Preprojective roots and cycle graphs

TL;DR

This paper characterizes affine Coxeter groups by the property that every positive root is preprojective for some Coxeter element, focusing on cyclic Coxeter graphs and their roots.

Contribution

It establishes a precise criterion linking the cyclic structure of Coxeter graphs to the preprojectivity of roots in affine Coxeter groups.

Findings

All positive roots are c-preprojective iff W is affine.

Cyclic Coxeter graphs correspond to affine Coxeter groups.

Characterization of roots in relation to Coxeter elements.

Abstract

We study $c$-preprojective roots for a Coxeter element $c$ of infinite Coxeter group $W$. In particular, we consider the case when any positive root is $c$-preprojective for some Coxeter element $c$. In this paper, by assuming that the Coxeter graph of $W$ is cyclic, we establish that any positive root is $c$-preprojective for some Coxeter element $c$ if and only if $W$ is an affine Coxeter group.