Word Measures on Unitary Groups: Improved Bounds for Small Representations

TL;DR

This paper refines the understanding of how measures derived from free group words behave on unitary groups, linking their moments to algebraic invariants like primitivity rank and confirming aspects of a related conjecture.

Contribution

It provides a more precise asymptotic analysis of moments of measures on unitary groups, connecting them to the primitivity rank of words and proving a special case of a conjecture on character expectations.

Findings

Moments of measures relate to primitivity rank of words.

Asymptotic bounds improve previous results.

Partial proof of Hanany and Puder's conjecture.

Abstract

Let $F$ be a free group of rank $r$ and fix some $w\in F$. For any compact group $G$ we can define a measure $\mu_{w,G}$ on $G$ by (Haar-)uniformly sampling $g_1,...,g_r\in G$ and evaluating $w(g_1,...,g_r)$. In [arXiv:1802.04862], Magee and Puder studied the case where $G$ is the unitary group $U(n)$, and analyzed how the moments of $\mu_{w,U(n)}$ behave as a function of $n$. In particular, they obtained asymptotic bounds on those moments, related to the commutator length and the stable commutator length of $w$. We continue their line of work and give a more precise analysis of the asymptotic behavior of the moments of $\mu_{w, U(n)}$, showing that it is related to another algebraic invariant of $w$: its primitivity rank. In addition, we prove a special case of a conjecture of Hanany and Puder ([arXiv:2009.00897, Conjecture 1.13]) regarding the asymptotic behaviour of expected values of irreducible characters under $\mu_{w, U(n)}$.