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On the geometricity of lattices generated by orbits of subspaces under finite classical groups. (English) Zbl 1024.51004

Let \(V\) be \(n\)-dimensional vector space over the finite field \(F= \text{GF}(q)\), and let \(G\) be a classical group of degree \(n\) over \(F\). Furthermore, let \({\mathcal M}\) be an orbit in \(V\) of subspaces under \(G\), and denote by \({\mathcal L}\) the set of subspaces obtained as intersections of subspaces in \({\mathcal M}\). (Assume \(F\) is the intersection of the empty set of subspaces of \(V\).) The set \({\mathcal L}\) can be ordered by ordinary inclusion or by reverse inclusion. Each gives rise to a lattice. The article under review discusses when these lattices form geometric lattices.
For example, let \(G= \text{GL}(n,F)\), \({\mathcal M}\) the set of subspaces of \(V\) of dimension \(m\) for some \(m\) with \(1\leq m\leq n-1\), and let \({\mathcal L}\) be the set of subspaces generated by \({\mathcal M}\). If \({\mathcal L}_0\) is the lattice obtained by ordering using ordinary inclusion, then \({\mathcal L}_0\) is a finite geometric lattice. On the other hand, if \({\mathcal L}_R\) is the lattice obtained by using reverse inclusion, then \({\mathcal L}_R\) is an atomic lattice but not a geometric lattice.
The authors also investigate the cases when the group \(G\) is symplectic, unitary, and orthogonal. The interested reader is refered to the article for their conclusions.

MSC:

51D25 Lattices of subspaces and geometric closure systems
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References:

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