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Lattices generated by orbits of subspaces under finite singular orthogonal groups. II. (English) Zbl 1255.51002

Summary: Let \(\mathbb F_q^{(2v + \delta + l)}\) be a \((2v + \delta + l)\)-dimensional vector space over the finite field \(\mathbb F_q\). In this paper we assume that \(\mathbb F_q\) is a finite field of odd characteristic, and \(O_{2v + \delta + l, \Delta}(\mathbb F_q)\) are the singular orthogonal groups of degree \(2v + \delta + l\) over \(\mathbb F_q\). Let \(\mathcal M\) be any orbit of subspaces under \(O_{2v + \delta + l, \Delta}(\mathbb F_q)\). Denote by \(\mathcal L\) the set of subspaces which are intersections of subspaces in \(\mathcal M\), where we make the convention that the intersection of an empty set of subspaces of \(\mathbb F_q^{(2v + \delta + l)}\) is assumed to be \(\mathbb F_q^{(2v + \delta + l)}\). By ordering \(\mathcal L\) by ordinary or reverse inclusion, two lattices are obtained. This paper studies the questions when these lattices \(\mathcal L\) are geometric lattices.
For Part I by the authors see [Finite Fields Appl. 16, No. 6, 385–400 (2010; Zbl 1214.51002)].

MSC:

51D25 Lattices of subspaces and geometric closure systems
20G40 Linear algebraic groups over finite fields
06C10 Semimodular lattices, geometric lattices

Citations:

Zbl 1214.51002
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References:

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