Lattices associated with vector spaces over a finite field

AbstractLet 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

Lattices generated by orbits of totally isotropic flats under finite affine-classical groups

AbstractLet ACG(n,Fq) be one of n-dimensional affine-classical spaces over a finite field Fq, and let M be an orbit of totally isotropic flats under the corresponding affine-classical group. Denote by L the set of all the intersections of totally isotropic flats in M. If we partially order L by ordinary or reverse inclusion, two families of finite atomic lattices are obtained. This paper discusses their geometricity and computes their characteristic polynomials