Oriented matroid structures on rank 3 root systems

TL;DR

This paper proves the uniqueness of oriented matroid structures on rank 3 root systems that are compatible with Weyl group actions and restrict correctly to rank 2 subsystems, extending known results to finite and certain rank 3 systems.

Contribution

It establishes the uniqueness of oriented matroid structures on rank 3 root systems under specific symmetry and restriction conditions, including finite and 'clean' systems.

Findings

Unique oriented matroid structure on rank 3 affine root systems with Weyl group symmetry.

Extension of uniqueness results to finite and 'clean' rank 3 root systems.

Contrasts with non-uniqueness in higher rank cases.

Abstract

We show that, given a rank 3 affine root system $\Phi$ with Weyl group $W$, there is a unique oriented matroid structure on $\Phi$ which is $W$-equivariant and restricts to the usual matroid structure on rank 2 subsystems. Such oriented matroids were called oriented matroid root systems in Dyer-Wang (2021), and are known to be non-unique in higher rank. We also show uniqueness for any finite root system or "clean" rank 3 root system (which conjecturally includes all rank 3 root systems).