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Central aspects of skew translation quadrangles, 1

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Abstract

Modulo a combination of duality, translation duality or Payne integration, every known finite generalized quadrangle except for the Hermitian quadrangles \(\mathcal {H}(4,q^2)\), 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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Notes

  1. This is a term coined by the author of the present paper. Given a generalized quadrangle \(\Gamma \) of finite order s and with regular point x, the Payne-derived quadrangle with respect to x is a quadrangle \(\mathcal {P}(\Gamma ,x)\) of order \((s - 1,s + 1)\). I define a “Payne integral” of \(\mathcal {P}(\Gamma ,x)\) as any quadrangle \(\Gamma '\) of order s and with regular point \(x'\) for which \(\mathcal {P}(\Gamma ',x') \cong \mathcal {P}(\Gamma ,x)\). Two such integrals do not have to be isomorphic. And \(\int \mathcal {P}(\Gamma ,x)\mathrm{d}x\) denotes the set of all such integrals.

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Acknowledgements

I presented some of the main results of this paper as a series of (then partly conjectural) results in a lecture, at the conference “Buildings and Groups” (Ghent, May 2007), which was organized by Peter Abramenko, Bernhard Mühlherr and Hendrik Van Maldeghem. Some of the ideas presented at that conference were developed when I was hosted, together with S. De Winter and E. E. Shult, by the Mathematisches Forschungsinstitut Oberwolfach (MFO) in the Research in Pairs program (April 2007). A short synopsis of the chronology can be found in the note [40]. I want to thank an anonymous referee for a number of helpful comments and suggestions on a draft of this paper. Finally, I wish to thank Stanley E. Payne for sending me numerous remarks and questions, which helped me to put the paper into its final shape.

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  1. Department of Mathematics, Ghent University, Krijgslaan 281, S25, 9000, Ghent, Belgium

    Koen Thas

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  1. Koen Thas
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Correspondence to Koen Thas.

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Dedicated to my dear friend Ernie Shult.

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Thas, K. Central aspects of skew translation quadrangles, 1. J Algebr Comb 48, 429–479 (2018). https://doi.org/10.1007/s10801-017-0801-3

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