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Lectures on the Geometry of Flag Varieties

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

Keywords

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

  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

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