Aldous’s spectral gap conjecture for normal sets

Abstract

Let S n S_{n} denote the symmetric group on n n elements, and let Σ ⊆ S n \Sigma \subseteq 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 \Sigma is a set of transpositions, then the second eigenvalue of the Cayley graph C a y ( S n , Σ ) \mathrm {Cay\!}\left (S_{n},\Sigma \right ) is identical to the second eigenvalue of the Schreier graph on n n vertices depicting the action of S n S_{n} on { 1 , … , n } \left \{ 1,\ldots ,n\right \} . Inspired by this seminal result, we study similar questions for other types of sets in S n 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 n , if Σ ⊂ S n \Sigma \subset S_{n} is a full conjugacy class, then the second eigenvalue of C a y ( S n , Σ ) \mathrm {Cay}\!\left (S_{n},\Sigma \right ) is roughly identical to the second eigenvalue of the Schreier graph depicting the action of S n S_{n} on ordered 4 4 -tuples of elements from { 1 , … , n } \left \{ 1,\ldots ,n\right \} . We further show that this type of result does not hold when Σ \Sigma 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 \Sigma \subset S_{n} , which yields surprisingly strong consequences.