×

Four positive formulae for type \(A\) quiver polynomials. (English) Zbl 1107.14046

Consider a chain of vector bundle maps over a base space. The points in the source, over which the ranks of the vector bundle maps and their compositions have certain values, are called quiver degeneracy loci. The authors study the classes these degeneracy loci represent in the cohomology and \(K\)-theory of the base space.
Namely, they calculate the so-called \(K\)-polynomials and multidegrees of the quiver loci corresponding to equioriented quivers of type \(A_n\). Four formulas are presented in the paper, all having their own advantages. The formulas are combinatorial, i.e. they express the result in terms of certain combinatorial objects (Zelevinsky permutations, lacing diagrams, Young-tableaux, pipe dreams). All the formulas are positive, which means that the result is presented as a sum with positive coefficients. Three of the formulas are geometric, that is, the terms correspond bijectively to torus-invariant schemes. The last formula is very similar in nature to the conjectured formula of Buch and Fulton for the same loci.

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14M15 Grassmannians, Schubert varieties, flag manifolds
05E05 Symmetric functions and generalizations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[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) · Zbl 0449.16024
[2] Abeasis, S., Del Fra, A., Kraft, H.: The geometry of representations of A m . Math. Ann. 256, 401–418 (1981) · Zbl 0477.14027 · doi:10.1007/BF01679706
[3] Bergeron, N., Billey, S.: RC-graphs and Schubert polynomials. Exp. Math. 2, 257–269 (1993) · Zbl 0803.05054
[4] Billey, S.C., Jockusch, W., Stanley, R.P.: Some combinatorial properties of Schubert polynomials. J. Algebr. Comb. 2, 345–374 (1993) · Zbl 0790.05093 · doi:10.1023/A:1022419800503
[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) · Zbl 0504.22015
[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) · Zbl 0577.22014
[7] Brion, M.: Equivariant Chow groups for torus actions. Transform. Groups 2, 225–267 (1997) · Zbl 0916.14003 · doi:10.1007/BF01234659
[8] Buch, A.S., Fulton, W.: Chern class formulas for quiver varieties. Invent. Math. 135, 665–687 (1999) · Zbl 0942.14027 · doi:10.1007/s002220050297
[9] Buch, A.S., Fehér, L.M., Rimányi, R.: Positivity of quiver coefficients through Thom polynomials. Adv. Math. 197, 306–320 (2005) · Zbl 1076.14075 · doi:10.1016/j.aim.2004.10.019
[10] Buch, A.S., Kresch, A., Tamvakis, H., Yong, A.: Schubert polynomials and quiver formulas. Duke Math. J. 122, 125–143 (2004) · Zbl 1072.14067 · doi:10.1215/S0012-7094-04-12214-6
[11] Buch, A.S.: Stanley symmetric functions and quiver varieties. J. Algebra 235, 243–260 (2001) · Zbl 0981.05099 · doi:10.1006/jabr.2000.8478
[12] Buch, A.S.: On a conjectured formula for quiver varieties. J. Algebr. Comb. 13, 151–172 (2001) · Zbl 0982.05106 · doi:10.1023/A:1011245531325
[13] Buch, A.S.: Grothendieck classes of quiver varieties. Duke Math. J. 115, 75–103 (2002) · Zbl 1052.14056 · doi:10.1215/S0012-7094-02-11513-0
[14] Buch, A.S.: Alternating signs of quiver coefficients. J. Am. Math. Soc. 18, 217–237 (2005) · Zbl 1061.14050 · doi:10.1090/S0894-0347-04-00473-4
[15] Demazure, M.: Une nouvelle formule des caractères. Bull. Sci. Math. (2) 98, 163–172 (1974) · Zbl 0365.17005
[16] Edelman, P., Greene, C.: Balanced tableaux. Adv. Math. 63, 42–99 (1987) · Zbl 0616.05005 · doi:10.1016/0001-8708(87)90063-6
[17] Edidin, D, Graham, W.: Equivariant intersection theory. Invent. Math. 131, 595–634 (1998) · Zbl 0940.14003 · doi:10.1007/s002220050214
[18] Eisenbud, D.: Commutative algebra, with a view toward algebraic geometry. Graduate Texts in Mathematics, vol. 150. New York: Springer 1995 · Zbl 0819.13001
[19] Fehér, L., Rimányi, R.: Classes of degeneracy loci for quivers: the Thom polynomial point of view. Duke Math. J. 114, 193–213 (2002) · Zbl 1054.14010 · doi:10.1215/S0012-7094-02-11421-5
[20] Fehér, L., Rimányi, R.: Schur and Schubert polynomials as Thom polynomials – cohomology of moduli spaces. Cent. Eur. J. Math. 1, 418–434 (2003) · Zbl 1038.57008 · doi:10.2478/BF02475176
[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 1994
[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) · Zbl 0852.05078
[23] Fomin, S., Stanley, R.P.: Schubert polynomials and the nil-Coxeter algebra. Adv. Math. 103, 196–207 (1994) · Zbl 0809.05091 · doi:10.1006/aima.1994.1009
[24] Fulton, W.: Flags, Schubert polynomials, degeneracy loci, and determinantal formulas. Duke Math. J. 65, 381–420 (1992) · Zbl 0788.14044 · doi:10.1215/S0012-7094-92-06516-1
[25] Fulton, W.: Young tableaux, with applications to representation theory and geometry. London Mathematical Society Student Texts, vol. 35. Cambridge: Cambridge University Press 1997 · Zbl 0878.14034
[26] Fulton, W.: Intersection theory, 2nd edn. Berlin: Springer 1998 · Zbl 0885.14002
[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 1998
[28] Fulton, W.: Universal Schubert polynomials. Duke Math. J. 96, 575–594 (1999) · Zbl 0981.14022 · doi:10.1215/S0012-7094-99-09618-7
[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)
[30] Haiman, M.: Dual equivalence with applications, including a conjecture of Proctor. Discrete Math. 99, 79–113 (1992) · Zbl 0760.05093 · doi:10.1016/0012-365X(92)90368-P
[31] Joseph, A.: On the variety of a highest weight module. J. Algebra 88, 238–278 (1984) · Zbl 0539.17006 · doi:10.1016/0021-8693(84)90100-5
[32] Kalkbrener, M., Sturmfels, B.: Initial complexes of prime ideals. Adv. Math. 116, 365–376 (1995) · Zbl 0877.13025 · doi:10.1006/aima.1995.1071
[33] Kazarian, M.É.: Characteristic classes of singularity theory, The Arnold-Gelfand mathematical seminars, pp. 325–340. Boston, MA: Birkhäuser 1997 · Zbl 0872.57034
[34] Knutson, A., Miller, E.: Subword complexes in Coxeter groups. Adv. Math. 184, 161–176 (2004) · Zbl 1069.20026 · doi:10.1016/S0001-8708(03)00142-7
[35] Knutson, A., Miller, E.: Gröbner geometry of Schubert polynomials. Ann. Math (2) 161, 1245–1318 (2005) · Zbl 1089.14007
[36] Kogan, M.: Schubert geometry of flag varieties and Gel’fand–Cetlin theory. Ph.D. thesis. Massachusetts Institute of Technology 2000
[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 1990 · Zbl 0742.14041
[38] Lascoux, A.: Transition on Grothendieck polynomials, Physics and Combinatorics, 2000 (Nagoya), pp. 164–179. River Edge, NJ: World Sci. Publishing 2001 · Zbl 1052.14059
[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 2003
[40] Lenart, C.: A unified approach to combinatorial formulas for Schubert polynomials. J. Algebr. Comb. 20, 263–299 (2004) · Zbl 1056.05146 · doi:10.1023/B:JACO.0000048515.00922.47
[41] Lakshmibai, V., Magyar, P.: Degeneracy schemes, quiver schemes, and Schubert varieties. Int. Math. Res. Not. 12, 627–640 (1998) · Zbl 0936.14001 · doi:10.1155/S1073792898000397
[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) · Zbl 0542.14030
[43] Lascoux, A., Schützenberger, M.-P.: Schubert polynomials and the Littlewood–Richardson rule. Lett. Math. Phys. 10, 111–124 (1985) · Zbl 0586.20007 · doi:10.1007/BF00398147
[44] Lascoux, A., Schützenberger, M.-P.: Noncommutative Schubert polynomials. Funct. Anal. Appl. 23, 223–225 (1990) · Zbl 0717.20017 · doi:10.1007/BF01079531
[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 1990
[46] Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3, 447–498 (1990) · Zbl 0703.17008 · doi:10.1090/S0894-0347-1990-1035415-6
[47] Macdonald, I.G.: Notes on Schubert polynomials. Publ. LACIM, vol. 6. Montréal: UQAM 1991 · Zbl 0784.05061
[48] Magyar, P.: Schubert polynomials and Bott-Samelson varieties. Comment. Math. Helv. 73, 603–636 (1998) · Zbl 0951.14036 · doi:10.1007/s000140050071
[49] Miller, E.: Alternating formulae for K-theoretic quiver polynomials. Duke Math. J. 128, 1–17 (2005) · Zbl 1099.05079 · doi:10.1215/S0012-7094-04-12811-8
[50] Miller, E., Sturmfels, B.: Combinatorial commutative algebra. Graduate Texts in Mathematics, vol. 227. New York: Springer 2005 · Zbl 1066.13001
[51] Ramanathan, A.: Schubert varieties are arithmetically Cohen-Macaulay. Invent. Math. 80, 283–294 (1985) · Zbl 0555.14021 · doi:10.1007/BF01388607
[52] Ramanan, S., Ramanathan, A.: Projective normality of flag varieties and Schubert varieties. Invent. Math. 79, 217–224 (1985) · Zbl 0553.14023 · doi:10.1007/BF01388970
[53] Rossmann, W.: Equivariant multiplicities on complex varieties, Orbites unipotentes et représentations, III. Astérisque 11, 313–330 (1989)
[54] Reiner, V., Shimozono, M.: Plactification. J. Algebr. Comb. 4, 331–351 (1995) · Zbl 0922.05049 · doi:10.1023/A:1022434000967
[55] Reiner, V., Shimozono, M.: Key polynomials and a flagged Littlewood-Richardson rule. J. Comb. Theory, Ser. A 70, 107–143 (1995) · Zbl 0819.05058 · doi:10.1016/0097-3165(95)90083-7
[56] Reiner, V., Shimozono, M.: Percentage-avoiding, northwest shapes and peelable tableaux. J. Comb. Theory, Ser. A 82, 1–73 (1998) · Zbl 0909.05049 · doi:10.1006/jcta.1997.2841
[57] Stanley, R.P.: On the number of reduced decompositions of elements of Coxeter groups. Eur. J. Comb. 5, 359–372 (1984) · Zbl 0587.20002
[58] Thom, R.: Les singularités des applications différentiables. Ann. Inst. Fourier 6, 43–87 (1955–1956)
[59] Yong, A.: On combinatorics of quiver component formulas. J. Algebr. Comb. 21, 351–371 (2005) · Zbl 1078.05087 · doi:10.1007/s10801-005-6916-y
[60] Zelevinskiĭ, A.V.: Two remarks on graded nilpotent classes. Usp. Mat. Nauk 40, 199–200 (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.