Lattices Generated by Two Orbits of Subspaces Under Finite Singular Classical Groups

Abstract

Let be the n + l-dimensional vector space over a finite field 𝔽 q , and let G n+l, n be the singular symplectic group Sp n+l, n (𝔽 q ) where n = 2ν; or the singular unitary group U n+l, n (𝔽 q ) where . For any two orbits M 1 and M 2 of subspaces under G n+l, n , let L 1 (resp., L 2) be the set of all subspaces which are sums (resp., intersections) of subspaces in M 1 (resp., M 2) such that M 2 ⊆ L 1 (resp., M 1 ⊆ L 2). Suppose ℒ is the intersection of L 1 and L 2 containing {0} and . By ordering ℒ by ordinary or reverse inclusion, two families of atomic lattices are obtained. This article characterizes the subspaces in the two lattices and classifies geometricity of these lattices.