Oriented Matroid Structures From Realized Root Systems

TL;DR

This paper studies the uniqueness of oriented matroid structures derived from root systems of Coxeter groups, establishing results for various classes and identifying exceptions in affine types.

Contribution

It proves the uniqueness of the oriented matroid structure for finite and certain affine Coxeter groups, and characterizes the exceptions in type tenilde{A}_n.

Findings

Unique oriented matroid structure for finite Coxeter groups

Unique structure for irreducible affine Weyl groups and rank three Coxeter groups

Three possible structures for tenilde{A}_n, n 3

Abstract

This paper investigates the question of uniqueness of the reduced oriented matroid structure arising from root systems of a Coxeter group in real vector spaces. We settle the question for finite Coxeter groups, irreducible affine Weyl groups and all rank three Coxeter groups. In these cases, the oriented matroid structure is unique unless $W$ is of type $\widetilde{A}_n, n\geq 3$, in which case there are three possibilities.