Polarized and homogeneous embeddings of dual polar spaces
type: Item
instance of: scholarly article
title: Polarized and homogeneous embeddings of dual polar spaces
zbMATH Open document ID: 1204.51003
published in: Journal of Algebraic Combinatorics
publication date: 2009-12-09T00:00:00Z
review text: An embedding \(e: \Gamma \rightarrow PG(V)\) of a dual polar space \(\Gamma\) is \textit{polarized} if, for every point \(x\in \Gamma\), the image \(e(H)_x\) of the hyperplane \(H_x\) of \(\Gamma\), formed by the points at non-maximal distance from \(x\), spans a hyperplane of \(PG(V)\). The main result of the paper under review is the following theorem. Theorem. Let \(\Gamma\) be the dual of a classical polar space and let \(e\) be a projective embedding of \(\Gamma\), defined over a commutative division ring. If \(e\) is \(\Aut(\Gamma)_0\)-homogeneous then it is polarized. Sections 2 and 3 of the paper contain definitions and basics on point-line geometries, dual polar spaces, embeddings and some preliminary results on embeddings of near-polar spaces, on generalized quadrangles and projective spaces. Finally, in Section 4 the main result is proved, and it is also proved a consequence of such a theorem on the absolutely universal embedding of \(\Gamma\).
reviewed by: V. Napolitano
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51A50
zbMATH DE Number: 5644671
zbMATH Keywords: dual polar spaces
zbMATH Keywords: polarized embeddings
zbMATH Keywords: homogeneous embeddings
author: Rieuwert J. Blok
author: Ilaria Cardinali
author: Bart De Bruyn
author: Antonio Pasini
MaRDI profile type: Publication
full work available at URL: s10801-008-0166-8
OpenAlex ID: W1983246169
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DOI: 10.1007/S10801-008-0166-8
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description: scientific article; zbMATH DE number 5644671