Reflection Representations of Coxeter Groups and Homology of Coxeter Graphs

TL;DR

This paper classifies a broad class of Coxeter group representations, called generalized geometric representations, using homology of related graphs, and describes when generators act as reflections.

Contribution

It introduces a classification of generalized geometric representations of Coxeter groups via homology of associated graphs, extending geometric representation theory.

Findings

Classified generalized geometric representations using graph homology

Provided explicit descriptions of reflection actions of generators

Connected homology of graphs to Coxeter group representations

Abstract

We study and classify a class of representations (called generalized geometric representations) of a Coxeter group of finite rank. These representations can be viewed as a natural generalization of the geometric representation. The classification is achieved by using characters of the integral homology group of certain graphs closely related to the Coxeter graph. On this basis, we also provide an explicit description of those representations on which the defining generators of the Coxeter group act by reflections.