On Combinatorics and Topology of Pairwise Intersections of Schubert Cells in SLn/B

  • Boris Shapiro
  • Michael Shapiro
  • Alek Vainshtein
Conference paper

Abstract

Topological properties of intersections of pairs and, more generally, of k-tuples of Schubert cells belonging to distinct Schubert cell decompositions of a flag space are of particular importance in representation theory and have been intensively studied during the last 15 years, see e.g. [BB, KL1, KL2, Del, GS]. Intersections of certain special arrangements of Schubert cells are related directly to the representability problem for matroids, see [GS]. Most likely, for a somewhat general class of arrangements of Schubert cells their intersections are too complicated to analyze. Even the nonemptyness problem for such intersections in complex flag varieties is very hard. However, in the case of pairs of Schubert cells in the space of complete flags one can obtain a special decomposition of such intersections, and of the whole space of complete flags, into products of algebraic tori and linear subspaces. This decomposition generalizes the standard Schubert cell decomposition. The above strata can be also obtained as intersections of more than two Schubert cells originating from the initial pair. The decomposition considered is used to calculate (algorithmically) natural additive topological characteristics of the intersections in question, namely, their Euler E p,q -characteristics (see [DK]). Generally speaking, this decomposition of the space of complete flags does not stratify all pairwise intersections of Schubert cells, i.e. the closure of a stratum is not necessary a union of strata of lower dimensions. Still there exists a natural analog of adjacency, and its combinatorial description is available, see Theorem D. We discuss combinatorics of this special decomposition and some rather simple consequences for the cohomology and the mixed Hodge structure of intersections of Schubert cells in SL n /B.

Keywords

Filtration Manifold Stratification Posite Tral 

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Copyright information

© Birkhäuser Boston 1997

Authors and Affiliations

  • Boris Shapiro
    • 1
  • Michael Shapiro
    • 2
  • Alek Vainshtein
    • 3
  1. 1.Department of MathematicsUniversity of StockholmSweden
  2. 2.Department of Theoretical MathematicsThe Weizmann Institute of ScienceRehovotIsrael
  3. 3.School of Mathematical SciencesTel Aviv UniversityRamat AvivIsrael

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