Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Aldous’s spectral gap conjecture for normal sets
HTML articles powered by AMS MathViewer

by Ori Parzanchevski and Doron Puder PDF
Trans. Amer. Math. Soc. 373 (2020), 7067-7086 Request permission

Abstract:

Let $S_{n}$ denote the symmetric group on $n$ elements, and let $\Sigma \subseteq S_{n}$ 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 $\Sigma$ is a set of transpositions, then the second eigenvalue of the Cayley graph $\mathrm {Cay\!}\left (S_{n},\Sigma \right )$ is identical to the second eigenvalue of the Schreier graph on $n$ vertices depicting the action of $S_{n}$ on $\left \{ 1,\ldots ,n\right \}$. 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 $\Sigma \subset S_{n}$ is a full conjugacy class, then the second eigenvalue of $\mathrm {Cay}\!\left (S_{n},\Sigma \right )$ is roughly identical to the second eigenvalue of the Schreier graph depicting the action of $S_{n}$ on ordered $4$-tuples of elements from $\left \{ 1,\ldots ,n\right \}$. We further show that this type of result does not hold when $\Sigma$ 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 $\Sigma \subset S_{n}$, which yields surprisingly strong consequences.
References
Similar Articles
Additional Information
  • Ori Parzanchevski
  • Affiliation: Einstein School of Mathematics, Edmond J. Safra Campus, The Hebrew University, Jerusalem, Israel
  • MR Author ID: 879748
  • ORCID: 0000-0003-1596-215X
  • Email: parzan@math.huji.ac.il
  • Doron Puder
  • Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
  • MR Author ID: 903080
  • ORCID: 0000-0003-2793-7525
  • Email: doronpuder@gmail.com
  • Received by editor(s): July 1, 2018
  • Received by editor(s) in revised form: December 31, 2019, and January 6, 2020
  • Published electronically: July 29, 2020
  • Additional Notes: 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.
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 7067-7086
  • MSC (2010): Primary 20C30, 05C81; Secondary 05C50, 20B20, 20B30, 60B15
  • DOI: https://doi.org/10.1090/tran/8155
  • MathSciNet review: 4155200