Duke Mathematical Journal

Three combinatorial formulas for type A quiver polynomials and K-polynomials

Ryan Kinser, Allen Knutson, and Jenna Rajchgot

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We provide combinatorial formulas for the multidegree and K-polynomial of an arbitrarily oriented type A quiver locus. These formulas are generalizations of three formulas by Knutson, Miller, and Shimozono from the equioriented setting. In particular, we prove the K-theoretic component formula conjectured by Buch and Rimányi.

Article information

Source
Duke Math. J., Volume 168, Number 4 (2019), 505-551.

Dates
Received: 19 March 2015
Revised: 12 July 2018
First available in Project Euclid: 4 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1549270814

Digital Object Identifier
doi:10.1215/00127094-2018-0043

Mathematical Reviews number (MathSciNet)
MR3916063

Zentralblatt MATH identifier
07055150

Subjects
Primary: 14M12: Determinantal varieties [See also 13C40]
Secondary: 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80] 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] 19E08: $K$-theory of schemes [See also 14C35]

Keywords
quiver locus representation variety orbit closure K-polynomial multidegree degeneracy locus pipe dream lacing diagram matrix Schubert variety

Citation

Kinser, Ryan; Knutson, Allen; Rajchgot, Jenna. Three combinatorial formulas for type $A$ quiver polynomials and $K$ -polynomials. Duke Math. J. 168 (2019), no. 4, 505--551. doi:10.1215/00127094-2018-0043. https://projecteuclid.org/euclid.dmj/1549270814


Export citation

References

  • [1] S. Abeasis and A. Del Fra, Degenerations for the representations of a quiver of type ${A}_{m}$, J. Algebra 93 (1985), no. 2, 376–412.
  • [2] J. Allman, Grothendieck classes of quiver cycles as iterated residues, Michigan Math. J. 63 (2014), no. 4, 865–888.
  • [3] H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of Quantum Groups at a $p$th Root of Unity and of Semisimple Groups in Characteristic $p$: Independence of $p$, Astérisque 220, Soc. Math. France, Paris, 1994.
  • [4] I. Assem, D. Simson, and A. Skowroński, Elements of the Representation Theory of Associative Algebras, Vol. 1, London Math. Soc. Stud. Texts 65, Cambridge Univ. Press, Cambridge, 2006.
  • [5] N. Bergeron and S. Billey, RC-graphs and Schubert polynomials, Exp. Math. 2 (1993), no. 4, 257–269.
  • [6] S. C. Billey, Kostant polynomials and the cohomology ring for $G/B$, Duke Math. J. 96 (1999), no. 1, 205–224.
  • [7] G. Bobiński and G. Zwara, Schubert varieties and representations of Dynkin quivers, Colloq. Math. 94 (2002), no. 2, 285–309.
  • [8] K. Bongartz, On degenerations and extensions of finite-dimensional modules, Adv. Math. 121 (1996), no. 2, 245–287.
  • [9] K. Bongartz, “Some geometric aspects of representation theory” in Algebras and Modules, I (Trondheim, 1996), CMS Conf. Proc. 23, Amer. Math. Soc., Providence, 1998, 1–27.
  • [10] A. S. Buch, Grothendieck classes of quiver varieties, Duke Math. J. 115 (2002), no. 1, 75–103.
  • [11] A. S. Buch, Alternating signs of quiver coefficients, J. Amer. Math. Soc. 18 (2005), no. 1, 217–237.
  • [12] A. S. Buch, Quiver coefficients of Dynkin type, Michigan Math. J. 57 (2008), 93–120.
  • [13] A. S. Buch, L. M. Fehér, and R. Rimányi, Positivity of quiver coefficients through Thom polynomials, Adv. Math. 197 (2005), no. 1, 306–320.
  • [14] A. S. Buch and W. Fulton, Chern class formulas for quiver varieties, Invent. Math. 135 (1999), no. 3, 665–687.
  • [15] A. S. Buch and R. Rimányi, Specializations of Grothendieck polynomials, C. R. Math. Acad. Sci. Paris 339 (2004), no. 1, 1–4.
  • [16] A. S. Buch and R. Rimányi, A formula for non-equioriented quiver orbits of type $A$, J. Algebraic Geom. 16 (2007), no. 3, 531–546.
  • [17] D. Eisenbud, Commutative Algebra, Grad. Texts in Math. 150, Springer, New York, 1995.
  • [18] L. Fehér and R. Rimányi, Classes of degeneracy loci for quivers: The Thom polynomial point of view, Duke Math. J. 114 (2002), no. 2, 193–213.
  • [19] S. Fomin and A. N. Kirillov, “Grothendieck polynomials and the Yang-Baxter equation” in Formal Power Series and Algebraic Combinatorics/Séries formelles et combinatoire algébrique, DIMACS, Piscataway, NJ, 1994, 183–189.
  • [20] S. Fomin and A. N. Kirillov, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math. 153 (1996), no. 1–3, 123–143.
  • [21] W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), no. 3, 381–420.
  • [22] W. Fulton and A. Lascoux, A Pieri formula in the Grothendieck ring of a flag bundle, Duke Math. J. 76 (1994), no. 3, 711–729.
  • [23] W. Graham, Equivariant $K$-theory and Schubert varieties, preprint, 2002.
  • [24] J. Herzog and T. Hibi, Monomial Ideals, Grad. Texts in Math. 260, Springer, London, 2011.
  • [25] B. Huisgen-Zimmermann, “Fine and coarse moduli spaces in the representation theory of finite dimensional algebras” in Expository Lectures on Representation Theory, Contemp. Math. 607, Amer. Math. Soc., Providence, 2014, 1–34.
  • [26] R. Kinser, “K-polynomials of type $A$ quiver orbit closures and lacing diagrams” in Representations of Algebras, Contemp. Math. 705, Amer. Math. Soc., Providence, 2018, 99–114.
  • [27] R. Kinser and J. Rajchgot, Type $A$ quiver loci and Schubert varieties, J. Commut. Algebra 7 (2015), no. 2, 265–301.
  • [28] A. Knutson, Frobenius splitting and Möbius inversion, preprint, arXiv:0902.1930v1 [math.AG].
  • [29] A. Knutson, Frobenius splitting, point counting, and degeneration, preprint, arXiv:0911.4941v1 [math.AG].
  • [30] A. Knutson and E. Miller, Subword complexes in Coxeter groups, Adv. Math. 184 (2004), no. 1, 161–176.
  • [31] A. Knutson and E. Miller, Gröbner geometry of Schubert polynomials, Ann. of Math. (2) 161 (2005), no. 3, 1245–1318.
  • [32] A. Knutson, E. Miller, and M. Shimozono, Four positive formulae for type $A$ quiver polynomials, Invent. Math. 166 (2006), no. 2, 229–325.
  • [33] M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Number Theory Phys. 5 (2011), no. 2, 231–352.
  • [34] V. Lakshmibai and P. Magyar, Degeneracy schemes, quiver schemes, and Schubert varieties, Int. Math. Res. Not. IMRN 1998, no. 12, 627–640.
  • [35] 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), no. 11, 629–633.
  • [36] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498.
  • [37] E. Miller, Alternating formulas for $K$-theoretic quiver polynomials, Duke Math. J. 128 (2005), no. 1, 1–17.
  • [38] E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Grad. Texts in Math. 227, Springer, New York, 2005.
  • [39] C. Riedtmann and G. Zwara, Orbit closures and rank schemes, Comment. Math. Helv. 88 (2013), no. 1, 55–84.
  • [40] R. Rimányi, Quiver polynomials in iterated residue form, J. Algebraic Combin. 40 (2014), no. 2, 527–542.
  • [41] R. Rimányi, On the cohomological Hall algebra of Dynkin quivers, preprint, arXiv:1303.3399v1 [math.AG].
  • [42] C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591.
  • [43] R. Schiffler, Quiver Representations, CMS Books Math./Ouvrages Math. SMC, Springer, Cham, 2014.
  • [44] M. Willems, $K$-théorie équivariante des tours de Bott. Application à la structure multiplicative de la $K$-théorie équivariante des variétés de drapeaux, Duke Math. J. 132 (2006), no. 2, 271–309.
  • [45] A. Woo and A. Yong, Governing singularities of Schubert varieties, J. Algebra 320 (2008), no. 2, 495–520.
  • [46] A. Woo and A. Yong, A Gröbner basis for Kazhdan-Lusztig ideals, Amer. J. Math. 134 (2012), no. 4, 1089–1137.
  • [47] A. Yong, On combinatorics of quiver component formulas, J. Algebraic Combin. 21 (2005), no. 3, 351–371.
  • [48] A. V. Zelevinsky, Two remarks on graded nilpotent classes (in Russian), Uspekhi Mat. Nauk 40 (1985), no. 1(241), 199–200; English translation in Russian Math. Surveys 40 (1985), no. 1, 249–250.
  • [49] G. Zwara, “Singularities of orbit closures in module varieties” in Representations of Algebras and Related Topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, 661–725.