Lattices Generated by Subspaces under Symplectic Group over a Finite Field

Abstract

For $1\le d\le v-1$, let $V$ denote the $2v$-dimensional symplectic space over a finite field $F_q$, and fix a $(v-d)$-dimensional totally isotropic subspace $W$ of $V$. Let $L(d,2v)=P\cup \{V\}$, where $P=\{A\mid A\text{ is a subspace of }V,A\cap W=\{0\}\text{ and }A\subset W^{\perp }\}$. Partially ordered by ordinary or reverse inclusion, two families of finite atomic lattices are obtained. This article discusses their geometricity, and computes their characteristic polynomials.