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Lattices generated by two orbits of subspaces under finite classical groups. (English) Zbl 1169.51010

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.

MSC:

51D25 Lattices of subspaces and geometric closure systems
20G40 Linear algebraic groups over finite fields

Citations:

Zbl 1158.51003
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References:

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