Word maps and surface relations in symmetric groups

Abstract

We study the expected number of fixed points of a random permutation obtained via a word map, with surface group constraints imposed. For stable irreducible characters $\ddot{I}‡$ of the symmetric group $S_n$ and with $R_g=[a_1,b_1]\dots [a_g,b_g]$ and $w\in F_{2g}$, we compute ${\mathbb{E}}_{S_n^{2g}}\left[\ddot{I}‡\left(R_g(h)\right)\#\mathrm{fix}\left(w(h)\right)\right]$. We show that, if $w$ is a shortest representative for the conjugacy class of ${\hat{I}}^3\in \hat{I}{“}_g=⟨a_1,b_1,\dots ,a_g,b_g:R_g⟩$, then this expectation is $O\left(1/\dim \ddot{I}‡\right)$. As an application, we recover a boundedness statement of Magee--Puder on the large $n$ limit of the expected number of fixed points of $\ddot{I}{•}_n({\hat{I}}^3)$, where ${\hat{I}}^3\in \hat{I}{“}_g$ is fixed and $\ddot{I}{•}_n\in \mathrm{hom}\left(\hat{I}{“}_g,S_n\right)$ is chosen uniformly at random.

32 pages, 6 figures, comments welcome!