  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.
