对称群中的字映射与曲面关系
Word maps and surface relations in symmetric groups
AI总结：
该研究聚焦带曲面群约束的字映射随机置换不动点期望，通过计算对称群稳定不可约特征标相关期望，证明最短代表元下的期望阶，并重得Magee--Puder的有界性结论。
AI中文摘要：
我们研究了施加曲面群约束时，通过字映射得到的随机置换的不动点期望数量。对于对称群$S_{n}$的稳定不可约特征标$χ$，令$R_{g}=[a_{1},b_{1}]\dots[a_{g},b_{g}]$且$w\in F_{2g}$，我们计算了$\mathbb{E}_{S_{n}^{2g}}\left[χ\left(R_{g}(h)\right)\\#\mathrm{fix}\left(w(h)\right)\right]$。我们证明，若$w$是$γ\inΓ_{g}=\left\langle a_{1},b_{1},\dots,a_{g},b_{g}:R_{g}\right\rangle$共轭类的最短代表元，则该期望为$O\left(1/\dimχ\right)$。作为应用，我们重新得到了Magee--Puder关于$ϕ_{n}(γ)$不动点期望数的大$n$极限的有界性结论，其中$γ\inΓ_{g}$固定，$ϕ_{n}\in\hom\left(Γ_{g},S_{n}\right)$为均匀随机选取。
英文摘要：
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 $χ$ 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 $\mathbb{E}_{S_{n}^{2g}}\left[χ\left(R_{g}(h)\right)\#\mathrm{fix}\left(w(h)\right)\right]$. We show that, if $w$ is a shortest representative for the conjugacy class of $γ\inΓ_{g}=\left\langle a_{1},b_{1},\dots,a_{g},b_{g}:R_{g}\right\rangle$, then this expectation is $O\left(1/\dimχ\right)$. As an application, we recover a boundedness statement of Magee--Puder on the large $n$ limit of the expected number of fixed points of $ϕ_{n}(γ)$, where $γ\inΓ_{g}$ is fixed and $ϕ_{n}\in\hom\left(Γ_{g},S_{n}\right)$ is chosen uniformly at random.
