The complex Wishart distribution and the symmetric group

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 γp,σ if E(exp \, (θ S))=( (Iᵣ-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.