Auto-correlation functions for unitary groups and identities involving Schur functions.

We compute the auto-correlations functions of order $m\ge 1$ for the characteristic polynomials of random matrices from certain subgroups of the unitary groups $U(2)$ and $U(3)$ to express them as sums of Schur functions of $U(m)$. This computation yields new symmetric polynomial identities with $m$-variables involving Schur functions for all $m \ge 1$ in an explicit, uniform way.