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Central aspects of skew translation quadrangles, 1

Authors
  • Thas, Koen1
  • 1 Ghent University, Department of Mathematics, Krijgslaan 281, S25, Ghent, 9000, Belgium , Ghent (Belgium)
Type
Published Article
Journal
Journal of Algebraic Combinatorics
Publisher
Springer US
Publication Date
Dec 04, 2017
Volume
48
Issue
3
Pages
429–479
Identifiers
DOI: 10.1007/s10801-017-0801-3
Source
Springer Nature
Keywords
License
Yellow

Abstract

Modulo a combination of duality, translation duality or Payne integration, every known finite generalized quadrangle except for the Hermitian quadrangles H(4,q2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}(4,q^2)$$\end{document}, is an elation generalized quadrangle for which the elation point is a center of symmetry—that is, is a “skew translation generalized quadrangle” (STGQ). In this series of papers, we classify and characterize STGQs. In the first installment of the series, (1) we obtain the rather surprising result that any skew translation quadrangle of finite odd order (s, s) is a symplectic quadrangle; (2) we determine all finite skew translation quadrangles with distinct elation groups (a problem posed by Payne in a less general setting); (3) we develop a structure theory for root elations of skew translation quadrangles which will also be used in further parts, and which essentially tells us that a very general class of skew translation quadrangles admits the theoretical maximal number of root elations for each member, and hence, all members are “central” (the main property needed to control STGQs, as which will be shown throughout); and (4) we show that finite “generic STGQs,” a class of STGQs which generalizes the class of the previous item (but does not contain it by definition), have the expected parameters. We conjecture that the classes of (3) and (4) contain all STGQs.

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