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2 results

  • 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.

  • A unified formula for Steenrod operations in flag manifolds

  • Haibao Duan, Xuezhi Zhao
  • Journal:
    Compositio Mathematica / Volume 143 / Issue 1 / January 2007
    Published online by Cambridge University Press:
    19 January 2007, pp. 257-270
    Print publication:
    January 2007
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  • The classical Schubert cells on a flag manifold $G/P$ give a cell decomposition for $G/P$ whose Kronecker duals (known as Schubert classes) form an additive base for the integral cohomology $H^{\ast}(G/P)$. We present a formula that expresses Steenrod mod-$p$ operations on Schubert classes in $G/P$ in terms of Cartan numbers of $G$.