Aldous-type Spectral Gaps in Generalized Symmetric Groups

TL;DR

This paper proves an analog of Aldous' spectral gap conjecture for generalized symmetric groups and extends Caputo's hypergraph conjecture to these groups, broadening the scope of spectral gap results.

Contribution

It establishes a spectral gap result for generalized symmetric groups and connects hypergraph conjecture extensions to these groups, advancing spectral gap theory.

Findings

Proved an Aldous-type spectral gap conjecture for $G\wr S_n$ groups.

Extended Caputo's hypergraph conjecture to generalized symmetric groups.

Connected spectral gap properties between groups and hypergraphs.

Abstract

We prove an analog of Aldous' spectral gap conjecture in the generalized symmetric groups $G\wr S_n$ where $G$ is an arbitrary finite group. Moreover, we show that Caputo's extension of the conjecture to hypergraphs transfers to these groups whenever it holds in the ordinary symmetric group.