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Multiplying Schubert Classes

  • Piotr Pragacz
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

We show how to compute the structure constants for cohomological multiplication of Schubert classes by exploiting the action of the Weyl group and that of BGG-operators, on the cohomology ring of a flag variety. We illustrate this method with simple proofs of the Chevalley and Pieri formulas.

Keywords

Structure Constant Weyl Group Simple Root Parabolic Subgroup Chern Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    I.N. Bernstein, I.M. Gelfand and S.I. Gelfand, Schubert cells and cohomology of the spaces G/P, Russian Math. Surveys, 28:3 (1973), 1–26.Google Scholar
  2. [2]
    A. Borel, Sur la cohomologie des spaces fibrés principaux et des spaces homogènes de groups de Lie compacts, Ann. of Math. 57 (1953), 115–207.Google Scholar
  3. [3]
    M. Brion, Lectures on geometry of flag varieties, this volume.Google Scholar
  4. [4]
    A.S. Buch, Combinatorial K-theory, this volume.Google Scholar
  5. [5]
    C. Chevalley, Sur les décompositions cellulaires des spaces G/B, Algebraic Groups and their generalizations, (W.S. Haboush and B.J. Parshall, eds.), Proc. Symp. Pure Math. 56, Part 1 (1994), AMS, 1–23.Google Scholar
  6. [6]
    M. Demazure, Invariants symétriques entiers des groups de Weyl et torsion, Invent. Math., 21 (1973), 287–301.CrossRefGoogle Scholar
  7. [7]
    M. Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. cole Norm. Sup. 7 (1974), 53–88.Google Scholar
  8. [8]
    H. Duan, Morse functions and cohomology of homogeneous spaces, this volume.Google Scholar
  9. [9]
    H. Duan, P. Pragacz, Divided differences of type D and the Grassmannian of complex structures, in: “Algebraic Combinatorics and Quantum Groups”, N. Jing ed., World Scientific (2003), 31–60.Google Scholar
  10. [10]
    W. Fulton, Young tableaux, Cambridge University Press, 1997.Google Scholar
  11. [11]
    W. Fulton, P. Pragacz, Schubert varieties and degeneracy loci, LNM 1689, Springer, 1998.Google Scholar
  12. [12]
    G.Z. Giambelli, Risoluzione del problema degli spazi secanti. Mem. R. Accad. Sci. Torino (2) 52 (1902), 171–211.Google Scholar
  13. [13]
    G.Z. Giambelli, Alcune proprietà delle funzioni simmetriche caratteristiche, Atti Torino 38 (1903), 823–844.Google Scholar
  14. [14]
    A. Grothendieck, La théorie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137–154.Google Scholar
  15. [15]
    H. Hiller, Geometry of Coxeter groups, Pitman, 1982.Google Scholar
  16. [16]
    W.V.D. Hodge, The intersection formulae for a Grassmannian variety, J. London Math. Soc. 17 (1942), 48–64.Google Scholar
  17. [17]
    J.E. Humphreys, Linear Algebraic Groups, Springer, 1975.Google Scholar
  18. [18]
    J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, 1990.Google Scholar
  19. [19]
    B. Kostant, S. Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G. Adv. in Math. 62 (1986), 187–237.CrossRefGoogle Scholar
  20. [20]
    I.G. Macdonald, Notes on Schubert polynomials. Publ. LACIM 6, UQUAM, Publ. Montréal, 1991.Google Scholar
  21. [21]
    D. Monk, The geometry of flag manifolds, Proc. London Math. Soc. 9 (1959), 253–286.Google Scholar
  22. [22]
    M. Pieri, Sul problema degli spazi secanti, Rend. Inst. Lombardo 2 (1893), 534–556 and 27 (1894), 258–273.Google Scholar
  23. [23]
    P. Pragacz, Symmetric polynomials and divided differences in formulas of intersection theory, in: Parameter Spaces, Banach Center Publications 36 (1996), 125–177.Google Scholar
  24. [24]
    P. Pragacz, J. Ratajski, Pieri type formula for isotropic Grassmannians; the operator approach. Manuscripta Math. 79 (1993), 127–151.Google Scholar
  25. [25]
    P. Pragacz, J. Ratajski, A Pieri-type theorem for Lagrangian and odd orthogonal Grassmannians, J. R.eine Angew. Math. 476 (1996), 143–189.Google Scholar
  26. [26]
    P. Pragacz, J. Ratajski, A Pieri-type formula for even orthogonal Grassmannians, Fund. Math. 178 (2003), 49–96.Google Scholar
  27. [27]
    H. Tamvakis, Gromov-Witten invariants and quantum cohomology of Grassmannians, this volume.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Piotr Pragacz
    • 1
  1. 1.Institute of MathematicsPolish Academy of SciencesWarszawaPoland

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