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.
You Gao. XinZhi Fu. "Lattices Generated by Orbits of Subspaces under Finite Singular Orthogonal Groups II." J. Appl. Math. 2012 1 - 16, 2012. https://doi.org/10.1155/2012/387132