Copyright © 2012 You Gao and XinZhi Fu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a -dimensional vector space over the finite field . In this paper we assume that is a finite field of odd characteristic, and the singular orthogonal groups of degree over . Let be any orbit of subspaces under . Denote by the set of subspaces which are intersections of subspaces in , where we make the convention that the intersection of an empty set of subspaces of is assumed to be . By ordering by ordinary or reverse inclusion, two lattices are obtained. This paper studies the questions when these lattices are geometric lattices.