Lattices Generated by Orbits of Flats under Finite Affine-Symplectic Groups

Abstract

Let $\text{ASG}(2v,{\mathbb{F}}_q)$ be the $2v$-dimensional affine-symplectic space over the finite field ${\mathbb{F}}_q$, and let ${\text{ASp}}_{2v}({\mathbb{F}}_q)$ be the affine-symplectic group of degree $2v$ over ${\mathbb{F}}_q$. For any two orbits $M^{\prime }$ and $M”$ of flats under ${\text{ASp}}_{2v}({\mathbb{F}}_q)$, let ${\mathcal{L}}^{\prime }$ (resp. $\mathcal{L}”$) be the set of all flats which are joins (resp. intersections) of flats in $M^{\prime }$ (resp. $M”$) such that $M”\subseteq L^{\prime }$ (resp. $M^{\prime }\subseteq \mathcal{L}”$) and assume the join (resp. intersection) of the empty set of flats in $\text{ASG}(2v,{\mathbb{F}}_q)$ is $\mathrm{\varnothing }$ (resp. ${\mathbb{F}}_q^{(2v)}$). Let $\mathcal{L}={\mathcal{L}}^{\prime }\cap \mathcal{L}”$. By ordering ${\mathcal{L}}^{\prime },\mathcal{L}”,\mathcal{L}$ by ordinary or reverse inclusion, six lattices are obtained. This article discusses the relations between different lattices, and computes their characteristic polynomial.