Spectra of Cayley Graphs of Complex Reflection Groups

TL;DR

This paper extends the understanding of eigenvalues in Cayley graphs from real to complex reflection groups, proving integrality and providing combinatorial formulas for spectra.

Contribution

It proves the integrality of eigenvalues for various matrices of Cayley graphs of complex reflection groups and offers combinatorial formulas for their spectra.

Findings

Eigenvalues of distance, adjacency, and codimension matrices are integral.

Provides combinatorial formulas for spectra of certain complex reflection groups.

Extends results from real to complex reflection groups.

Abstract

Renteln proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. We prove the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provide a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups.