Automorphism groups of root systems matroids

TL;DR

This paper characterizes and explicitly computes the automorphism groups of matroids derived from irreducible root systems, showing they are uniquely determined by their size-3 independent sets, thus completing their classification.

Contribution

It proves that automorphism groups of all irreducible root systems matroids are uniquely determined by their size-3 independent sets and provides explicit computations for these groups.

Findings

Automorphism groups are uniquely determined by size-3 independent sets.

Explicit computation of automorphism groups for all irreducible root systems.

Complete classification of automorphism groups of root systems matroids.

Abstract

Given a root system $\mathsf{R}$, the vector system $\tilde{\mathsf{R}}$ is obtained by taking a representative $v$ in each antipodal pair $\{v, -v\}$. The matroid $M(\mathsf{R})$ is formed by all independent subsets of $\tilde{\mathsf{R}}$. The automorphism group of a matroid is the group of permutations preserving its independent subsets. We prove that the automorphism groups of all irreducible root systems matroids $M(\mathsf{R})$ are uniquely determined by their independent sets of size 3. As a corollary, we compute these groups explicitly, and thus complete the classification of the automorphism groups of root systems matroids.