Duke Mathematical Journal

Alternating formulas for K -theoretic quiver polynomials

Ezra Miller

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The main theorem here is the K -theoretic analogue of the cohomological ``stable double component formula'' for quiver polynomials in [KMS]. This K -theoretic version is still in terms of lacing diagrams, but nonminimal diagrams contribute terms of higher degree. The motivating consequence is a conjecture of Buch [B1] on the sign alternation of the coefficients appearing in his expansion of quiver K -polynomials in terms of stable Grothendieck polynomials for partitions.

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Duke Math. J. Volume 128, Number 1 (2005), 1-17.

First available in Project Euclid: 17 May 2005

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Zentralblatt MATH identifier

Primary: 05E05: Symmetric functions and generalizations
Secondary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]


Miller, Ezra. Alternating formulas for $K$ -theoretic quiver polynomials. Duke Math. J. 128 (2005), no. 1, 1--17. doi:10.1215/S0012-7094-04-12811-8. https://projecteuclid.org/euclid.dmj/1116361225

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  • S. Abeasis and A. Del\thinspace\thinspaceFra, Degenerations for the representations of an equioriented quiver of type $A_m$, Boll. Un. Mat. Ital. Suppl. 1980, 157--171.
  • N. Bergeron and S. Billey, RC-graphs and Schubert polynomials, Experiment. Math. 2 (1993), 257--269.
  • A. S. Buch, Grothendieck classes of quiver varieties, Duke Math. J. 115 (2002), 75--103.
  • --. --. --. --., A Littlewood-Richardson rule for the $K$-theory of Grassmannians, Acta Math. 189 (2002), 37--78.
  • --. --. --. --., Alternating signs of quiver coefficients, J. Amer. Math. Soc. 18 (2005), 217--237.
  • A. S. Buch, L. M. Fehér, and R. Rimányi, Positivity of quiver coefficients through Thom polynomials, to appear in Adv. Math., preprint.
  • 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, to appear in Amer. J. Math., preprint.
  • S. Fomin and A. N. Kirillov, ``The Yang-Baxter equation, symmetric functions, and Schubert polynomials'' in Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993), Discrete Math. 153, North-Holland, Amsterdam, 1996, 123--143.
  • --------, Grothendieck polynomials and the Yang-Baxter equation, extended abstract for the 6th Conference on Formal Power Series and Algebraic Combinatorics (New Brunswick, N.J., 1994), available from http://www.math.lsa.umich.edu/$\tilde\ \,$fomin/papers.html
  • A. Knutson and E. Miller, Subword complexes in Coxeter groups, Adv. Math. 184 (2004), 161--176.
  • --------, Gröbner geometry of Schubert polynomials, to appear in Ann. of Math (2), preprint.
  • A. Knutson, E. Miller, and M. Shimozono, Four positive formulae for type $A$ quiver polynomials, preprint.
  • A. Lascoux, ``Transition on Grothendieck polynomials'' in Physics and Combinatorics, 2000 (Nagoya, Japan), World Sci., River Edge, N.J., 2001. 164--179.
  • A. Lascoux and M.-P. Schützenberger, 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.
  • A. Yong, On combinatorics of quiver component formulas, to appear in J. Algebraic Combin., preprint.