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.

Matroid automorphisms of the H_4 root system

“…This incidence structure allows us to compute the stabilizer of a point of the matroid (Lemma 4.1), a fact we need to understand the structure of the group. The connection between the geometric and combinatorial symmetry of certain root systems has been explored in [4,5,6,7]. In [7], matroid automorphism groups are computed for the root systems A n , B n and D n , while [5] considers the root system H 3 associated with the icosahedron and [6] examines the matroid associated with the root system F 4 .…”

“…The connection between the geometric and combinatorial symmetry of certain root systems has been explored in [4,5,6,7]. In [7], matroid automorphism groups are computed for the root systems A n , B n and D n , while [5] considers the root system H 3 associated with the icosahedron and [6] examines the matroid associated with the root system F 4 . The general case is treated in [4], where a computer program is employed to show that Aut(M (S)) ∼ = G S /W for all root systems S except F 4 , H 3 and H 4 , where G S is the Coxeter/Weyl group associated with S and W is either the 2-element group Z 2 (when G has central inversion) or W is trivial (when G does not have central inversion).…”