All invariant moments of the Wishart distribution

In this paper, we compute moments of a Wishart matrix variate U of the form E(Q(U)) where Q(u) is a polynomial with respect to the entries of the symmetric matrix u, invariant in the sense that it depends only on the eigenvalues of the matrix u. This gives us in particular the expected value of any power of the Wishart matrix U or its inverse U- 1. For our proofs, we do not rely on traditional combinatorial methods but rather on the interplay between two bases of the space of invariant polynomials in U. This means that all moments can be obtained through the multiplication of three matrices with known entries. Practically, the moments are obtained by computer with an extremely simple Maple program.