Cayley graphs on the symmetric group generated by initial reversals have unit spectral gap

TL;DR

This paper proves that the Cayley graph on the symmetric group generated by initial reversals has a spectral gap of 1, confirming empirical observations with a representation-theoretic proof.

Contribution

It provides a simple, representation-theoretic proof that the spectral gap of these Cayley graphs is always 1, extending previous empirical findings.

Findings

Spectral gap of Cayley graphs is always 1

Representation theory decomposes Laplacian into irreducible components

Confirms empirical evidence with a rigorous proof

Abstract

In a recent paper Gunnells, Scott and Walden have determined the complete spectrum of the Schreier graph on the symmetric group corresponding to the Young subgroup $S_{n-2}\times S_2$ and generated by initial reversals. In particular they find that the first nonzero eigenvalue, or spectral gap, of the Laplacian is always 1, and report that "empirical evidence" suggests that this also holds for the corresponding Cayley graph. We provide a simple proof of this last assertion, based on the decomposition of the Laplacian of Cayley graphs, into a direct sum of irreducible representation matrices of the symmetric group.