The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups
Yuxuan Li, Binzhou Xia, Sanming Zhou
TL;DR
The paper determines the second largest eigenvalue behavior (Aldous property) for nonnormal Cayley graphs on the symmetric group $S_n$ with connection sets $C(n,k;r)$, the $k$-cycles moving the first $r$ points. Using an induction framework that leverages a recurrence for $\mu_2(n,k;r)$, a decomposition of $H$ into smaller components, Weyl inequalities, and deep representation-theoretic tools (including Specht modules, Branching Rule, and normalized characters), the authors obtain precise eigenvalue attainments and identify the representations responsible for the strictly second largest eigenvalue. They prove the Aldous property for all $n\ge5$ with $1\le r<k<n$ except for the two exceptional cases $(n,k,r)=(6,5,1)$ and, when $n$ is odd, $k=n-1$ with $1\le r<n/2$, and they fully analyze the $k=n-1$ case to describe the spectrum and multiplicities. For $4\le k\le n-2$, they establish the Aldous property as well, showing $\alpha_2(\mathrm{Cay}(S_n,C(n,k;r)))=\mu_2(n,k;r)$, while outlining a sharp conjecture and confirming key subcases. The results extend Aldous-type spectral-gap phenomena to broad families of nonnormal Cayley graphs on $S_n$, providing exact eigenvalue characterizations and a unified inductive strategy.
Abstract
A Cayley graph on the symmetric group $S_n$ is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of $S_n$. For $1\leq r < k < n$, let $C(n,k;r)$ be the set of $k$-cycles of $S_n$ moving every point in $\{1, \ldots, r\}$. Recently, Siemons and Zalesski [J. Algebraic Combin. 55 (2022) 989--1005] posed a conjecture which is equivalent to saying that for any $n \ge 5$ and $1\leq r<k<n$ the nonnormal Cayley graph $\mathrm{Cay}(S_n, C(n,k;r))$ on $S_n$ with connection set $C(n,k;r)$ has the Aldous property. Solving this conjecture, we prove that all these graphs have the Aldous property except when (i) $(n, k, r) = (6, 5, 1)$ or (ii) $n$ is odd, $k = n-1$, and $1 \le r < \frac{n}{2}$. Along the way we determine all irreducible representations of $S_n$ that can achieve the strictly second largest eigenvalue of $\mathrm{Cay}(S_n, C(n,n-1;r))$ as well as the smallest eigenvalue of this graph.