The second largest eigenvalue of normal Cayley graphs on symmetric groups generated by cycles

TL;DR

This paper analyzes the second largest eigenvalue of normal Cayley graphs on symmetric groups generated by cycles, identifying which irreducible representations attain this eigenvalue and exploring conditions for the Aldous property.

Contribution

It characterizes the irreducible representations achieving the second largest eigenvalue and determines when the Cayley graphs have the Aldous property, resolving a recent conjecture.

Findings

Second largest eigenvalue achieved by at most four irreducible representations.

Exact conditions for the Aldous property based on the generating set.

Proof of a conjecture on eigenvalues for specific cycle lengths.

Abstract

We study the normal Cayley graphs $\mathrm{Cay}(S_n, C(n,I))$ on the symmetric group $S_n$, where $I\subseteq \{2,3,\ldots,n\}$ and $C(n,I)$ is the set of all cycles in $S_n$ with length in $I$. We prove that the strictly second largest eigenvalue of $\mathrm{Cay}(S_n,C(n,I))$ can only be achieved by at most four irreducible representations of $S_n$, and we determine further the multiplicity of this eigenvalue in several special cases. As a corollary, in the case when $I$ contains neither $n-1$ nor $n$ we know exactly when $\mathrm{Cay}(S_n, C(n,I))$ has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of $S_n$, and we obtain that $\mathrm{Cay}(S_n, C(n,I))$ does not have the Aldous property whenever $n \in I$. As another corollary of our main results, we prove a recent conjecture on the second largest eigenvalue of $\mathrm{Cay}(S_n, C(n,\{k\}))$ where $2 \le k \le n-2$.