On the Geometricity of Lattices Generated by Orbits of Subspaces Under Finite Classical Groups,

Let nq be the n-dimensional vector space over the finite field q and let Gn be one of the classical groups of degree n over q. Let be any orbit of subspaces under Gn. Denote by the set of subspaces which are intersections of subspaces in and assume the intersection of the empty set of subspaces of nq is nq itself. By ordering by ordinary or reverse inclusion, two lattices are obtained. This paper discusses when they form geometric lattices.