Auto-Correlation Functions for Unitary Groups

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 \({\text {U}}(2)\) and \({\text {U}}(3)\) by establishing new branching rules. These subgroups can be understood as certain analogues of Sato–Tate groups of \({\text {USp}}(4)\) in our previous paper. Our computation yields symmetric polynomial identities with m-variables involving irreducible characters of \({\text {U}}(m)\) for all \(m \ge 1\) in an explicit, uniform way.