Weighted Coxeter graphs and generalized geometric representations of Coxeter Groups

TL;DR

This paper introduces weighted Coxeter graphs and a generalized geometric representation of Coxeter groups, providing conditions for faithfulness and exploring connections to the numbers game, advancing understanding of Coxeter group representations.

Contribution

It defines weighted Coxeter graphs and develops a generalized geometric representation, including criteria for faithfulness and links to the numbers game.

Findings

Provided sufficient conditions for faithfulness of the representation

Established the relationship between balanced graphs and the numbers game

Extended the standard geometric representation to weighted graphs

Abstract

We introduce the notion of weighted Coxeter graph and associate to it a certain generalization of the standard geometric representation of a Coxeter group. We prove sufficient conditions for faithfulness and non-faithfulness of such a representation. In the case when the weighted Coxeter graph is balanced we discuss how the generalized geometric representation is related to the numbers game played on the Coxeter graph.