Coxeter groups and the proper joint spectrums of their faithful representations

TL;DR

This paper demonstrates that Coxeter groups can be uniquely identified by analyzing the proper joint spectrum of their faithful representations, focusing on dihedral groups as a key example.

Contribution

It establishes a spectral characterization of Coxeter groups based on the proper joint spectrum of their faithful representations, providing a new method for group identification.

Findings

Coxeter groups are determined by their proper joint spectrum.

Faithful representations of dihedral groups are analyzed.

Spectral methods can distinguish Coxeter groups from other groups.

Abstract

In this paper, we analyze the faithful representations of the dihedral groups, and prove that the Coxeter groups can be determined by the proper joint spectrum of their faithful representations.