Lattices generated by two orbits of subspaces under finite classical groups (Q1011453)

The two authors continue their work on lattices generated by orbits of classical groups in finite vector spaces [Finite Fields Appl. 14, 571--578 (2008; Zbl 1158.51003)]. Let \(\mathbb F_q^{(n)}\) be the \(n\)-dimensional vector space over a finite field \(\mathbb F_q\), and let \(G_n\) be the symplectic group \(Sp_n(\mathbb F_q)\) where \(n=2\nu \); or the unitary group \(U_n(\mathbb F_q)\) where \(q=q_0^2\). For any two orbits \(M_{1}\) and \(M_{2}\) of subspaces under \(G_n\), let \(L_{1}\) (resp. \(L_{2}\)) be the set of all subspaces which are sums (resp. intersections) of subspaces in \(M_{1}\) (resp. \(M_{2}\)) such that \(M_{2}\subseteq L_{1}\) (resp. \(M_{1}\subseteq L_{2}\)). Suppose \(\mathcal L\) is the intersection of \(L_{1}\) and \(L_{2}\) containing \(\{0\}\) and \(\mathbb F_q^{(n)}\). By ordering \(\mathcal L\) by ordinary or reverse inclusion, two families of atomic lattices are obtained. This article characterizes the subspaces in these lattices and classifies their geometricity.