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

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.

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

References in corpus (1)

• The second largest eigenvalues of some Cayley graphs on alternating groups

Cited by in corpus (1)

• The second eigenvalue of some normal Cayley graphs of high transitive groups