Word Measures on Unitary Groups: Improved Bounds for Small Representations

Yaron Brodsky

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arXiv · 2208.11957

Word Measures on Unitary Groups: Improved Bounds for Small Representations

Yaron Brodsky

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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 $\mu_{w, U(n)}$.

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Yaron Brodsky. 2022-08-25. Word Measures on Unitary Groups: Improved Bounds for Small Representations. https://arxiv.org/abs/2208.11957

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