LAUSR

Let $G$ be a finite group and $\Sigma\subseteq G$ a symmetric subset. Every eigenvalue of the adjacency matrix of the Cayley graph $Cay\left(G,\Sigma\right)$ is naturally associated with some irreducible representation of $G$. Aldous' spectral gap conjecture, proved in 2009 by Caputo, Liggett and Richthammer, states that if $\Sigma$ is a set of transpositions in the symmetric group $S_{n}$, then the second eigenvalue of $Cay\left(S_{n},\Sigma\right)$ is always associated with the standard representation of $S_{n}$. Inspired by this seminal result, we study similar questions for other types of sets in $S_{n}$. Specifically, we consider normal sets: sets that are invariant under conjugation. Relying on character bounds due to Larsen and Shalev [2008], we show that for large enough $n$, if $\Sigma\subset S_{n}$ is a full conjugacy class, then the largest non-trivial eigenvalue is always associated with one of eight low-dimensional representations. We further show that this type of result does not hold when $\Sigma$ is an arbitrary normal set, but a slightly weaker result does hold. We state a conjecture in the same spirit regarding an arbitrary symmetric set $\Sigma\subset S_{n}$.