Oriented Matroid Structures From Realized Root Systems

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.