arXiv subjects

Ewan Cassidy

Publications and source records attributed to Ewan Cassidy.

3 recordsLinked to original sources

arXiv2026

Word maps and surface relations in symmetric groups

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 $\chi$ 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 $\mathbb{E}_{S_{n}^{2g}}\left[\chi\left(R_{g}(h)\right)\#\mathrm{fix}\left(w(h)\right)\right]$. We show that, if $w$ is a shortest representative for the conjugacy class of $\gamma\in\Gamma_{g}=\left\langle a_{1},b_{1},\dots,a_{g},b_{g}:R_{g}\right\rangle$, then this expectation is $O\left(1/\dim\chi\right)$. As an application, we recover a boundedness statement of Magee--Puder on the large $n$ limit of the expected number of fixed points of $\phi_{n}(\gamma)$, where $\gamma\in\Gamma_{g}$ is fixed and $\phi_{n}\in\hom\left(\Gamma_{g},S_{n}\right)$ is chosen uniformly at random.

math.GR↗

arXiv2025

Random permutations acting on $k$--tuples have near--optimal spectral gap for $k=\mathrm{poly}(n)$

We extend Friedman's theorem to show that, for any fixed $r>1$, a random $2r$--regular Schreier graph associated with the action of $r$ uniformly random permutations of $[n]$ on $k_{n}$--tuples of distinct elements in $[n]$ has a near--optimal spectral gap with high probability, provided $k_{n}\leq n^{\frac{1}{20}-ε}.$ Previously this was known only for $k$--tuples where $k$ is fixed. In fact, we prove the stronger result of strong convergence of random permutations in irreducible representations of quasi--exponential dimension. Along the way, we give a new bound for the expected stable irreducible character of a random permutation obtained via a word map, showing that $\mathbb{E}\left[χ^μ\left(w(σ_{1},\dots,σ_{r})\right)\right]=O\left(\frac{1}{\dimχ^μ}\right)=O\left(n^{-k}\right)$, where $k$ is the number of boxes outside the first row of the Young diagram $μ,$ solving one aspect of a conjecture of Hanany and Puder. We obtain this bound using an extension of Wise's $w$--cycle conjecture.

math.RT↗

arXiv2025

Projection formulas and a refinement of Schur--Weyl--Jones duality for symmetric groups

Schur--Weyl--Jones duality establishes the connection between the commuting actions of the symmetric group $S_{n}$ and the partition algebra $P_{k}(n)$ on the tensor space $\left(\mathbb{C}^n\right)^{\otimes k}.$ We give a refinement of this, determining a subspace of $\left(\mathbb{C}^n\right)^{\otimes k}$ on which we have a version of Schur--Weyl duality for the symmetric groups $S_{n}$ and $S_{k}.$ We use this refinement to construct subspaces of $\left(\mathbb{C}^n\right)^{\otimes k}$ that are isomorphic to certain irreducible representations of $S_{n}\times S_{k}.$ We then use the Weingarten calculus for the symmetric group to obtain an explicit formula for the orthogonal projection from $\left(\mathbb{C}^n\right)^{\otimes k}$ to each subspace.

math.RT↗