On the generation of dual polar spaces of symplectic type over finite fields
type: Item
instance of: scholarly article
title: On the generation of dual polar spaces of symplectic type over finite fields
zbMATH Open document ID: 0914.51002
author: B. N. Cooperstein
published in: Journal of Combinatorial Theory. Series A
publication date: 1999-06-14T00:00:00Z
review text: Let \(\Gamma=(P,L)\) be a point-line geometry. We define the generating rank, \(\text{gr}(\Gamma)\) to be \(\min\{| X|\mid X\subset P,\langle X\rangle_{\Gamma}=P\}\). By a projective embedding of \(\Gamma\) we mean an injective mapping \(e:P\to PG(V)\), \(V\) a vector space over some division ring, such that (i) the space spanned by \(e(P)\) is all of \(PG(V)\) and (ii) for \(l\in L\), \(e(l)\) is a full line of \(PG(V)\). We define the embedding rank, \(\text{er}(\Gamma)\) to be the maximal dimension of a vector space \(V\) for which there exist an embedding into \(PG(V)\). An embedding \(\widehat{e}\) is said to be a universal relative to \(e\) if there is a morphism \(\widehat{\alpha}:\widehat{e} \to e\) such that for any other morphism \(\gamma:\widehat{e} \to e\), \(\widehat{\alpha}\) factors through \(\gamma\). An embedding \(e:P\to PG(W)\) is relatively universal, if it is universal relative to itself. Let \(V\) be a vector space of dimension \(2n\) over \(F_q\) and let \(f:V\times V \to F_q\) be a non-degenerate alternative form. We denote the isotropic subspaces of dimension \(l\) by \(P(l,q)\). Dual polar space \(DSp_{2n}(q)=({\mathcal P},{\mathcal L})\) is the point-line geometry with \({\mathcal P}=P(n,q)\) and \({\mathcal L}=\{l(A)=\{M\in {\mathcal P} | \;A\subset M\} | \;A\in P(n-1,q)\}\). In this paper it is proved that for \(q>2\) the generating rank and embedding rank of \(DSp_{2n}(q)\) is \({2n\choose n}-{2n\choose n-2}\). Moreover, embedding into \(PG(M)\) with \(dim(M)={2n\choose n}-{2n\choose n-2}\) is relatively universal.
Mathematics Subject Classification ID: 51A50
zbMATH DE Number: 1212979
zbMATH Keywords: point-line geometry
zbMATH Keywords: relatively universal embedding
zbMATH Keywords: dual polar space
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
OpenAlex ID: W1965124955
cites work: Q4763522
cites work: Dual polar spaces
cites work: On the generation of some dual polar spaces of symplectic type over \(GF(2)\)
cites work: Frames and bases of Lie incidence geometries
cites work: Subgroups of Classical Groups Generated by Long Root Elements
cites work: Some groups generated by transvections
cites work: Q3344217
cites work: The weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights
cites work: Near \(n\)-gons and line systems
DOI: 10.1006/JCTA.1998.2875
Recommended article: On the generation of some dual polar spaces of symplectic type over \(GF(2)\)
Recommended article: On the generation of dual polar spaces of unitary type over finite fields
Recommended article: Generating symplectic and Hermitian dual polar spaces over arbitrary fields nonisomorphic to \(\mathbb F_{2}\)
Recommended article: On symplectic polar spaces over non-perfect fields of characteristic 2
Recommended article: A note on symplectic polar spaces over non-perfect fields of characteristic 2
Recommended article: Hyperplanes of symplectic dual polar spaces: a survey
Recommended article: The hyperplanes of finite symplectic dual polar spaces which arise from projective embeddings
Recommended article: On a class of hyperplanes of the symplectic and Hermitian dual polar spaces
Recommended article: On certain forms and quadrics related to symplectic dual polar spaces in characteristic 2
Recommended article: The structure of full polarized embeddings of symplectic and Hermitian dual polar spaces
description: scientific article; zbMATH DE number 1212979