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Some combinatorial aspects of the Schubert calculus

Fonctions Symétriques

Part of the Lecture Notes in Mathematics book series (LNM,volume 579)

Keywords

  • Symmetric Function
  • Cohomology Class
  • Betti Number
  • Young Tableau
  • Maximal Chain

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References

  1. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass., 1969.

    MATH  Google Scholar 

  2. E. A. Bender and D. E. Knuth, Enumeration of plane partitions, J. Combinatorial Theory 13 (1972), 40–54.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. J. B. Carrell and D. I. Lieberman, Chern classes of the Grassmannians and the Schubert calculus, preprint.

    Google Scholar 

  4. P. Doubilet, G.-C. Rota, and J. Stein, On the foundations of combinatorial theory: IX, Combinatorial methods in invariant theory, Studies in Applied Math. 53 (1974), 185–216.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. P. Hall, The algebra of partitions, Proc. 4th Canad. Math. Congr. Banff. 1957 (1959), 147–159.

    Google Scholar 

  6. M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytones, Annals of Math. 96 (1972), 318–337.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry, vols. I and II, Cambridge University Press, 1953.

    Google Scholar 

  8. G. Horrocks, On the relation of S-functions to Schubert varieties, Proc. London Math. Soc. (3) 7 (1957), 265–280.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. S. L. Kleiman, Problem 15. Rigorous foundation of Schubert's enumerative calculus, to appear in the Proceedings of a Symposium on Mathematical Developments Arising from the Hilbert Problems, held at DeKalb, Illinois, May 13–17, 1974.

    Google Scholar 

  10. S. L. Kleiman and D. Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061–1082.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. D. E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 (1970), 709–727.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du BURO 6 (1965).

    Google Scholar 

  13. A. Lascoux, Puissances extérieures, déterminants et cycles de Schubert, Bull. Soc. Math. France 102 (1974), 161–179.

    MathSciNet  MATH  Google Scholar 

  14. A. Lascoux, Polynômes symétriques et coefficients d'intersection de cycles de Schubert, C. R. Acad. Sc. Paris, 279 A (1974), 201–204.

    MathSciNet  MATH  Google Scholar 

  15. A. Lascoux, Fonctions de Schur et grassmanniennes, C. R. Acad. Sc. Paris 281 A (1975), 813–815, 851–854.

    MathSciNet  MATH  Google Scholar 

  16. L. Lesieur, Les problèmes d'intersections sur une variété de Grassmann, C. R. Acad. Sc. Paris 225 (1947), 916–917.

    MathSciNet  MATH  Google Scholar 

  17. D. E. Littlewood, The Theory of Group Characters, second ed., Oxford University Press, 1950.

    Google Scholar 

  18. P. A. MacMahon, Combinatory Analysis, vols. 1–2, Cambridge University Press, 1915, 1916; reprinted by Chelsea, New York, 1960.

    Google Scholar 

  19. G.-C. Rota, Combinatorial theory, old and new, Actes du Congres International des Mathématiciens (Nice, 1970), Gauthier-Villars, Paris, 1971, pp. 229–233.

    MATH  Google Scholar 

  20. M. Schützenberger, La correspondance de Robinson, in these Lecture Notes.

    Google Scholar 

  21. R. Stanley, Theory and application of plane partitions, Parts 1 and 2, Studies in Applied Math. 50 (1971), 167–188, 259–279.

    CrossRef  MATH  Google Scholar 

  22. R. Stanley, Ordered structures and partitions, Memoirs of the Amer. Math. Soc., no. 119, 1972.

    Google Scholar 

  23. R. Stanley, Combinatorial reciprocity theorems, Advances in Math. 14 (1974), 194–253.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. R. Stanley, Hilbert functions of graded algebras, Advances in Math., to appear.

    Google Scholar 

  25. N. White, The bracket ring of a combinatorial geometry, I, Trans. Amer. Math. Soc. 202 (1975), 79–95.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. N. White, The bracket ring of a combinatorial geometry, II: Unimodular geometries, Trans. Amer. Math. Soc. 214 (1975), 233–248.

    MathSciNet  MATH  Google Scholar 

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© 1977 Springer-Verlag

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Stanley, R.P. (1977). Some combinatorial aspects of the Schubert calculus. In: Foata, D. (eds) Combinatoire et Représentation du Groupe Symétrique. Lecture Notes in Mathematics, vol 579. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090019

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  • DOI: https://doi.org/10.1007/BFb0090019

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08143-2

  • Online ISBN: 978-3-540-37385-8

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