Lattices Associated with Vector Spaces Over A Finite Field

Abstract

Let \(V\) denote the \(n\)-dimensional row vector space over a finite field \(\mathbb{F}_q\), and let \(W\) be a subspace of dimension \(n-d\). Let \(L(n,d) = \mathcal{P} \cup \{0\}\), where \({P} = \{A | A \text{ is a subspace of } V, A + W = V\}\). 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.