Lattices generated by orbits of subspaces under finite singular pseudo-symplectic groups. I.
Let \(\mathbb{F}_q^{2\nu+1+l}\) be the \((2\nu+1+l)\)-dimensional vector space over the finite field \(\mathbb{F}_q\) of characteristic \(2\), and let \(Ps_{2\nu+1+l,2\nu+1}(\mathbb{F}_q)\) be the singular pseudo-symplectic group of degree \(2\nu+1+l\) over \(\mathbb{F}_q\). Let \(\mathcal{M}\) be any orbit of subspaces under \(Ps_{2\nu+1+l,2\nu+1}(\mathbb{F}_q)\). Denote by \(\mathcal{L}\) the set of subspaces that are intersections of subspaces in \(\mathcal{M}\). By ordering \(\mathcal{L}\) by ordinary or reverse inclusion, two lattices \(\mathcal{L_O}(\mathcal{M})\) and \(\mathcal{L_R}(\mathcal{M})\) are obtained. The authors investigate inclusion relations between different lattices, they give a characterization of subspaces contained in a given lattice \(\mathcal{L}\), and obtain the characteristic polynomial of \(\mathcal{L}\).
scientific article; zbMATH DE number 5603565