Symmetric function theory and unitary invariant ensembles

Representation theory and the theory of symmetric functions have played a central role in random matrix theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices drawn from the circular unitary ensemble and other circular ensembles related to the classical compact groups. The reason is that they enable the derivation of exact formulas, which then provide a route to calculating the large-matrix asymptotics of these quantities. We develop a parallel theory for the Gaussian Unitary Ensemble (GUE) of random matrices and other related unitary invariant matrix ensembles. This allows us to write down exact formulas in these cases for the joint moments of the traces and the joint moments of the characteristic polynomials in terms of appropriately defined symmetric functions. As an example of an application, for the joint moments of the traces, we derive explicit asymptotic formulas for the rate of convergence of the moments of polynomial functions of GUE matrices to those of a standard normal distribution when the matrix size tends to infinity.