Preprojective roots of Coxeter groups

TL;DR

This paper explores the concept of preprojective roots in Coxeter groups, establishing their properties and connections to quiver representations, and characterizing finiteness of the group through these roots.

Contribution

It introduces the notion of preprojective roots in Coxeter groups and relates their properties to the group's finiteness and quiver representation theory.

Findings

A Coxeter group is finite iff all positive roots are preprojective.

Preprojective roots relate to orientation-admissible words and reduced words.

The structure of the Coxeter graph influences the properties of roots and group finiteness.

Abstract

Certain results on representations of quivers have analogs in the structure theory of general Coxeter groups. A fixed Coxeter element turns the Coxeter graph into an acyclic quiver, allowing for the definition of a preprojective root. A positive root is an analog of an indecomposable representation of the quiver. The Coxeter group is finite if and only if every positive root is preprojective, which is analogous to the well-known result that a quiver is of finite representation type if and only if every indecomposable representation is preprojective. Combinatorics of orientation-admissible words in the graph monoid of the Coxeter graph relates strongly to reduced words and the weak order of the group.