Matroid automorphisms of the root system H 3

Abstract

Let H 3 be the root system associated with the icosahedron, and let M(H 3 ) be the linear dependence matroid corresponding to this root system. We prove Aut(M(H 3 )) ∼ = S 5 , and interpret these automorphisms geometrically.

On deformation and classification of ∨-systems

“…Here matroid M is defined on the set X = {1, 2, 3, 4, 5, 6, 7, 8, 9}, with 2-flats {(4, 1, 6), (6, 2, 3), (4, 5, 2), (1, 5, 3)} and { (3,4,8,9), (1,2,7,8), (5,6,7,9)} with three and four elements respectively. Together with the 3-flat X this gives the complete list of flats.…”

“…Remark. The existence of two projectively non-equivalent realisations is related to the existence of a symmetry of matroid M (H 3 ), which can not be realised geometrically, see [3]. These two realisations are related by re-ordering of the vectors and thus give rise to the equivalent ∨-systems.…”

“…Below is a schematic way to present all known ∨-systems in dimension three taken from [5]. 3 (AB (t), A ) 4 1 1 (AB (t), A ) (E , A x D ) 8 1 4 (E , A ) 7 4 (E , A x A ) 6 8 1 4…”

“…The ∨-systems of type AB 4 , G 3 and D 3 are related to the exceptional generalised root systems AB(1, 3), G(1, 2) and D(2, 1, λ) appeared in the theory of basic classical Lie superalgebras [16,17]. 6), (2,5), (3,4)} 9.2. ∨-system D 3 (t, s).…”

“…∨-system D 3 (t, s). (1, 4), (1,7), (2,5), (2,8), (3,6), (3,9)} Four 3-point lines: I 3 ={(4, 5, 9), (4, 6, 8), (5, 6, 7), (7,8,9) 9), (2, 7), (3, 5), (5, 10), (7, 10), (9, 10)} (3,4), (3,5), (4,9), (4, 11), (5,8), (5, 10)}…”