Auto-correlation functions for unitary groups

TL;DR

This paper computes auto-correlation functions of characteristic polynomials for specific subgroups of unitary groups, revealing symmetric polynomial identities involving irreducible characters, advancing understanding of random matrix behaviors.

Contribution

It introduces explicit, uniform formulas for auto-correlation functions of characteristic polynomials for certain unitary subgroup ensembles, linking to symmetric polynomial identities.

Findings

Derived explicit formulas for auto-correlation functions

Established connections to symmetric polynomial identities

Extended results to subgroups analogous to Sato--Tate groups

Abstract

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 $\U(3)$ by applying branching rules. These subgroups can be understood as analogs of Sato--Tate groups of $\USp(4)$ in our previous paper. This computation yields symmetric polynomial identities with $m$-variables involving irreducible characters of $\U(m)$ for all $m \ge 1$ in an explicit, uniform way.