Word measures on unitary groups: improved bounds for small representations (Q6629647)

Let \(F_{r}\) be the free group on \(r\) generators and \(G\) a compact group. For any word \(w \in F_{r}\) let \(\mu_{w,G}\) be the measure defined by choosing \(r\) elements of \(G\) Haar-uniformly and independently and plugging them into \(w\). Equivalently, the measure \(\mu_{w,G}\) is the pushforward of the Haar-measure on \(G^{r}\) along the map \(G^{r} \rightarrow G \;\;\; (g_{1},\ldots, g_{r}) \mapsto w(g_{1},\ldots, g_{r})\).\N\NWord measures on the unitary groups \(\mathbf{U}(n)\) were studied by \textit{M. Magee} and \textit{D. Puder} [Invent. Math. 218, No. 2, 341--411 (2019; Zbl 1484.60005)]. In particular, they examined the behavior of the moments of \(\mu_{w,\mathbf{U}(n)}\) as a function of \(n\), establishing a connection between their asymptotic behavior and certain algebraic invariants of \(w\), such as its commutator length.\N\NThe author employs geometric insights to refine their analysis and shows that the asymptotic behavior of the moments is also governed by the primitivity rank of \(w\). Furthermore, he applies his methods to prove a special case of a Conjecture 1.13 of [\textit{L. Hanany} and \textit{D. Puder}, Int. Math. Res. Not. 2023, No. 11, 9221--9297 (2023; Zbl 1522.20016)] regarding the asymptotic behavior of expected values of irreducible characters of \(\mathbf{U}(n)\) under \(\mu_{w,\mathbf{U}(n)}\).