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Cauchy Identities for Universal Schubert Polynomials

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Abstract

We prove the Cauchy type identities for the universal double Schubert polynomials recently introduced by W. Fulton. As a corollary, the determinantal formula for some specializations of the universal double Schubert polynomials corresponding to the Grassmannian permutations are obtained. We also introduce and study the universal Schur functions and a multiparameter deformation of Schubert polynomials. Bibliography: 13 titles.

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REFERENCES

  1. W. Fulton, “Universal Schubert polynomials,” Duke Math. J., 96, 575–594 (1999).

    Google Scholar 

  2. S. Fomin, S. Gelfand, and A. Postnikov, “Quantum Schubert polynomials,” J. Amer. Math. Soc., 10, 565–596 (1997).

    Google Scholar 

  3. S. Fomin and A. N. Kirillov, “Quadratic algebras, Dunkl elements, and Schubert calculus progress in mathematics,” Adv. Geometry, 172, 147–182 (1998).

    Google Scholar 

  4. A. Givental B. and Kim, “Quantum cohomology of flag manifolds and Toda lattices,” Comm. Math. Phys., 168, 609–641 (1995).

    Google Scholar 

  5. A. N. Kirillov, “Quantum Grothendieck polynomials, algebraic methods and q-special functions,” Lect. Notes, 22, 215–226 (1999).

    Google Scholar 

  6. A. N. Kirillov, “Quantum Schubert polynomials and quantum Schur functions,” Int. J. Algebra Computation, 9, 385–404 (1999).

    Google Scholar 

  7. A. N. Kirillov and T. Maeno, “Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula,” Discrete Mathematics, 217, 191–224 (2000).

    Google Scholar 

  8. D. E. Littlewood, The Theory of Group Characters, 2nd ed., Oxford Univ. Press (1950).

  9. A. Lascoux and M.-P. Schützenberger, “Functorialité des polynomes de Schubert,” Contemporary Math., 88, 585–598 (1989).

    Google Scholar 

  10. I. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Univ. Press (1995).

  11. I. Macdonald, “Schur functions: theme and variations,” Publ. I.R.M.A., 498/S-28, Actes 28e Seminaire Lotharingien, Strasbourg (1992), pp. 5–39.

    Google Scholar 

  12. I. Macdonald, “Notes on Schubert polynomials,” Publ. LCIM, Univ. Quebec Montreal (1991).

    Google Scholar 

  13. A. Postnikov, “On quantum version of Pieri's formulas, progress in mathematics,” Adv. Geometry, 172, 371–383 (1998).

    Google Scholar 

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  1. St.Petersburg Department of the Steklov Mathematical Institute, Russia

    A. N. Kirillov

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  1. A. N. Kirillov
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Kirillov, A.N. Cauchy Identities for Universal Schubert Polynomials. Journal of Mathematical Sciences 121, 2360–2370 (2004). https://doi.org/10.1023/B:JOTH.0000024617.88286.f2

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  • DOI: https://doi.org/10.1023/B:JOTH.0000024617.88286.f2

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