MacMahon's Master Theorem, Representation Theory, and Moments of Wishart Distributions

Abstract

D. Foata and D. Zeilberger (1988, SIAM J. Discrete Math.4, 425–433) and D. Vere-Jones (1988, Linear Algebra Appl.111, 119–124) independently derived a generalization of MacMahon's master theorem. In this article we apply their result to obtain an explicit formula for the moments of arbitrary polynomials in the entries of X, a real random matrix having a Wishart distribution. In the case of the complex Wishart distributions, the same method is applicable. Furthermore, we apply the representation theory of GL(d,C), the complex general linear group, to derive explicit formulas for the expectation of Kronecker products of any complex Wishart random matrix.