let Ω be the cone of the positive definite ones. We say that the random variable S, taking its values in Ω̄, has the complex Wishart distribution γ p,σ if E(exp trace(θS)) = (det(I r - σθ)) -p, where σ and σ -1 - θ are in Ω, and where p = 1, 2, . . . , r - 1 or p > r - 1. In this paper, we compute all moments of S and S -1. The techniques involve in particular the use of the irreducible characters of the symmetric group.