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Split buildings of type \(\mathsf {F_4}\) in buildings of type \(\mathsf {E_6}\)

  • Anneleen De Schepper
  • N. S. Narasimha Sastry
  • Hendrik Van Maldeghem
Article

Abstract

A symplectic polarity of a building \(\varDelta \) of type \(\mathsf {E_6}\) is a polarity whose fixed point structure is a building of type \(\mathsf {F_4}\) containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type \(\mathsf {F_4}\)). In this paper, we show in a geometric way that every building of type \(\mathsf {E_6}\) contains, up to conjugacy, a unique class of symplectic polarities. We also show that the natural point-line geometry of each split building of type \(\mathsf {F_4}\) fully embedded in the natural point-line geometry of \(\varDelta \) arises from a symplectic polarity.

Keywords

Buildings of exceptional type Metasymplectic spaces Point-line geometries Symplectic polarity 

Mathematics Subject Classification

51E24 

Notes

Acknowledgements

The work of the second author was carried when he was at Indian Statistical Institute, Bangalore centre, and during his visits to Department of Mathematics, Ghent University. He thanks both the institutions for extending their kind hospitality and excellent working conditions.

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Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Anneleen De Schepper
    • 1
  • N. S. Narasimha Sastry
    • 2
  • Hendrik Van Maldeghem
    • 1
  1. 1.Department of MathematicsGhent UniversityGhentBelgium
  2. 2.Indian Institute of Technology, DharwadDharwadIndia

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