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2 results

  • 2 - Polar spaces

  • Andries E. Brouwer, Technische Universiteit Eindhoven, The Netherlands, H. Van Maldeghem, Universiteit Gent, Belgium
  • Book:
    Strongly Regular Graphs
    Published online:
    06 January 2022
    Print publication:
    13 January 2022, pp 33-83

  • Summary

    The second chapter contains some theory of polar spaces. The latter provide the main examples of rank 3 graphs, and, more generally, many examples of strongly regular graphs. Polar spaces are introduced in general as embedded in a projective space, and a full proof of the classification of finite embedded polar spaces is provided.In this chapter, the emphasis is on the graphs whose vertices are objects of a polar space such as the point graph and graphs on (one type of) (maximal) singular subspaces. We review the three types of polar spaces, orthogonal, unitary and symplectic using reflexive forms and discuss Witt’s theorem. For the collinearity graphs of (finite) polar spaces, we discuss, sometimes in detail, sometimes merely surveying, maximal cliques, maximal co-cliques (ovoids), intriguing sets (tight sets and h-ovoids), spreads, partial spreads, hemisystems,…

  • THE THICKNESS OF SCHUBERT CELLS AS INCIDENCE STRUCTURES

  • Part of
  • JOHN BAMBERG, ARUN RAM, JON XU
  • Journal:
    Journal of the Australian Mathematical Society / Volume 109 / Issue 2 / October 2020
    Published online by Cambridge University Press:
    02 October 2019, pp. 145-156
    Print publication:
    October 2020

  • This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory.