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Eigenvalue problem and a new product in cohomology of flag varieties

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References

  1. Agnihotri, S., Woodward, C.: Eigenvalues of products of unitary matrices and quantum Schubert calculus. Math. Res. Lett. 5, 817–836 (1998)

    MATH  MathSciNet  Google Scholar 

  2. Belkale, P.: Local systems on ℙ1-S for S a finite set. Compos. Math. 129, 67–86 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Belkale, P.: Geometric proofs of Horn and saturation conjectures. J. Alg. Geom. 15, 133–173 (2006)

    MATH  MathSciNet  Google Scholar 

  4. Belkale, P.: Invariant theory of GL(n) and intersection theory of Grassmannians. Int. Math. Res. Not. 69, 3709–3721 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Belkale, P.: The quantum horn conjecture. Preprint, math.AG/0303013

  6. Belkale, P.: Extremal unitary representations of π1(ℙ1-{p 1,...,p s }). To appear in the proceedings of the International Colloquium on Algebraic Groups and Homogeneous Spaces (2004), Tata Institute of Fundamental Research, Mumbai

  7. Berenstein, A., Sjamaar, R.: Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion. J. Am. Math. Soc. 13, 433–466 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Schubert cells and cohomology of the spaces G/P. Russ. Math. Surv. 28, 1–26 (1973)

    Article  MATH  Google Scholar 

  9. Bourbaki, N.: Groupes et Algèbres de Lie, Chap. 4–6. Paris: Masson 1981

  10. Fulton, W.: Intersection Theory, 2nd edn. Springer 1998

  11. Fulton, W.: Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Am. Math. Soc., New. Ser. 37, 209–249 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press 1978

  13. Hesselink, W.: Uniform instability in reductive group. J. Reine Angew. Math. 303–304, 74–96 (1978)

    Google Scholar 

  14. Jantzen, J.C.: Representations of Algebraic Groups, 2nd edn. Am. Math. Soc. 2003

  15. Kapovich, M., Leeb, B., Millson, J.J.: Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity. Preprint 2005

  16. Kempf, G.: Instability in invariant theory. Ann. Math. 108, 299–316 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kirwan, F.: Cohomology of quotients in symplectic and algebraic geometry. Princeton University Press 1984

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

    Google Scholar 

  19. Klyachko, A.: Stable bundles, representation theory and Hermitian operators. Sel. Math. 4, 419–445 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Knutson, A., Tao, T.: The honeycomb model of GL n (ℂ) tensor products I: Proof of the saturation conjecture. J. Am. Math. Soc. 12, 1055–1090 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. Knutson, A., Tao, T., Woodward, C.: The honeycomb model of GL n (ℂ) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone. J. Am. Math. Soc. 17, 19–48 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. Math. 74, 329–387 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kostant, B.: Lie algebra cohomology and generalized Schubert cells. Ann. Math. 77, 72–144 (1963)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Kumar, S.: Geometry of Schubert cells and cohomology of Kac-Moody Lie algebras. J. Differ. Geom. 20, 389–431 (1984)

    MATH  Google Scholar 

  26. Kumar, S.: Kac-Moody Groups, their Flag Varieties and Representation Theory. Progress in Mathematics, vol. 204. Birkhäuser 2002

  27. Kumar, S., Leeb, B., Millson, J.: The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type. Contemp. Math. 332, 171–195 (2003)

    MATH  Google Scholar 

  28. Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3rd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34. Springer 1994

  29. Purbhoo, K.: Vanishing and non-vanishing criteria in Schubert calculus. Preprint, math. CO/0304070. To appear in Int. Math. Res. Not.

  30. Ramanan, S., Ramanathan, A.: Some remarks on the instability flag. Tôhoku Math. J. 36, 269–291 (1984)

    MATH  MathSciNet  Google Scholar 

  31. Shafarevich, I.R.: Basic Algebraic Geometry I, 2nd edn. Springer 1994

  32. Serre, J.-P.: Complex Semisimple Lie Algebras. Springer 1987

  33. Sjamaar, R.: Convexity properties of the moment mapping re-examined. Adv. Math. 138, 46–91 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  34. Teleman, C., Woodward, C.: Parabolic bundles, products of conjugacy classes and Gromov-Witten invariants. Ann. Inst. Fourier 53, 713–748 (2003)

    MATH  MathSciNet  Google Scholar 

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Belkale, P., Kumar, S. Eigenvalue problem and a new product in cohomology of flag varieties. Invent. math. 166, 185–228 (2006). https://doi.org/10.1007/s00222-006-0516-x

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