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Mathematics > Combinatorics

arXiv:1807.11833 (math)
[Submitted on 31 Jul 2018 (v1), last revised 11 Sep 2019 (this version, v2)]

Title:On the spectral gap of some Cayley graphs on the Weyl group $W(B_n)$

Authors:Filippo Cesi
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Abstract:The Laplacian of a (weighted) Cayley graph on the Weyl group $W(B_n)$ is a $N\times N$ matrix with $N = 2^n n!$ equal to the order of the group. We show that for a class of (weighted) generating sets, its spectral gap (lowest nontrivial eigenvalue), is actually equal to the spectral gap of a $2n \times 2n$ matrix associated to a $2n$-dimensional permutation representation of $W_n$. This result can be viewed as an extension to $W(B_n)$ of an analogous result valid for the symmetric group, known as `Aldous' spectral gap conjecture', proven in 2010 by Caputo, Liggett and Richthammer.
Comments: Version 1 (v1) contains a mistake. The main result is proved here under a less general hypothesis than in v1. Main result of v1 is left as a conjecture
Subjects: Combinatorics (math.CO); Group Theory (math.GR); Probability (math.PR)
MSC classes: 05C25, 05C50, 20C15, 20C30, 60K35
Cite as: arXiv:1807.11833 [math.CO]
  (or arXiv:1807.11833v2 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.1807.11833

Submission history

From: Filippo Cesi [view email]
[v1] Tue, 31 Jul 2018 14:19:30 UTC (24 KB)
[v2] Wed, 11 Sep 2019 14:31:30 UTC (25 KB)
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