Matroid automorphisms of the H_4 root system

TL;DR

This paper investigates the automorphism group of the matroid derived from the 4-dimensional H_4 root system, revealing its structure, transitivity, and geometric properties using incidence and orthoframe analysis.

Contribution

It determines the automorphism group of the H_4 root system matroid, showing its size, structure, and geometric versus non-geometric automorphisms.

Findings

Automorphism group has 14,400 elements, evenly split between geometric and non-geometric.

The group acts transitively and primitively on the matroid flats.

The study uses incidence properties and orthoframes to understand automorphisms.

Abstract

We study the rank 4 linear matroid $M(H_4)$ associated with the 4-dimensional root system $H_4$. This root system coincides with the vertices of the 600-cell, a 4-dimensional regular solid. We determine the automorphism group of this matroid, showing half of the 14,400 automorphisms are geometric and half are not. We prove this group is transitive on the flats of the matroid, and also prove this group action is primitive. We use the incidence properties of the flats and the {\it orthoframes} of the matroid as a tool to understand these automorphisms, and interpret the flats geometrically.