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

Abstract

Let ${𝔽}_q^{(2\nu +\delta +l)}$ be a $(2\nu +\delta +l)$-dimensional vector space over the finite field ${𝔽}_q$. In this paper we assume that ${𝔽}_q$ is a finite field of odd characteristic, and $O_{2\nu +\delta +l,\mathrm{\Delta }}({𝔽}_q)$ the singular orthogonal groups of degree $2\nu +\delta +l$ over ${𝔽}_q$. Let $ℳ$ be any orbit of subspaces under $O_{2\nu +\delta +l,\mathrm{\Delta }}({𝔽}_q)$. 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 ${𝔽}_q^{(2\nu +\delta +l)}$ is assumed to be ${𝔽}_q^{(2\nu +\delta +l)}$. By ordering $ℒ$ by ordinary or reverse inclusion, two lattices are obtained. This paper studies the questions when these lattices $ℒ$ are geometric lattices.