Duke Mathematical Journal

Schubert polynomials and quiver formulas

Anders S. Buch, Andrew Kresch, Harry Tamvakis, and Alexander Yong

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Fulton's universal Schubert polynomials [F3] represent degeneracy loci for morphisms of vector bundles with rank conditions coming from a permutation. The quiver formula of Buch and Fulton [BF] expresses these polynomials as an integer linear combination of products of Schur determinants. We present a positive, nonrecursive combinatorial formula for the coefficients. Our result is applied to obtain new expansions for the Schubert polynomials of Lascoux and Schützenberger [LS1] and explicit Giambelli formulas in the classical and quantum cohomology ring of any partial flag variety.

Article information

Duke Math. J., Volume 122, Number 1 (2004), 125-143.

First available in Project Euclid: 24 March 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05E15 14M15


Buch, Anders S.; Kresch, Andrew; Tamvakis, Harry; Yong, Alexander. Schubert polynomials and quiver formulas. Duke Math. J. 122 (2004), no. 1, 125--143. doi:10.1215/S0012-7094-04-12214-6. https://projecteuclid.org/euclid.dmj/1080137204

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  • N. Bergeron and F. Sottile, Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Duke Math. J. 95 (1998), 373--423.
  • S. C. Billey, W. Jockusch and R. P. Stanley, Some combinatorial properties of Schubert polynomials, J. Algebraic Combin. 2 (1993), 345--374.
  • A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115--207.
  • A. S. Buch, Stanley symmetric functions and quiver varieties, J. Algebra 235 (2001), 243--260.
  • A. S. Buch and W. Fulton, Chern class formulas for quiver varieties, Invent. Math. 135 (1999), 665--687.
  • A. S. Buch, A. Kresch, H. Tamvakis and A. Yong, Grothendieck polynomials and quiver formulas, preprint, 2003.
  • I. Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), 485--524.
  • P. Edelman and C. Greene, Balanced tableaux, Adv. Math. 63 (1987), 42--99.
  • S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), 565--596.
  • S. Fomin and C. Greene, Noncommutative Schur functions and their applications, Discrete Math. 193 (1998), 179--200.
  • S. Fomin and A. N. Kirillov, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math. 153 (1996), 123--143.
  • S. Fomin and R. P. Stanley, Schubert polynomials and the nil-Coxeter algebra, Adv. Math. 103 (1994), 196--207.
  • W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), 381--420.
  • --------, Young Tableaux, London Math. Soc. Student Texts 35, Cambridge University Press, Cambridge, 1997.
  • --. --. --. --., Universal Schubert polynomials, Duke Math. J. 96 (1999), 575--594.
  • A. N. Kirillov, Cauchy identities for universal Schubert polynomials, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 283 (2001), 123--139., 260.
  • W. Kraskiewicz and P. Pragacz, Foncteurs de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), 209--211.
  • A. Lascoux, Classes de Chern des variétés de drapeaux, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), 393--398.
  • --------, Notes on interpolation in one and several variables, preprint, 1996, http://phalanstere.univ-mlv.fr/$^\sim$al/
  • A. Lascoux and M.-P. Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 447--450.
  • --. --. --. --., Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), 629--633.
  • --. --. --. --., "Fonctorialité des polynômes de Schubert" in Invariant Theory (Denton, Tex., 1986), Contemp. Math. 88, Amer. Math. Soc., Providence, 1989, 585--598.
  • I. G. Macdonald, Notes on Schubert polynomials, Publ. LACIM 6, Univ. de Québec à Montréal, Montréal, 1991.
  • --------, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Univ. Press, New York, 1995.
  • F. Sottile, Pieri's formula for flag manifolds and Schubert polynomials, Ann. Inst. Fourier (Grenoble) 46 (1996), 89--110.
  • R. P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Combin. 5 (1984), 359--372.