TY - JOUR

T1 - Aldous's spectral gap conjecture for normal sets

AU - Parzanchevski, Ori

AU - Puder, Doron

N1 - Funding Information:
Received by the editors July 1, 2018, and, in revised form, December 31, 2019, and January 6, 2020. 2010 Mathematics Subject Classification. Primary 20C30, 05C81; Secondary 05C50, 20B20, 20B30, 60B15. This research was supported by the Israel Science Foundation, ISF grant 1031/17 awarded to the first author and ISF grant 1071/16 awarded to the second author.
Publisher Copyright:
© 2020 American Mathematical Society
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/10

Y1 - 2020/10

N2 - Let Sn denote the symmetric group on n elements, and let Σ ⊆ Sn 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(Sn, Σ) is identical to the second eigenvalue of the Schreier graph on n vertices depicting the action of Sn on {1, . . ., n}. Inspired by this seminal result, we study similar questions for other types of sets in Sn. 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 ⊂ Sn is a full conjugacy class, then the second eigenvalue of Cay(Sn, Σ) is roughly identical to the second eigenvalue of the Schreier graph depicting the action of Sn 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 ⊂ Sn, which yields surprisingly strong consequences.

AB - Let Sn denote the symmetric group on n elements, and let Σ ⊆ Sn 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(Sn, Σ) is identical to the second eigenvalue of the Schreier graph on n vertices depicting the action of Sn on {1, . . ., n}. Inspired by this seminal result, we study similar questions for other types of sets in Sn. 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 ⊂ Sn is a full conjugacy class, then the second eigenvalue of Cay(Sn, Σ) is roughly identical to the second eigenvalue of the Schreier graph depicting the action of Sn 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 ⊂ Sn, which yields surprisingly strong consequences.

UR - http://www.scopus.com/inward/record.url?scp=85092357210&partnerID=8YFLogxK

U2 - 10.1090/tran/8155

DO - 10.1090/tran/8155

M3 - מאמר

AN - SCOPUS:85092357210

VL - 373

SP - 7067

EP - 7086

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 10

ER -