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 \(M’\) and \(M”\) of flats under \(\text{ASp}_{2v}(\mathbb{F}_q)\), let \(\mathcal{L}’\) (resp. \(\mathcal{L}”\)) be the set of all flats which are joins (resp. intersections) of flats in \(M’\) (resp. \(M”\)) such that \(M” \subseteq L’\) (resp. \(M’ \subseteq \mathcal{L}”\)) and assume the join (resp. intersection) of the empty set of flats in \(\text{ASG}(2v,\mathbb{F}_q)\) is \(\emptyset\) (resp. \(\mathbb{F}_q^{(2v)}\)). Let \(\mathcal{L} =\mathcal{L}’ \cap \mathcal{L}”\). By ordering \(\mathcal{L}’,\mathcal{L}”, \mathcal{L}\) by ordinary or reverse inclusion, six lattices are obtained. This article discusses the relations between different lattices, and computes their characteristic polynomial.