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Inventiones mathematicae

, Volume 166, Issue 2, pp 229–325 | Cite as

Four positive formulae for type A quiver polynomials

Article

Abstract

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99 ]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient 1 and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula is a variant of the conjecture of Buch and Fulton in terms of factor sequences of Young tableaux [BF99 ]; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in “doubled” versions, two for double quiver polynomials, and the other two for their stable limits, the double quiver functions, where setting half the variables equal to the other half specializes to the ordinary case.

Our method begins by identifying quiver polynomials as multidegrees [BB82 , Jos84 , BB85 , Ros89 ] via equivariant Chow groups [EG98 ]. Then we make use of Zelevinsky’s map from quiver loci to open subvarieties of Schubert varieties in partial flag manifolds [Zel85 ]. Interpreted in equivariant cohomology, this lets us write double quiver polynomials as ratios of double Schubert polynomials [LS82 ] associated to Zelevinsky permutations; this is our first formula. In the process, we provide a simple argument that Zelevinsky maps are scheme-theoretic isomorphisms (originally proved in [LM98 ]). Writing double Schubert polynomials in terms of pipe dreams [FK96 ] then provides another geometric formula for double quiver polynomials, via [KM05 ]. The combinatorics of pipe dreams for Zelevinsky permutations implies an expression for limits of double quiver polynomials in terms of products of Stanley symmetric functions [Sta84 ]. A degeneration of quiver loci (orbit closures of GL on quiver representations) to unions of products of matrix Schubert varieties [Ful92 , KM05 ] identifies the summands in our Stanley function formula combinatorially, as lacing diagrams that we construct based on the strands of Abeasis and Del Fra in the representation theory of quivers [AD80 ]. Finally, we apply the combinatorial theory of key polynomials to pass from our lacing diagram formula to a double Schur function formula in terms of peelable tableaux [RS95a , RS98 ], and from there to our formula of Buch–Fulton type.

Keywords

Schubert Variety Quiver Representation Block Column Degeneracy Locus Positive Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abeasis, S., Del Fra, A.: Degenerations for the representations of an equioriented quiver of type A m. Boll. Unione. Mat. Ital. Suppl. 157–171 (1980)Google Scholar
  2. 2.
    Abeasis, S., Del Fra, A., Kraft, H.: The geometry of representations of A m. Math. Ann. 256, 401–418 (1981)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bergeron, N., Billey, S.: RC-graphs and Schubert polynomials. Exp. Math. 2, 257–269 (1993)MATHMathSciNetGoogle Scholar
  4. 4.
    Billey, S.C., Jockusch, W., Stanley, R.P.: Some combinatorial properties of Schubert polynomials. J. Algebr. Comb. 2, 345–374 (1993)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Borho, W., Brylinski, J.-L.: Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules. Invent. Math. 69, 437–476 (1982)MATHMathSciNetGoogle Scholar
  6. 6.
    Borho, B., Brylinski, J.-L.: Differential operators on homogeneous spaces. III. Characteristic varieties of Harish–Chandra modules and of primitive ideals. Invent. Math. 80, 1–68 (1985)MATHMathSciNetGoogle Scholar
  7. 7.
    Brion, M.: Equivariant Chow groups for torus actions. Transform. Groups 2, 225–267 (1997)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Buch, A.S., Fulton, W.: Chern class formulas for quiver varieties. Invent. Math. 135, 665–687 (1999)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Buch, A.S., Fehér, L.M., Rimányi, R.: Positivity of quiver coefficients through Thom polynomials. Adv. Math. 197, 306–320 (2005)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Buch, A.S., Kresch, A., Tamvakis, H., Yong, A.: Schubert polynomials and quiver formulas. Duke Math. J. 122, 125–143 (2004)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Buch, A.S.: Stanley symmetric functions and quiver varieties. J. Algebra 235, 243–260 (2001)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Buch, A.S.: On a conjectured formula for quiver varieties. J. Algebr. Comb. 13, 151–172 (2001)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Buch, A.S.: Grothendieck classes of quiver varieties. Duke Math. J. 115, 75–103 (2002)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Buch, A.S.: Alternating signs of quiver coefficients. J. Am. Math. Soc. 18, 217–237 (2005)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Demazure, M.: Une nouvelle formule des caractères. Bull. Sci. Math. (2) 98, 163–172 (1974)Google Scholar
  16. 16.
    Edelman, P., Greene, C.: Balanced tableaux. Adv. Math. 63, 42–99 (1987)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Edidin, D, Graham, W.: Equivariant intersection theory. Invent. Math. 131, 595–634 (1998)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Eisenbud, D.: Commutative algebra, with a view toward algebraic geometry. Graduate Texts in Mathematics, vol. 150. New York: Springer 1995Google Scholar
  19. 19.
    Fehér, L., Rimányi, R.: Classes of degeneracy loci for quivers: the Thom polynomial point of view. Duke Math. J. 114, 193–213 (2002)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Fehér, L., Rimányi, R.: Schur and Schubert polynomials as Thom polynomials – cohomology of moduli spaces. Cent. Eur. J. Math. 1, 418–434 (2003)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Fomin, S., Kirillov, A.N.: Grothendieck polynomials and the Yang–Baxter equation. Proceedings of the Sixth Conference in Formal Power Series and Algebraic Combinatorics, pp. 183–190. DIMACS 1994Google Scholar
  22. 22.
    Fomin, S., Kirillov, A.N.: The Yang-Baxter equation, symmetric functions, and Schubert polynomials. Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence 1993). Discrete Math. 153, 123–143 (1996)Google Scholar
  23. 23.
    Fomin, S., Stanley, R.P.: Schubert polynomials and the nil-Coxeter algebra. Adv. Math. 103, 196–207 (1994)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Fulton, W.: Flags, Schubert polynomials, degeneracy loci, and determinantal formulas. Duke Math. J. 65, 381–420 (1992)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Fulton, W.: Young tableaux, with applications to representation theory and geometry. London Mathematical Society Student Texts, vol. 35. Cambridge: Cambridge University Press 1997Google Scholar
  26. 26.
    Fulton, W.: Intersection theory, 2nd edn. Berlin: Springer 1998Google Scholar
  27. 27.
    Fulton, W., Pragacz, P.: Schubert varieties and degeneracy loci, Appendix J by the authors in collaboration with I. Ciocan-Fontanine. Lect. Notes Math., vol. 1689. Berlin: Springer 1998Google Scholar
  28. 28.
    Fulton, W.: Universal Schubert polynomials. Duke Math. J. 96, 575–594 (1999)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Giambelli, G.Z.: Ordine di una varietà più ampia di quella rappresentata coll’annullare tutti i minori di dato ordine estratti da una data matrice generica di forme. Mem. R. Ist. Lombardo, III. Ser. 11, 101–135 (1904)Google Scholar
  30. 30.
    Haiman, M.: Dual equivalence with applications, including a conjecture of Proctor. Discrete Math. 99, 79–113 (1992)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Joseph, A.: On the variety of a highest weight module. J. Algebra 88, 238–278 (1984)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Kalkbrener, M., Sturmfels, B.: Initial complexes of prime ideals. Adv. Math. 116, 365–376 (1995)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Kazarian, M.É.: Characteristic classes of singularity theory, The Arnold-Gelfand mathematical seminars, pp. 325–340. Boston, MA: Birkhäuser 1997Google Scholar
  34. 34.
    Knutson, A., Miller, E.: Subword complexes in Coxeter groups. Adv. Math. 184, 161–176 (2004)MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Knutson, A., Miller, E.: Gröbner geometry of Schubert polynomials. Ann. Math (2) 161, 1245–1318 (2005)Google Scholar
  36. 36.
    Kogan, M.: Schubert geometry of flag varieties and Gel’fand–Cetlin theory. Ph.D. thesis. Massachusetts Institute of Technology 2000Google Scholar
  37. 37.
    Lascoux, A.: Anneau de Grothendieck de la variété de drapeaux. The Grothendieck Festschrift, vol. III, pp. 1–34. Prog. Math., vol. 88. Boston, MA: Birkhäuser 1990Google Scholar
  38. 38.
    Lascoux, A.: Transition on Grothendieck polynomials, Physics and Combinatorics, 2000 (Nagoya), pp. 164–179. River Edge, NJ: World Sci. Publishing 2001Google Scholar
  39. 39.
    Lascoux, A.: Double crystal graphs, Studies in Memory of Issai Schur (Chevaleret/Rehovot 2000), pp. 95–114. Prog. Math., vol. 210. Boston, MA: Birkhäuser 2003Google Scholar
  40. 40.
    Lenart, C.: A unified approach to combinatorial formulas for Schubert polynomials. J. Algebr. Comb. 20, 263–299 (2004)MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Lakshmibai, V., Magyar, P.: Degeneracy schemes, quiver schemes, and Schubert varieties. Int. Math. Res. Not. 12, 627–640 (1998)MATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Lascoux, A., Schützenberger, M.-P.: 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, 629–633 (1982)MATHMathSciNetGoogle Scholar
  43. 43.
    Lascoux, A., Schützenberger, M.-P.: Schubert polynomials and the Littlewood–Richardson rule. Lett. Math. Phys. 10, 111–124 (1985)MATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Lascoux, A., Schützenberger, M.-P.: Noncommutative Schubert polynomials. Funct. Anal. Appl. 23, 223–225 (1990)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Lascoux, A., Schützenberger, M.-P.: Keys and standard bases, Invariant theory and tableaux (Minneapolis MN, 1988), pp. 125–144. IMA Vol. Math. Appl., vol. 19. New York: Springer 1990Google Scholar
  46. 46.
    Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3, 447–498 (1990)MATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    Macdonald, I.G.: Notes on Schubert polynomials. Publ. LACIM, vol. 6. Montréal: UQAM 1991Google Scholar
  48. 48.
    Magyar, P.: Schubert polynomials and Bott-Samelson varieties. Comment. Math. Helv. 73, 603–636 (1998)MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Miller, E.: Alternating formulae for K-theoretic quiver polynomials. Duke Math. J. 128, 1–17 (2005)MATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Miller, E., Sturmfels, B.: Combinatorial commutative algebra. Graduate Texts in Mathematics, vol. 227. New York: Springer 2005Google Scholar
  51. 51.
    Ramanathan, A.: Schubert varieties are arithmetically Cohen-Macaulay. Invent. Math. 80, 283–294 (1985)MATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Ramanan, S., Ramanathan, A.: Projective normality of flag varieties and Schubert varieties. Invent. Math. 79, 217–224 (1985)MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Rossmann, W.: Equivariant multiplicities on complex varieties, Orbites unipotentes et représentations, III. Astérisque 11, 313–330 (1989)MathSciNetGoogle Scholar
  54. 54.
    Reiner, V., Shimozono, M.: Plactification. J. Algebr. Comb. 4, 331–351 (1995)MATHMathSciNetCrossRefGoogle Scholar
  55. 55.
    Reiner, V., Shimozono, M.: Key polynomials and a flagged Littlewood-Richardson rule. J. Comb. Theory, Ser. A 70, 107–143 (1995)MATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Reiner, V., Shimozono, M.: Percentage-avoiding, northwest shapes and peelable tableaux. J. Comb. Theory, Ser. A 82, 1–73 (1998)MATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    Stanley, R.P.: On the number of reduced decompositions of elements of Coxeter groups. Eur. J. Comb. 5, 359–372 (1984)MATHMathSciNetGoogle Scholar
  58. 58.
    Thom, R.: Les singularités des applications différentiables. Ann. Inst. Fourier 6, 43–87 (1955–1956)Google Scholar
  59. 59.
    Yong, A.: On combinatorics of quiver component formulas. J. Algebr. Comb. 21, 351–371 (2005)MATHMathSciNetCrossRefGoogle Scholar
  60. 60.
    Zelevinskiĭ, A.V.: Two remarks on graded nilpotent classes. Usp. Mat. Nauk 40, 199–200 (1985)Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Mathematics DepartmentUC BerkeleyBerkeleyUSA
  2. 2.Mathematical Sciences Research InstituteBerkeleyUSA
  3. 3.Mathematics DepartmentVirginia TechBlacksburgUSA

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