Polarities of symplectic quadrangles
type: Item
instance of: scholarly article
title: Polarities of symplectic quadrangles
zbMATH Open document ID: 1040.51005
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2004-01-22T00:00:00Z
review text: \textit{J. Tits} [Arch. Math. 13, 187-198 (1962; Zbl 0109.39402)] showed that the symplectic quadrangle \(W(F)\) is self-dual if and only if the field \(F\) is perfect of characteristic 2 and that the existence of a polarity is equivalent to the existence of a square root of the Frobenius automorphism \(x\mapsto x^2\) of \(F\). These results were obtained in the context of ovoids related to the Suzuki simple groups. In the paper under review the author gives an elementary proof of these facts using simple computations in the elation and translation groups in the automorphism group of \(W(F)\) thus avoiding the use of simple groups. (The groups considered are nilpotent subgroups of the symplectic group and actually even abelian in the interesting characteristic 2 case.) Moreover, the author determines all polarities, characterizes the conjugacy classes of polarities and gives a system of representatives. Using these results he also shows that the centralizer of a polarity acts 2-transitively on the corresponding ovoid.
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51A10
Mathematics Subject Classification ID: 51A50
zbMATH DE Number: 2031035
zbMATH Keywords: generalized quadrangle
zbMATH Keywords: symplectic quadrangle
zbMATH Keywords: polarity
zbMATH Keywords: duality
zbMATH Keywords: ovoid
zbMATH Keywords: elation generalized quadrangle
zbMATH Keywords: translation generalized quadrangle
zbMATH Keywords: symplectic group
author: Markus Stroppel
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2031035