Auto-Correlation Functions for Unitary Groups

We compute the auto-correlations functions of order m≥1 for the characteristic polynomials of random matrices from certain subgroups of the unitary groups U(2) and U(3) by establishing new branching rules. These subgroups can be understood as certain analogues of Sato–Tate groups of USp(4) in our previous paper. Our computation yields symmetric polynomial identities with m-variables involving irreducible characters of U(m) for all m≥1 in an explicit, uniform way.