Lattices Generated by Subspaces under Symplectic Group over a Finite Field

Abstract

For \(1 \leq d \leq v-1\), 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.