Word Measures on Unitary Groups: Improved Bounds for Small Representations

2022/08/25 by Yaron Brodsky, Brodsky, Yaron
Mathematics · #Finite Group Theory Research #Limits and Structures in Graph Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2208.11957

Abstract

Let F be a free group of rank r and fix some w∈ F. For any compact group G we can define a measure μ_w,G on G by (Haar-)uniformly sampling g₁,...,g_r∈ G and evaluating w(g₁,...,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 μ_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 μ_{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 μ_{w, U(n)}.

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