Lattices generated by orbits of subspaces under finite singular orthogonal groups I

Abstract

Let F q ( 2 ν + δ + l ) be the ( 2 ν + δ + l ) -dimensional vector space over the finite field F q . In the paper we assume that F q is a finite field of odd characteristic, and O 2 ν + δ + l , Δ ( F q ) the singular orthogonal groups of degree 2 ν + δ + l over F q . Let M be any orbit of subspaces under O 2 ν + δ + l , Δ ( F q ) . Denote by L the set of subspaces which are intersections of subspaces in M and the intersection of the empty set of subspaces of F q ( 2 ν + δ + l ) is assumed to be F q ( 2 ν + δ + l ) . By ordering L by ordinary or reverse inclusion, two lattices are obtained. This paper studies the inclusion relations between different lattices, a characterization of subspaces contained in a given lattice L , and the characteristic polynomial of L .