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Two Murnaghan-Nakayama Rules in Schubert Calculus

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Abstract

The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. We establish a version of the Murnaghan- Nakayama rule for Schubert polynomials and a version for the quantum cohomology ring of the Grassmannian. These rules compute all intersections of Schubert cycles with tautological classes coming fromthe Chern character. Like the classical rule, both rules are multiplicity-free signed sums.

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References

  1. Bandlow, J., Schilling, A., Zabrocki, M.: The Murnaghan-Nakayama rule for \(k\)-Schur functions. J. Combin. Theory Ser. A 118(5), 1588–1607 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bergeron, N., Sottile, F.: Schubert polynomials, the Bruhat order, and the geometry of flag manifolds. Duke Math. J. 95(2), 373–423 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bergeron, N., Sottile, F.: A monoid for the Grassmannian Bruhat order. European J. Combin. 20(3), 197–211 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bergeron, N., Sottile, F.: A Pieri-type formula for isotropic flag manifolds. Trans. Amer. Math. Soc. 354(7), 2659–2705 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernstein, I.N., Gel’fand, I.M., Gel’fand, S.I.: Schubert cells and cohomology of the spaces G/P. Russian Math. Surveys 28(3), 1–26 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bertram, A.: Quantum Schubert calculus. Adv. Math. 128(2), 289–305 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bertram, A., Ciocan-Fontanine, I., Fulton, W.: Quantum multiplication of Schur polynomials. J. Algebra 219(2), 728–746 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Demazure, M.: Désingularization des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. 4(7), 53–88 (1974)

    Article  MATH  Google Scholar 

  9. Fomin, S., Greene, C.: Noncommutative Schur functions and their applications. Discrete Math. 193(1–3), 179–200 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology. In: Kollár, J., Lazarsfeld, R., Morrison, D.R. (eds.) Algebraic geometry — Santa Cruz 1995, Proc. Sympos. Pure Math., Vol. 62, pp. 45-96. Amer. Math. Soc., Providence, RI (1997)

  11. Goodman, F., Wenzl, H.: Littlewood-Richardson coefficients for Hecke algebras at roots of unity. Adv. Math. 82(2), 244–265 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Intriligator, K.: Fusion residues. Modern Phys. Lett. A 6(38), 3543–3556 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kac, V.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  14. Konvalinka, M.: Skew quantum Murnaghan-Nakayama rule. J. Algebraic Combin. 35(4), 519–545 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lascoux, A.: Schützenberger, M.-P.: Polynômes de Schubert. C. R. Acad. Sci. Paris Sér. I Math. 294(13), 447–450 (1982)

    MathSciNet  MATH  Google Scholar 

  16. Macdonald, I.G.: Notes on Schubert Polynomials. Laboratoire de combinatoire et d’informatique mathématique (LACIM), Univ. du Québec à Montréal, Montréal (1991)

  17. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. The Clarendon Press, Oxford University Press, New York (1995)

    MATH  Google Scholar 

  18. McNamara, P.: A Pieri rule for skew shapes. Talk at FPSAC 2010, available online at http://www.facstaff.bucknell.edu/pm040/Slides/fpsac10.pdf (2010)

  19. Monk, D.: The geometry of flag manifolds. Proc. London Math. Soc. 3(9), 253–286 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  20. Morrison, A.: A Murnaghan-Nakayama rule for Schubert polynomials. In: Billera, L.J., Novik, I. (eds.) 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), Discrete Math. Theor. Comput. Sci. Proc., AT, pp. 525- 536. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2014)

  21. Murnaghan, F.: The characters of the symmetric group. Amer. J. Math. 59(4), 739–753 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nakayama, T.: On some modular properties of irreducible representations of a symmetric group. I. Jap. J. Math. 18, 89–108 (1941)

    MathSciNet  Google Scholar 

  23. Nakayama, T.: On some modular properties of irreducible representations of a symmetric group. II. Jap. J. Math. 17, 411 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pieri, M.: Sul problema degli spazi secanti. Rend. Reale Ist. lombardo 26, 534–546 (1893)

    MATH  Google Scholar 

  25. Ross, D.: The loop Murnaghan-Nakayama rule. J. Algebraic Combin. 39(1), 3–15 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ross, D., Zong, Z.: The gerby Gopakumar-Mariño-Vafa formula. Geom. Topol. 17(5), 2935–2976 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Siebert, B., Tian, G.: On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intrilligator. Asian J. Math. 1(4), 679–695 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sottile, F.: Pieri’s formula for flag manifolds and Schubert polynomials. Ann. Inst. Fourier (Grenoble) 46(1), 89–110 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tewari, V.: AMurnaghan-Nakayama rule for noncommutative Schur functions. European J. Combin. 58, 118–143 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Vafa, C.: Topological mirrors and quantum rings. In: Yau, S.-T. (ed.) Essays on Mirror Manifolds, pp. 96–119. International Press, Hong Kong (1992)

    Google Scholar 

  31. Walton, M.: Fusion rules in Weiss-Zumino-Witten models. Nuclear Phys. B 340(2–3), 777–790 (1990)

    Article  MathSciNet  Google Scholar 

  32. Wildon, M.: A Combinatorial proof of a plethystic Murnaghan-Nakayama rule. SIAM J. Discrete Math. 30(3), 1526–1533 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Witten, E.: The Verlinde algebera and the cohomology of the Grassmannian. In: Yau, S.- T. (ed.) Geometry, Topology, and Physics, Lecture Notes in Geometric Topology, Vol. IV, pp. 357–422. Int. Press, Cambridge, MA (1995)

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Authors and Affiliations

  1. Department of Mathematics, ETH Zürich, CH-8092, Zürich, Switzerland

    Andrew Morrison

  2. Department of Mathematics, Texas A&M University, College Station, Texas, 77843, USA

    Frank Sottile

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  1. Andrew Morrison
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  2. Frank Sottile
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Correspondence to Frank Sottile.

Additional information

Research of Morrison was supported by the Swiss National Science Foundation through grant SNF-200021-143274 and theMSRI.

Research of Sottile was supported by the NSF through grants DMS-1001615 and DMS-1501370 and the MSRI.

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Morrison, A., Sottile, F. Two Murnaghan-Nakayama Rules in Schubert Calculus. Ann. Comb. 22, 363–375 (2018). https://doi.org/10.1007/s00026-018-0387-z

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  • DOI: https://doi.org/10.1007/s00026-018-0387-z

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