Automorphism groups of root system matroids

Matroid automorphisms of the H_4 root system

“…In [4], Aut(M (H 4 )) is obtained as follows: First extend the root system H 4 by adding an isomorphic copy H 4 of H 4 . Then Aut(M (H 4 )) ∼ = W (H 4 ∪ H 4 )/Z, where Z ∼ = Z 2 is the subgroup generated by central inversion (Z is the center of W ).…”

“…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 .…”

“…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). No attempt is made to understand the structure of these matroids in [4], however.…”

“…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). No attempt is made to understand the structure of these matroids in [4], however.…”