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 formulae, which
then provide a route to calculating the large-matrix asymptotics of these quantities.
We develop a parallel theory for the Gaussian Unitary Ensemble of random matrices,
and other related unitary invariant matrix ensembles. This allows us to write down
exact formulae 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 formulae 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