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Lattices generated by transitive sets of subspaces under finite classical groups. II: The orthogonal case of odd characteristic. (English) Zbl 0821.20030

Let \(V\) be a finite dimensional space over a finite field \(F\) of odd characteristic with a nondegenerate symmetric form \(f\). Let \(z \in F^*\) be a non-square element. Any subspace \(P \subset V\) can be represented as \(U \oplus U_ 1 \oplus \text{rad }P\) where \(U\) is a nondegenerate subspace of dimension \(2s\) and index \(s\), \(U_ 1\) is a nondegenerate subspace without nonzero isotropic vectors or \(U_ 1=0\). Set \(m=\dim P\), \(\gamma=\dim U_ 1\). It is well known that \(\gamma \leq 2\) and the form \(f\) has the matrix \(\Gamma=\emptyset\), (1), \((z)\) or \(\text{diag}(1,z)\) in some basis of \(U_ 1\); moreover, \(s\), \(\gamma\) and \(\Gamma\) are uniquely determined. We say that \(P\) is a subspace of type \((m, 2s+\gamma, s, \Gamma)\). Let \({\mathcal M}={\mathcal M}(m, 2s+\gamma, s, \Gamma)\) be the set of all subspaces of type \((m, 2s+\gamma, s, \Gamma)\) and \({\mathcal L}(m, 2s+\gamma, s, \Gamma)\) be the set of subspaces of \(V\) which are intersections of subspaces in \(\mathcal M\).
The goal of the article under review is to investigate the lattices \({\mathcal L}(m, 2s+\gamma, s, \Gamma)\). The inclusion relations between them are studied. An explicit characterization of the subspaces contained in a fixed \({\mathcal L}(m, 2s+\gamma, s, \Gamma)\) is given. The characteristic polynomial of this lattice is computed. If \(| F|=3\), there are some peculiarities which do not occur in the general case.
A closed formula for \(| {\mathcal M}|\) was given by Z. Dai and X. Feng [Acta Math. Sin. 15, 545-558 (1965; Zbl 0147.200)].

MSC:

20G40 Linear algebraic groups over finite fields
51D25 Lattices of subspaces and geometric closure systems
51N30 Geometry of classical groups

Citations:

Zbl 0147.200
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References:

[1] Dai Z., Some\(quot:Anzahl\)quot:theorems in orthogonal geometry over finite fields of characteristic!!2, Acta Maihematica Sinica 15 pp 533– (1965)
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