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The Grassmannian Variety pp 155–162Cite as

Related Topics

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Part of the book series: Developments in Mathematics ((DEVM,volume 42))

Abstract

In this chapter, we give a brief account of some of the topics that are related to flag and Grassmannian varieties.

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References

  1. Anderson, D.: Okounkov bodies and toric degenerations. Math. Ann. 356(3), 1183–1202 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Balan, M.: Standard monomial theory for desingularized Richardson varieties in the flag variety GL(n)∕B. Transform. Groups 18(2), 329–359 (2013)

    Google Scholar 

  3. Berenstein, A., Zelevinsky, A.: Tensor product multiplicities, canonical bases and totally positive varieties. Invent. Math. 143(1), 77–128 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Billey, S., Lakshmibai, V.: Singular Loci of Schubert Varieties. Progress in Mathematics, vol. 182. Birkh\(\ddot{\text{a}}\) user, Boston (2000)

    Google Scholar 

  5. Bott, R., Samelson, H.: Applications of the theory of morse to symmetric spaces. Am. J. Math. 80, 964–1029 (1958)

    Article  MathSciNet  Google Scholar 

  6. Brion, M., Kumar, S.: Frobenius Splitting Methods in Geometry and Representation Theory. Progress in Mathematics, vol. 231. Birkh\(\ddot{\text{a}}\) user, Boston (2005)

    Google Scholar 

  7. Caldero, P.: Toric degenerations of Schubert varieties. Transform. Groups 7(1), 51–60 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Carrell, J.B., Kuttler, J.: Smooth points of T-stable varieties in G∕B and the Peterson map. Invent. Math. 151(2), 353–379 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chirivi, R.: LS algebras and applications to Schubert varieties. Transform. groups 5(3), 245–264 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Demazure, M.: Désingularisation des variétés de Schubert généralisées. In: Annales scientifiques de l’École Normale Supérieure, vol. 7, pp. 53–88. Société mathématique de France, France (1974)

    Google Scholar 

  11. Fulton, W.: Young Tableaux: With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  12. Gaussent, S., Littelmann, P.: LS galleries, the path model, and MV cycles. Duke Math. J. 127(1), 35–88 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Grossberg, M., Karshon, Y.: Bott towers, complete integrability, and the extended character of representations. Duke Math. J. 76(1), 23–58 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  15. Harada, M., Yang, J.J.: Newton-Okounkov bodies of Bott-Samelson varieties and Grossberg-Karshon twisted cubes. arXiv preprint. arXiv:1504.00982 (2015)

    Google Scholar 

  16. Hartshone, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1997)

    Google Scholar 

  17. Hodges, R., Laskhmibai, V.: Free resolutions of Schubert singularities in the Lagrangian Grassmannian. Preprint (2015)

    Google Scholar 

  18. Kaveh, K., Khovanskii, A.G.: Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math. 176(2), 925–978 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Khovanskii, A.G.: Newton polyhedron, Hilbert polynomial, and sums of finite sets. Funct. Anal. Appl. 26(4), 276–281 (1992)

    Article  MathSciNet  Google Scholar 

  20. Kreiman, V., Lakshmibai, V., Magyar, P., Weyman, J.: Standard bases for affine SL(n)-modules. Int. Math. Res. Not. 2005(21), 1251–1276 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kreiman, V., Lakshmibai, V., Magyar, P., Weyman, J.: On ideal generators for affine Schubert varieties. In: Algebraic Groups and Homogeneous Spaces. Tata Institute Fundamental Research Studies in Mathematics, pp. 353–388. Tata Institute Fundamental Research, Mumbai (2007)

    Google Scholar 

  22. Kumar, S.: The nil Hecke ring and singularity of schubert varieties. Invent. Math. 123(3), 471–506 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kummini, M., Lakshmibai, V., Sastry, P., Seshadri, C.: Free resolutions of some schubert singularities. arXiv preprint arXiv:1504.04415 [special volume of PJM dedicated to Robert Steinberg] (2015, to appear)

    Google Scholar 

  24. Lakshmibai, V.: Singular loci of Schubert varieties for classical groups. Bull. Am. Math. Soc. (N.S.) 16(1), 83–90 (1987)

    Google Scholar 

  25. Lakshmibai, V.: Tangent spaces to Schubert varieties. Math. Res. Lett. 2(4), 473–477 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lakshmibai, V., Littelmann, P., Magyar, P.: Standard monomial theory for Bott–Samelson varieties. Compos. Math. 130(03), 293–318 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Lakshmibai, V., Seshadri, C.: Singular locus of a schubert variety. Bull. A.M.S. 11, 363–366 (1984)

    Google Scholar 

  28. Lakshmibai, V., Seshadri, C.S.: Geometry of G∕P. V. J. Algebra 100(2), 462–557 (1986)

    Google Scholar 

  29. Lakshmibai, V., Weyman, J.: Multiplicities of points on a Schubert variety in a minuscule G∕P. Adv. Math. 84(2), 179–208 (1990)

    Google Scholar 

  30. Lascoux, A.: Syzygies des variétés déterminantales. Adv. Math. 30(3), 202–237 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lazarsfeld, R., Mustaţă, M.: Convex bodies associated to linear series. Ann. Sci. de lÉcole Norm. Supèrieure (4) 42(5), 783–835 (2009)

    Google Scholar 

  32. Lenart, C., Postnikov, A.: Affine Weyl groups in K-theory and representation theory. Int. Math. Res. Not. IMRN 12 65 (2007) (Art. ID rnm038)

    Google Scholar 

  33. Li, L., Yong, A.: Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties. Adv. Math. 229(1), 633–667 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  34. Littelmann, P.: A generalization of the Littlewood-Richardson rule. J. Algebra 130(2), 328–368 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  35. Littelmann, P.: A Littlewood Richardson rule for symmetrizable Kac-Moody algebras. Invent. Math. 116(1–3), 329–346 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  36. Littelmann, P.: Cones, crystals, and patterns. Transform. Groups 3(2), 145–179 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  37. Littelmann, P.: Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras. J. Am. Math. Soc. 11(3), 551–567 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  38. Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3(2), 447–498 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  39. Mehta, V.B., Ramanathan, A.: Frobenius splitting and cohomology vanishing for Schubert varieties. Ann. Math. (2) 122(1), 27–40 (1985)

    Google Scholar 

  40. Mirkovic, I., Vilonen, K.: Perverse sheaves on affine Grassmannians and Langlands duality (1999)

    Google Scholar 

  41. Pasquier, B.: Vanishing theorem for the cohomology of line bundles on Bott-Samelson varieties. J. Algebra 323(10), 2834–2847 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  42. Spanier, E.H.: Algebraic Topology. McGraw-Hill Book Co., New York (1966)

    MATH  Google Scholar 

  43. Weyman, J.: Cohomology of Vector Bundles and Syzygies. Cambridge Tracts in Mathematics, vol. 149. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  44. Woo, A., Yong, A.: A Gröbner basis for Kazhdan-Lusztig ideals. Am. J. Math. 134(4), 1089–1137 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Northeastern University, Boston, MA, USA

    V. Lakshmibai

  2. Department of Mathematics, Olivet Nazarene University, Bourbonnais, IL, USA

    Justin Brown

Authors
  1. V. Lakshmibai
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  2. Justin Brown
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Lakshmibai, V., Brown, J. (2015). Related Topics. In: The Grassmannian Variety. Developments in Mathematics, vol 42. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3082-1_11

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