## Abstract

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.

Original language | American English |
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Pages (from-to) | 7067-7086 |

Number of pages | 20 |

Journal | Transactions of the American Mathematical Society |

Volume | 373 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2020 |

### Bibliographical note

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