Lattices Generated by Orbits of Subspaces in \(t\)-Singular Linear Spaces

Abstract

For non-negative integers \(n_1, n_2, \ldots, n_t\), let \(GL_{n_1, n_2, \ldots, n_t}(\mathbb{F}_q)\) denote the \(t\)-singular general linear group of degree \(n = n_1 + n_2 + \cdots + n_t\) over the finite field \(\mathbb{F}_q^{n_1+n_2+\ldots+n_t}\) denote the \((n_1+n_2+\ldots+n_t)\)-dimensional \(t\)-singular linear space over the finite \(\mathbb{F}\). Let \(\mathcal{M}\) be any orbit of subspaces under \(GL_{n_1, n_2, \ldots, n_t}(\mathbb{F}_q)\). Denote by \(\mathcal{L}\) the set of all intersections of subspaces in \(M\). Ordered by ordinary or reverse inclusion, two posets are obtained. This paper discusses their geometricity and computes their characteristic polynomials.