Lectures on the Geometry of Flag Varieties

  • Michel Brion
Part of the Trends in Mathematics book series (TM)


Line Bundle Rational Singularity Schubert Variety Ample Line Bundle Dominant Weight 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Billey and V. Lakshmibai: Singular loci of Schubert varieties, Progr. Math. 182, Birkhäuser, Boston (2000).Google Scholar
  2. [2]
    S. Billey and G. Warrington: Maximal singular loci of Schubert varieties in SL(n)/B, Trans. Amer. Math. Soc. 335 (2003), 3915–3945.CrossRefGoogle Scholar
  3. [3]
    A. Borel and J.-P. Serre: Le théorbmè de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97–136.Google Scholar
  4. [4]
    R. Bott and H. Samelson: Application of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964–1029.Google Scholar
  5. [5]
    M. Brion and P. Polo: Large Schubert varieties, Represent. Theory 4 (2000), 97–126.CrossRefGoogle Scholar
  6. [6]
    M. Brion: Positivity in the Grothendieck group of complex flag varieties, J. Algebra 258 (2002), 137–159.CrossRefGoogle Scholar
  7. [7]
    M. Brion: Group completions via Hilbert schemes, J. Algebraic Geom. 12 (2003), 605–626.Google Scholar
  8. [8]
    M. Brion: Multiplicity-free subvarieties of flag varieties, in: Commutative algebra, interactions with algebraic geometry, Contemporary Math. 331, AMS, Providence (2003), 13–24.Google Scholar
  9. [9]
    M. Brion and V. Lakshmibai: A geometric approach to standard monomial theory, Represent. Theory 7 (2003), 651–680.CrossRefGoogle Scholar
  10. [10]
    A.S. Buch: A Littlewood-Richardson rule for the K-theory of Grassmannians, Acta Math. 189 (2002), 37–78.Google Scholar
  11. [11]
    A.S. Buch: Combinatorial K-theory, this volume.Google Scholar
  12. [12]
    C. Chevalley: Sur les décompositions cellulaires des spaces G/B, in: Proceedings of Symposia in Pure Mathematics 56, Amer. Math. Soc., Providence (1994), 1–25.Google Scholar
  13. [13]
    A. Cortez: Singularités génériques et quasi-résolutions des variétés de Schubert pour le group linéaire, Adv. Math. 178 (2003), 396–445.CrossRefGoogle Scholar
  14. [14]
    M. Demazure: Désingularisations des variétés de Schubert généralisées, Ann. Sci. Éc. Norm. Supér. 7 (1974), 53–88.Google Scholar
  15. [15]
    V.V. Deodhar: Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function, Invent. math. 39 (1977), 187–198.CrossRefGoogle Scholar
  16. [16]
    V.V. Deodhar: On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. math. 79 (1985), 499–511.CrossRefGoogle Scholar
  17. [17]
    H. Duan: Multiplicative rule of Schubert classes, preprint available on arXiv: math.AG/0306227.Google Scholar
  18. [18]
    H. Duan: Morse functions and cohomology of homogeneous spaces, this volume.Google Scholar
  19. [19]
    H. Esnault and E. Viehweg: Lectures on vanishing theorems, DMV Seminar Band 13, Birkhäuser (1992).Google Scholar
  20. [20]
    W. Fulton and A. Lascoux: A Pieri formula in the Grothendieck ring of a flag bundle, Duke Math. J. 76 (1994), 711–729.CrossRefGoogle Scholar
  21. [21]
    W. Fulton: Young tableaux with applications to representation theory and geometry, London Mathematical Society Student Texts 35, Cambridge University Press (1997).Google Scholar
  22. [22]
    W. Fulton: Intersection theory (second edition), Springer-Verlag (1998).Google Scholar
  23. [23]
    W. Fulton and P. Pragacz: Schubert varieties and degeneracy loci, Lecture Notes in Mathematics 1689, Springer-Verlag (1998).Google Scholar
  24. [24]
    R. Goldin: The cohomology ring of weight varieties and polygon spaces, Adv. Math. 160 (2001), 175–204.CrossRefGoogle Scholar
  25. [25]
    W. Graham: Positivity in equivariant Schubert calculus, Duke Math. J. 109 (2001), 599–614.CrossRefGoogle Scholar
  26. [26]
    M. Greenberg and J. Harper: Algebraic topology. A first course, Benjamin/Cummings Publishing Co. (1981).Google Scholar
  27. [27]
    S. Griffeth and A. Ram: Affine Hecke algebras and the Schubert calculus, preprint available on arXiv: math.RT/0405333.Google Scholar
  28. [28]
    A. Grothendieck: Sur quelques points d’algèbre homologique, Tohoku Math. J. 9 (1957), 119–221.Google Scholar
  29. [29]
    H.L. Hansen: On cycles in flag manifolds, Math. Scand. 33 (1973), 269–274.Google Scholar
  30. [30]
    R. Hartshorne: Algebraic Geometry, Graduate Texts in Math. 52, Springer-Verlag (1977).Google Scholar
  31. [31]
    H. Hiller and B. Boe: Pieri formula for SO2n+ 1/Un and Spn/Un, Adv. in Math. 62 (1986), 49–67.CrossRefGoogle Scholar
  32. [32]
    W.V.D. Hodge: The intersection formulae for a Grassmannian variety, J. London Math. Soc. 17 (1942), 48–64.Google Scholar
  33. [33]
    C. Kassel, A. Lascoux and C. Reutenauer: The singular locus of a Schubert variety, J. Algebra 269 (2003), 74–108.CrossRefGoogle Scholar
  34. [34]
    G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat: Toroidal embeddings, Lecture Notes in Math. 339, Springer-Verlag (1973).Google Scholar
  35. [35]
    S. Kleiman: The transversality of a general translate, Compos. Math. 28 (1974), 287–297.Google Scholar
  36. [36]
    A. Knutson and E. Miller: Gröbner geometry of Schubert polynomials, preprint available on arXiv: math.AG/011058.Google Scholar
  37. [37]
    B. Kostant and S. Kumar: The nil Hecke ring and cohomology of G/P for a Kac-Moody group G, Adv. Math. 62 (1986), 187–237.CrossRefGoogle Scholar
  38. [38]
    B. Kostant and S. Kumar: T-equivariant K-theory of generalized flag varieties, J. Differential Geom. 32 (1990), 549–603.Google Scholar
  39. [39]
    S. Kumar: Kac-Moody groups, their flag varieties and representation theory, Progr. Math. 204, Birkhäuser (2002).Google Scholar
  40. [40]
    V. Kreiman and V. Lakshmibai: Richardson varieties in the Grassmannian, in: Contributions to Automorphic Forms, Geometry and Number Theory: Shalikafest 2002, Johns Hopkins University Press (2003), 573–597.Google Scholar
  41. [41]
    V. Lakshmibai and P. Littelmann: Richardson varieties and equivariant K-theory, J. Algebra 260 (2003), 230–260.CrossRefGoogle Scholar
  42. [42]
    A. Lascoux and M.-P. Schützenberger: Polynômes de Schubert, C. R. Acad. Sci. Paris Séer. 1 Math. 294 (1982), 447–450.Google Scholar
  43. [43]
    A. Lascoux and M.-P. Schützenberger: Symmetry and flag manifolds, in: Invariant Theory (Proceedings. Montecatini 1982), Lecture Note in Math. 996, Springer-Verlag (1983), 118–144.Google Scholar
  44. [44]
    A. Lascoux and M.-P. Schützenberger: Interpolation de Newton à plusieurs variables, in: Séminaire d’algèbre Paul Dubreil et Marie-Paule Malliavin, Lecture Notes in Math. 1146, Springer-Verlag (1985), 161–175.Google Scholar
  45. [45]
    N. Lauritzen and J.F. Thomsen: Line bundles on Bott-Samelson varieties, J. Algebraic Geom. 13 (2004), 461–473.Google Scholar
  46. [46]
    P. Littelmann: Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras, J. Amer. Math. Soc. 11 (1998), 551–567.CrossRefGoogle Scholar
  47. [47]
    P. Littelmann: The path model, the quantum Frobenius map and standard monomial theory, in: Algebraic groups and their representations (Cambridge. 1997), Kluwer Acad. Publ. (1998), 175–212.Google Scholar
  48. [48]
    P. Littelmann and C.S. Seshadri: A Pieri-Chevalley type formula for K(G/B) and standard monomial theory, in: Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), Progr. Math. 210, Birkhäuser (2003), 155–176.Google Scholar
  49. [49]
    L. Manivel: Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, Cours Spécialisés 3, Soc. Math. France (1998).Google Scholar
  50. [50]
    L. Manivel: Le lieu singulier des variétés de Schubert, Internat. Math. Res. Notices 16 (2001), 849–871.CrossRefGoogle Scholar
  51. [51]
    L. Manivel: Generic singularities of Schubert varieties, preprint available on arXiv: math.AG/0105239.Google Scholar
  52. [52]
    R. Marlin: Anneaux de Grothendieck des variétés de drapeaux, Bull. Soc. Math. France 104 (1976), 337–348.Google Scholar
  53. [53]
    O. Mathieu: Formules de caractères pour les algèbres de Kac-Moody générales, Astérisque 159–160 (1988).Google Scholar
  54. [54]
    O. Mathieu: Positivity of some intersections in K0(G/B), J. Pure Appl. Algebra 152 (2000), 231–243.CrossRefGoogle Scholar
  55. [55]
    V. Mehta and A. Ramanathan: Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2) 122 (1985), 27–40.Google Scholar
  56. [56]
    D. Monk: The geometry of flag manifolds, Proc. London Math. Soc. 9 (1959), 253–286.Google Scholar
  57. [57]
    H. Pittie and A. Ram: A Pieri-Chevalley formula in the K-theory of a G/B-bundle, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 102–107.CrossRefGoogle Scholar
  58. [58]
    P. Pragacz and J. Ratajski: Pieri type formula for isotropic Grassmannians; the operator approach, Manuscripta Math. 79 (1993), 127–151.Google Scholar
  59. [59]
    P. Pragacz and J. Ratajski: A Pieri-type theorem for Lagrangian and odd orthogonal Grassmannians, J. Reine Angew. Math. 476 (1996), 143–189.Google Scholar
  60. [60]
    P. Pragacz and J. Ratajski: A Pieri-type formula for even orthogonal Grassmannians, Fund. Math. 178 (2003), 49–96.Google Scholar
  61. [61]
    P. Pragacz: Multiplying Schubert classes, this volume.Google Scholar
  62. [62]
    S. Ramanan and A. Ramanathan: Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), 217–224.CrossRefGoogle Scholar
  63. [63]
    A. Ramanathan: Equations defining Schubert varieties and Frobenius splitting of diagonals, Pub. Math. IHES 65 (1987), 61–90.Google Scholar
  64. [64]
    R.W. Richardson: Intersections of double costs in algebraic groups, Indag. Math. (N.S.) 3 (1992), 69–77.CrossRefGoogle Scholar
  65. [65]
    C.S. Seshadri: Line bundles over Schubert varieties, in: Vector Bundles on Algebraic Varieties, Bombay Colloquium 1984, Oxford University Press (1987), 499–528.Google Scholar
  66. [66]
    T.A. Springer: Linear algebraic groups (second edition), Progr. Math. 9, Birkhäuser (1998).Google Scholar
  67. [67]
    T.A. Springer: Schubert varieties and generalizations, in: Representation theories and algebraic geometry (Montreal. PQ. 1997), Kluwer Acad. Publ. (1998), 413–440.Google Scholar
  68. [68]
    H. Tamvakis: Gromov-Witten invariants and quantum cohomology of Grassmannians, this volume.Google Scholar
  69. [69]
    M. Willems: Cohomologie et K-théorie équivariantes des tours de Bott et des variétés de drapeaux, Bull. Soc. Math. France 132 (2004), 569–589.MathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Michel Brion
    • 1
  1. 1.Institute FourierSaint-Martin d’Hères CedexFrance

Personalised recommendations