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 $\Sigma\subset S_{n}$.

中文翻译：

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正规集的 Aldous 谱隙猜想

令 $G$ 是一个有限群，$\Sigma\subseteq G$ 是一个对称子集。Cayley 图 $Cay\left(G,\Sigma\right)$ 的邻接矩阵的每个特征值自然与 $G$ 的一些不可约表示相关联。2009 年由 Caputo、Liggett 和 Richthammer 证明的 Aldous 谱隙猜想指出，如果 $\Sigma$ 是对称群 $S_{n}$ 中的一组换位，则 $Cay\left(S_ {n},\Sigma\right)$ 始终与 $S_{n}$ 的标准表示相关联。受这一开创性结果的启发，我们研究了 $S_{n}$ 中其他类型集合的类似问题。具体来说，我们考虑正规集：在共轭下不变的集。依靠 Larsen 和 Shalev [2008] 的字符边界，我们证明对于足够大的 $n$，如果 $\Sigma\subset S_{n}$ 是一个完整的共轭类，那么最大的非平凡特征值总是与八个低维表示之一相关联。我们进一步表明，当 $\Sigma$ 是任意正态集时，这种类型的结果不成立，但稍弱的结果确实成立。我们以同样的精神陈述一个关于任意对称集 $\Sigma\subset S_{n}$ 的猜想。