The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups

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.