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The nil Hecke ring and singularity of Schubert varieties

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References

  1. Andersen, H.H.: Schubert varieties and Demazure’s character formula. Invent. Math. 79 (1985) 611–618

    Article  MATH  MathSciNet  Google Scholar 

  2. Bernstein, I.N., Gel’fand, I.M., Gel’fand, S.I.: Schubert cells and cohomology of the spaces G/P. Russian Math. Surveys 28 (1973) 1–26

    Article  MATH  Google Scholar 

  3. Boe, B.D.: Kazhdan-Lusztig polynomials for hermitian symmetric spaces. Trans. A.M.S. 309 (1988) 279–294

    Article  MATH  MathSciNet  Google Scholar 

  4. Bourbaki, N.: Groupes et Algèbres de Lie, Chaps. IV–VI. Hermann, Paris 1968

    Google Scholar 

  5. Carrell, J.B.: The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties. Proc. Symp. Pure Math. 56 (1994) (edited by W. J. Haboush and B.J. Parshall) 53–61

    MathSciNet  Google Scholar 

  6. Carrell, J.B.: On the singular locus of a Schubert variety: A survey. (1994) Preprint

  7. Carrell, J.B.: The span of the tangent cone of a Schubert variety. (1995) Preprint

  8. Cline, E., Parshall, B., Scott, L.: Cohomology, hyperalgebras and representations. J. Algebra 63 (1980) 98–123

    Article  MATH  MathSciNet  Google Scholar 

  9. Deodhar, V.V.: Local Poincaré duality and non-singularity of Schubert varieties. Comm. Algebra 13 (1985) 1379–1388

    Article  MATH  MathSciNet  Google Scholar 

  10. Deodhar, V.V.: A brief survey of Kazhdan-Lusztig theory and related topics, Proc. Symp. Pure Math. 56 (1994) (edited by W.J. Haboush and B.J. Parshall) 105–124

    MathSciNet  Google Scholar 

  11. Dyer, M.J.: The nil Hecke ring and Deodhar’s conjecture on Bruhat intervals. Invent. Math. 111 (1993) 571–574

    Article  MATH  MathSciNet  Google Scholar 

  12. Fulton, W.: Introduction to Intersection Theory in Algebraic Geometry. CBMS regional conference series in Mathematics no. 54 (1984) Am. Math. Soc.

  13. Harris, J.: Algebraic Geometry. Springer, Berlin Heidelberg New York 1992

    Google Scholar 

  14. Hartshorne, R.: Algebraic Geometry. Springer, Berlin Heidelberg New York 1977

    MATH  Google Scholar 

  15. Jantzen, J.C.: Moduln mit einem höchsten Gewicht. LNM vol. 750, Springer, Berlin Heidelberg New York 1979

    MATH  Google Scholar 

  16. Joseph, A.: On the variety of a highest weight module. J. Algebra 88 (1984) 238–278

    Article  MATH  MathSciNet  Google Scholar 

  17. Joseph, A.: On the Demazure character formula. Ann. Sci. Éc. Norm. Sup. 18 (1985) 389–419

    MATH  Google Scholar 

  18. Kac, V.G.: Infinite Dimensional Lie Algebras. Third edition, Cambridge University Press 1990

  19. Kac, V.G., Peterson, D.H.: Regular functions on certain infinite-dimensional groups. In: Arithmetic and Geometry-II, M. Artin, J. Tate (eds.), Birkhäuser (1983) 141–166

  20. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53 (1979) 165–184

    Article  MATH  MathSciNet  Google Scholar 

  21. Kostant, B., Kumar, S.: The nil Hecke ring and cohomology of G/P for a Kac-Moody group G. Adv. Math. 62 (1986) 187–237

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  23. Kumar, S.: Demazure character formula in arbitrary Kac-Moody setting. Invent. Math. 89 (1987) 395–423

    Article  MATH  MathSciNet  Google Scholar 

  24. Kumar, S.: The nil Hecke ring and singularity of Schubert varieties. In: Lie Theory and Geometry (in honor of Bertram Kostant), J.-L. Brylinski et. al. (eds.), Progress in Math. vol. 123, Birkhäuser (1994) 497–507

  25. Lakshmibai, V.: Singular loci of Schubert varieties for classical groups. Bull. A.M.S. 16 (1987) 83–90

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Mathieu, O.: Formules de caractères pour les algèbres de Kac-Moody générales. Astérisque no. 159–160 (1988) 1–267

  28. Mumford, D.: The Red Book of Varieties and Schemes. LNM vol. 1358, Springer, Berlin Heidelberg New York (1988)

    MATH  Google Scholar 

  29. Polo, P.: On Zariski tangent spaces of Schubert varieties, and a proof of a conjecture of Deodhar. Indag. Math., n.s., 5 (1994) 483–493

    Google Scholar 

  30. Ramanathan, A.: Schubert varieties are arithmetically Cohen-Macaulay. Invent. Math. 80 (1985) 283–294

    Article  MATH  MathSciNet  Google Scholar 

  31. Ramanathan, A.: Equations defining Schubert varieties and Frobenius splitting of diagonals. Publ Math. IHES no. 65 (1987) 61–90

  32. Rossmann, W.: Equivariant multiplicities on complex varieties. Astérisque no. 173– 174 (1989) 313–330

  33. Ryan, K.M.: On Schubert varieties in the flag manifold of SL(n,ℂ). Math. Ann. 276 (1987) 205–224

    Article  MATH  MathSciNet  Google Scholar 

  34. Seshadri, CS.: Line bundles on Schubert varieties. In: Vector Bundles on Algebraic Varieties, Tata Institute of Fundamental Research, Bombay Colloquium, Oxford University Press (1987) 499–528

  35. Slodowy, P.: On the geometry of Schubert varieties attached to Kac-Moody Lie-algebras. Can. Math. Soc. Conf. Proc. on ‘Algebraic Geometry’ (Vancouver) vol. 6 (1984) 405–442

    MathSciNet  Google Scholar 

  36. Wolper, J.S.: A combinatorial approach to the singularities of Schubert varieties. Adv. Maths. 76 (1989) 184–193

    Article  MATH  MathSciNet  Google Scholar 

  37. Wolper, J.S.: The Riccati flow and singularities of Schubert varieties. Proceedings of the AMS 123 (1995) 703–709

    Article  MATH  MathSciNet  Google Scholar 

  38. Whitney, H.: Complex Analytic Varieties. Addison-Wesley Publishing Company 1972

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  1. Department of Mathematics, University of North Carolina, 27599-3250, Chapel Hill, NC, USA

    Shrawan Kumar

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  1. Shrawan Kumar
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Oblatum 23-111-1995 ⇐p; 26-VIII-1995

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Kumar, S. The nil Hecke ring and singularity of Schubert varieties. Invent. math. 123, 471–506 (1996). https://doi.org/10.1007/s002220050038

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