Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-27T14:30:50.533Z Has data issue: false hasContentIssue false
  • English
  • Français

Regular Embeddings of Generalized Hexagons

Published online by Cambridge University Press:  20 November 2018

Anja Steinbach  and
Hendrik Van Maldeghem
Anja Steinbach
Affiliation:
Justus-Liebig-Universität Gießen, Mathematisches Institut, Arndtstraße 2, D 35392 Gießen, Germany e-mail: Anja.Steinbach@math.uni-giessen.de
Hendrik Van Maldeghem
Affiliation:
Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B-9000 Gent, Belgium e-mail: hvm@cage.rug.ac.be
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We classify the generalized hexagons which are laxly embedded in projective space such that the embedding is flat and polarized. Besides the standard examples related to the hexagons defined over the algebraic groups of type ${{\text{G}}_{2}}$, ${}^{3}{{\text{D}}_{4}}$ and ${}^{6}{{\text{D}}_{\text{4}}}$ (and occurring in projective dimensions 5, 6, 7), we find new examples in unbounded dimension related to the mixed groups of type ${{\text{G}}_{2}}$ .

Keywords

51E1251A45Moufang generalized hexagonsembeddingsmixed hexagonsclassical hexagons
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 5 , 01 October 2004 , pp. 1068 - 1093
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Bosch, S., Algebra. Springer-Verlag, Berlin, 2001.Google Scholar
[2] Cohen, A. M., Point-line spaces related to buildings. In: Handbook of Incidence Geometry, Buildings and Foundations, (ed. Buekenhout, F.), North-Holland, Amsterdam, 1995, pp. 647–737.Google Scholar
[3] Cuypers, H. and Steinbach, A., Weak embeddings of generalized hexagons and groups of type G 2 .. J. Group Theory 1(1998), 225–236.Google Scholar
[4] Hua, L. K., On the automorphisms of a field. Proc. Nat. Acad. Sci. U.S.A. 35(1949), 386–389.Google Scholar
[5] Lang, S., Algebra. 3rd revised ed. Graduate Texts in Mathematics, 211, Springer-Verlag, New York, 2002.Google Scholar
[6] Ronan, M., A geometric characterization of Moufang hexagons. Invent.Math. 57(1980), 227–262.Google Scholar
[7] Timmesfeld, F. G., Abstract root subgroups and simple groups of Lie-type. Monographs in Mathematics, 95, Birkhäuser Verlag, Basel, 2001.Google Scholar
[8] Thas, J., and Van Maldeghem, H., Embedded thick finite generalized hexagons in projective space. J. London Math. Soc. 54(1996), 566–580.Google Scholar
[9] Thas, J., and Van Maldeghem, H., Flat lax and weak lax embeddings of finite generalized hexagons. European J. Combin. 19(1998), 733–751.Google Scholar
[10] Thas, J., and Van Maldeghem, H., Full embeddings of the finite dual split Cayley hexagons, to appear in Combinatorica.Google Scholar
[11] Tits, J., Sur la trialité et certains groupes qui s’en déduisent. Publ. Math., Inst. Hautes Étud. Sci. 2(1959), 13–60.Google Scholar
[12] Tits, J., Buildings of spherical type and finite BN-pairs. Lecture Notes in Mathematics, 386, Springer-Verlag, Berlin, 1974.Google Scholar
[13] Tits, J., Moufang polygons. I: Root data. Bull. Belg. Math. Soc. - Simon Stevin 1(1994), 455–468.Google Scholar
[14] Tits, J. and Weiss, R., Moufang polygons. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.Google Scholar
[15] Van Maldeghem, H., Generalized polygons. Monographs in Mathematics, 93, Birkhäuser Verlag, Basel, 1998.Google Scholar