Aldous's spectral gap conjecture for normal sets

Let S_n denote the symmetric group on n elements, and let Σ ⊆ S_n be a symmetric subset of permutations. Aldous's spectral gap conjecture, proved by Caputo, Liggett, and Richthammer [J. Amer. Math. Soc. 23 (2010), no. 3, 831-851], states that if Σ is a set of transpositions, then the second eigenvalue of the Cayley graph Cay(S_n, Σ) is identical to the second eigenvalue of the Schreier graph on n vertices depicting the action of S_n on {1, . . ., 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 [Invent. Math. 174 (2008), no. 3, 645-687], we show that for large enough n, if Σ ⊂ S_n is a full conjugacy class, then the second eigenvalue of Cay(S_n, Σ) is roughly identical to the second eigenvalue of the Schreier graph depicting the action of S_n on ordered 4-tuples of elements from {1, . . ., n}. We further show that this type of result does not hold when Σ is an arbitrary normal set, but a slightly weaker one does hold. We state a conjecture in the same spirit regarding an arbitrary symmetric set Σ ⊂ S_n, which yields surprisingly strong consequences.