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Oriented matroid structures on rank 3 root systems

Grant Barkley, Katherine Tung

Abstract

We show that, given a rank 3 affine root system $Φ$ with Weyl group $W$, there is a unique oriented matroid structure on $Φ$ 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).

Oriented matroid structures on rank 3 root systems

Abstract

We show that, given a rank 3 affine root system with Weyl group , there is a unique oriented matroid structure on which is -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).

Paper Structure

This paper contains 6 sections, 4 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dictionary between hyperplane arrangement and oriented matroid language
  4. Oriented matroids
  5. Biclosed sets and oriented matroid root systems
  6. Proofs

Key Result

Theorem 1

Let $\Phi$ be a clean rank 3 root system. Then $\Phi$ has a unique oriented matroid root system structure.

Figures (1)

  • Figure 1: Configuration of roots in $I$ for $R$ having least 4 sides.

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Remark 5
  • Definition 6
  • Remark 7
  • Definition 8
  • Remark 9
  • Definition 10
  • ...and 9 more