Group character averages via a single Laguerre

TL;DR

This paper explores group character averages in Gaussian matrix models, revealing sum rules that simplify complex trace convolutions using Laguerre polynomials.

Contribution

It introduces generic sum rules that express traces via convolutions of a single Laguerre polynomial, simplifying calculations in matrix models.

Findings

Sum rules relate traces to Laguerre polynomial convolutions.

Simplification of complex trace calculations in Gaussian matrix models.

Extension of character averages using Laguerre polynomial properties.

Abstract

Average of exponential ${\rm Tr}_R e^X$, i.e. of a group rather than an algebra character, in Gaussian matrix model is known to be an amusing generalization of Schur polynomial, where time variables are substituted by traces of products of non-commuting matrices ${\rm Tr} \left(\prod_i A_{k_i}\right)$ and are thus labeled by weak compositions. The entries of matrices $A_k$ are made from extended Laguerre polynomials, what introduces additional difficulties. We describe the generic sum rules, which express arbitrary traces through convolutions of a single Laguerre polynomial $L_{N-1}^1(z_{k_i})$, what is a considerable simplification.