Lattices Generated by Orbits of Subspaces under Finite Singular Orthogonal Groups II

Abstract EN

Let ?q(2ν+δ+l) be a (2ν+δ+l)-dimensional vector space over the finite field ?q.

In this paper we assume that ?q is a finite field of odd characteristic, and O2ν+δ+l, Δ(?q) the singular orthogonal groups of degree 2ν+δ+l over ?q.

Let ℳ be any orbit of subspaces under O2ν+δ+l, Δ(?q).

Denote by ℒ the set of subspaces which are intersections of subspaces in ℳ, where we make the convention that the intersection of an empty set of subspaces of ?q(2ν+δ+l) is assumed to be ?q(2ν+δ+l).

By ordering ℒ by ordinary or reverse inclusion, two lattices are obtained.

This paper studies the questions when these lattices ℒ are geometric lattices.