Huo, Yuanji; Liu, Yingsheng; Wan, Zhexian Lattices generated by transitive sets of subspaces under finite classical groups. II: The orthogonal case of odd characteristic. (English) Zbl 0821.20030 Commun. Algebra 20, No. 9, 2685-2727 (1992). 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)]. Reviewer: I.Suprunenko (Minsk) Cited in 1 ReviewCited in 29 Documents MSC: 20G40 Linear algebraic groups over finite fields 51D25 Lattices of subspaces and geometric closure systems 51N30 Geometry of classical groups Keywords:finite orthogonal groups; lattices of subspaces; nondegenerate symmetric forms; characteristic polynomials Citations:Zbl 0147.200 PDFBibTeX XMLCite \textit{Y. Huo} et al., Commun. Algebra 20, No. 9, 2685--2727 (1992; Zbl 0821.20030) Full Text: DOI 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) [2] Dai Z., English translation: Chincsc Mathematics 7 pp 265– (1965) [3] Dickson L.E., Lattices generated by ansitive sets of subspaces. II [4] Hou Y., Comm.in Algebra 20 pp 1123– (1992) · Zbl 0763.51002 · doi:10.1080/00927879208824395 [5] Wan Z., Tr. Vsesoyuzn. matem. s’ezda [6] Wan, Z. ”Finite geometries and block designs, to appear in the Proceedings of R. G Bo8e Memorial Conference on Combinatorics”. Edited by: Bo8e, R.G. · Zbl 0882.05039 [7] Wan Z., Science Press [8] Witt E., J.Reine Angew.Math 176 pp 31– (1937) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.