Matroid Automorphisms of the $F_4$ Root System

Abstract

Let $F_4$ be the root system associated with the 24-cell, and let $M(F_4)$ be the simple linear dependence matroid corresponding to this root system. We determine the automorphism group of this matroid and compare it to the Coxeter group $W$ for the root system. We find non-geometric automorphisms that preserve the matroid but not the root system.