Stein's Method and Characters of Compact Lie Groups

TL;DR

This paper applies Stein's method to analyze the distribution of traces of random elements in compact Lie groups and symmetric spaces, proving central limit theorems with minimal information and illustrating the approach on classical groups and ensembles.

Contribution

It introduces a novel application of Stein's method using character values and irreducible decomposition to study trace distributions in compact Lie groups.

Findings

Central limit theorems established for traces of random group elements

Method applicable to classical Lie groups and Dyson's ensembles

Framework useful for higher-dimensional characters where normal approximation fails

Abstract

Stein's method is used to study the trace of a random element from a compact Lie group or symmetric space. Central limit theorems are proved using very little information: character values on a single element and the decomposition of the square of the trace into irreducible components. This is illustrated for Lie groups of classical type and Dyson's circular ensembles. The approach in this paper will be useful for the study of higher dimensional characters, where normal approximations need not hold.