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.
展开▼
