Skip to main content

Part of the book series: Trends in Mathematics ((TM))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Billey and V. Lakshmibai: Singular loci of Schubert varieties, Progr. Math. 182, Birkhäuser, Boston (2000).

    Google Scholar 

  2. S. Billey and G. Warrington: Maximal singular loci of Schubert varieties in SL(n)/B, Trans. Amer. Math. Soc. 335 (2003), 3915–3945.

    Article  Google Scholar 

  3. A. Borel and J.-P. Serre: Le théorbmè de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97–136.

    Google Scholar 

  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. M. Brion and P. Polo: Large Schubert varieties, Represent. Theory 4 (2000), 97–126.

    Article  Google Scholar 

  6. M. Brion: Positivity in the Grothendieck group of complex flag varieties, J. Algebra 258 (2002), 137–159.

    Article  Google Scholar 

  7. M. Brion: Group completions via Hilbert schemes, J. Algebraic Geom. 12 (2003), 605–626.

    Google Scholar 

  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. M. Brion and V. Lakshmibai: A geometric approach to standard monomial theory, Represent. Theory 7 (2003), 651–680.

    Article  Google Scholar 

  10. A.S. Buch: A Littlewood-Richardson rule for the K-theory of Grassmannians, Acta Math. 189 (2002), 37–78.

    Google Scholar 

  11. A.S. Buch: Combinatorial K-theory, this volume.

    Google Scholar 

  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. A. Cortez: Singularités génériques et quasi-résolutions des variétés de Schubert pour le group linéaire, Adv. Math. 178 (2003), 396–445.

    Article  Google Scholar 

  14. M. Demazure: Désingularisations des variétés de Schubert généralisées, Ann. Sci. Éc. Norm. Supér. 7 (1974), 53–88.

    Google Scholar 

  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.

    Article  Google Scholar 

  16. V.V. Deodhar: On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. math. 79 (1985), 499–511.

    Article  Google Scholar 

  17. H. Duan: Multiplicative rule of Schubert classes, preprint available on arXiv: math.AG/0306227.

    Google Scholar 

  18. H. Duan: Morse functions and cohomology of homogeneous spaces, this volume.

    Google Scholar 

  19. H. Esnault and E. Viehweg: Lectures on vanishing theorems, DMV Seminar Band 13, Birkhäuser (1992).

    Google Scholar 

  20. W. Fulton and A. Lascoux: A Pieri formula in the Grothendieck ring of a flag bundle, Duke Math. J. 76 (1994), 711–729.

    Article  Google Scholar 

  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. W. Fulton: Intersection theory (second edition), Springer-Verlag (1998).

    Google Scholar 

  23. W. Fulton and P. Pragacz: Schubert varieties and degeneracy loci, Lecture Notes in Mathematics 1689, Springer-Verlag (1998).

    Google Scholar 

  24. R. Goldin: The cohomology ring of weight varieties and polygon spaces, Adv. Math. 160 (2001), 175–204.

    Article  Google Scholar 

  25. W. Graham: Positivity in equivariant Schubert calculus, Duke Math. J. 109 (2001), 599–614.

    Article  Google Scholar 

  26. M. Greenberg and J. Harper: Algebraic topology. A first course, Benjamin/Cummings Publishing Co. (1981).

    Google Scholar 

  27. S. Griffeth and A. Ram: Affine Hecke algebras and the Schubert calculus, preprint available on arXiv: math.RT/0405333.

    Google Scholar 

  28. A. Grothendieck: Sur quelques points d’algèbre homologique, Tohoku Math. J. 9 (1957), 119–221.

    Google Scholar 

  29. H.L. Hansen: On cycles in flag manifolds, Math. Scand. 33 (1973), 269–274.

    Google Scholar 

  30. R. Hartshorne: Algebraic Geometry, Graduate Texts in Math. 52, Springer-Verlag (1977).

    Google Scholar 

  31. H. Hiller and B. Boe: Pieri formula for SO2n+ 1/Un and Spn/Un, Adv. in Math. 62 (1986), 49–67.

    Article  Google Scholar 

  32. W.V.D. Hodge: The intersection formulae for a Grassmannian variety, J. London Math. Soc. 17 (1942), 48–64.

    Google Scholar 

  33. C. Kassel, A. Lascoux and C. Reutenauer: The singular locus of a Schubert variety, J. Algebra 269 (2003), 74–108.

    Article  Google Scholar 

  34. G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat: Toroidal embeddings, Lecture Notes in Math. 339, Springer-Verlag (1973).

    Google Scholar 

  35. S. Kleiman: The transversality of a general translate, Compos. Math. 28 (1974), 287–297.

    Google Scholar 

  36. A. Knutson and E. Miller: Gröbner geometry of Schubert polynomials, preprint available on arXiv: math.AG/011058.

    Google Scholar 

  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.

    Article  Google Scholar 

  38. B. Kostant and S. Kumar: T-equivariant K-theory of generalized flag varieties, J. Differential Geom. 32 (1990), 549–603.

    Google Scholar 

  39. S. Kumar: Kac-Moody groups, their flag varieties and representation theory, Progr. Math. 204, Birkhäuser (2002).

    Google Scholar 

  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. V. Lakshmibai and P. Littelmann: Richardson varieties and equivariant K-theory, J. Algebra 260 (2003), 230–260.

    Article  Google Scholar 

  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. 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. 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. N. Lauritzen and J.F. Thomsen: Line bundles on Bott-Samelson varieties, J. Algebraic Geom. 13 (2004), 461–473.

    Google Scholar 

  46. P. Littelmann: Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras, J. Amer. Math. Soc. 11 (1998), 551–567.

    Article  Google Scholar 

  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. 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. L. Manivel: Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, Cours Spécialisés 3, Soc. Math. France (1998).

    Google Scholar 

  50. L. Manivel: Le lieu singulier des variétés de Schubert, Internat. Math. Res. Notices 16 (2001), 849–871.

    Article  Google Scholar 

  51. L. Manivel: Generic singularities of Schubert varieties, preprint available on arXiv: math.AG/0105239.

    Google Scholar 

  52. R. Marlin: Anneaux de Grothendieck des variétés de drapeaux, Bull. Soc. Math. France 104 (1976), 337–348.

    Google Scholar 

  53. O. Mathieu: Formules de caractères pour les algèbres de Kac-Moody générales, Astérisque 159–160 (1988).

    Google Scholar 

  54. O. Mathieu: Positivity of some intersections in K0(G/B), J. Pure Appl. Algebra 152 (2000), 231–243.

    Article  Google Scholar 

  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. D. Monk: The geometry of flag manifolds, Proc. London Math. Soc. 9 (1959), 253–286.

    Google Scholar 

  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.

    Article  Google Scholar 

  58. P. Pragacz and J. Ratajski: Pieri type formula for isotropic Grassmannians; the operator approach, Manuscripta Math. 79 (1993), 127–151.

    Google Scholar 

  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. P. Pragacz and J. Ratajski: A Pieri-type formula for even orthogonal Grassmannians, Fund. Math. 178 (2003), 49–96.

    Google Scholar 

  61. P. Pragacz: Multiplying Schubert classes, this volume.

    Google Scholar 

  62. S. Ramanan and A. Ramanathan: Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), 217–224.

    Article  Google Scholar 

  63. A. Ramanathan: Equations defining Schubert varieties and Frobenius splitting of diagonals, Pub. Math. IHES 65 (1987), 61–90.

    Google Scholar 

  64. R.W. Richardson: Intersections of double costs in algebraic groups, Indag. Math. (N.S.) 3 (1992), 69–77.

    Article  Google Scholar 

  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. T.A. Springer: Linear algebraic groups (second edition), Progr. Math. 9, Birkhäuser (1998).

    Google Scholar 

  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. H. Tamvakis: Gromov-Witten invariants and quantum cohomology of Grassmannians, this volume.

    Google Scholar 

  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.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Brion, M. (2005). Lectures on the Geometry of Flag Varieties. In: Pragacz, P. (eds) Topics in Cohomological Studies of Algebraic Varieties. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7342-3_2

Download citation

Publish with us

Policies and ethics