Lattices associated with vector spaces over a finite field

Let V denote the n-dimensional row vector space over a finite field Fq, and fix a subspace W of dimension n-d. Let L(n,d)=P∪{V}, where P is the set of all the subspaces of V intersecting trivially with W. Partially ordered by ordinary or reverse inclusion, two families of finite atomic lattices are obtained. This article discusses their geometricity, and computes their characteristic polynomials.