×

zbMATH — the first resource for mathematics

Stable Grothendieck polynomials and \(K\)-theoretic factor sequences. (English) Zbl 1157.14036
Schubert polynomials are the building blocks of several cohomological degeneracy locus formulas. Their role in \(K\)-theory is played by so-called Grothendieck polynomials. In particular, “stable” versions of Schubert and Grothendieck polynomials turn up naturally, for example in degeneracy locus problems associated with certain quiver representations.
The paper under review studies the expansion of stable Grothendieck polynomials in the basis of stable Grothendieck polynomials for partitions. This generalizes the result of Fomin-Green in the cohomological setting, where the analogues of stable Grothendick polynomials for partitions are the Schur polynomials. The existence of such a finite, integer linear combination expansion was proved by Buch, and the sign of the coefficients were determined by Lascoux. In this paper the authors give a new, non-recursive combinatorial rule for the coefficients. Namely, they prove that the coefficients, up to explicit sign, count the number of increasing tableau of a given shape, with an associated word having explicit combinatorial properties stemming from the combinatorics of the 0-Hecke monoid. The main ingredient of the proof is a generalized, so-called Hecke insertion algorithm.
The main application showed in the paper is a \(K\)-theoretic analogue of the factor sequence formula of Buch-Fulton for the cohomological quiver polynomials (of equioriented type A).

MSC:
14M15 Grassmannians, Schubert varieties, flag manifolds
05E15 Combinatorial aspects of groups and algebras (MSC2010)
19E08 \(K\)-theory of schemes
05E05 Symmetric functions and generalizations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Billey S., Jockusch W and Stanley R.P. (1993). Some combinatorial properties of Schubert polynomials. J. Algebraic Comb. 2(4): 345–374 · Zbl 0790.05093 · doi:10.1023/A:1022419800503
[2] Buch A.S. (2002). Grothendieck classes of quiver varieties. Duke Math. J. 115(1): 75–103 · Zbl 1052.14056 · doi:10.1215/S0012-7094-02-11513-0
[3] Buch A.S. (2002). A Littlewood–Richardson rule for the K-theory of Grassmannians. Acta Math. 189(1): 37–78 · Zbl 1090.14015 · doi:10.1007/BF02392644
[4] Buch A.S. (2005). Alternating signs of quiver coefficients. J. Am. Math. Soc. 18(1): 217–237 · Zbl 1061.14050 · doi:10.1090/S0894-0347-04-00473-4
[5] Buch A.S., Fehér L.M. and Rimányi R. (2005). Positivity of quiver coefficients through Thom polynomials. Adv. Math. 197(1): 306–320 · Zbl 1076.14075 · doi:10.1016/j.aim.2004.10.019
[6] Buch A.S. and Fulton W. (1999). Chern class formulas for quiver varieties. Invent. Math. 135(3): 665–687 · Zbl 0942.14027 · doi:10.1007/s002220050297
[7] Buch A.S., Kresch A., Tamvakis H and Yong A. (2005). Grothendieck polynomials and quiver formulas. Am. J. Math. 127(3): 551–567 · Zbl 1084.14048 · doi:10.1353/ajm.2005.0017
[8] Buch A.S., Kresch A., Tamvakis H and Yong A. (2004). Schubert polynomials and quiver formulas. Duke Math. J. 122: 125–143 · Zbl 1072.14067 · doi:10.1215/S0012-7094-04-12214-6
[9] Buch A.S., Sottile F and Yong A. (2005). Quiver coefficients are Schubert structure constants. Math. Res. Lett. 12(4): 567–574 · Zbl 1112.14055
[10] Edelman P. and Greene C. (1987). Balanced tableaux. Adv. Math. 63(1): 42–99 · Zbl 0616.05005 · doi:10.1016/0001-8708(87)90063-6
[11] Fomin S., Gelfand S. and Postnikov A. (1997). Quantum Schubert polynomials. J. Am. Math. Soc 10: 565–596 · Zbl 0912.14018 · doi:10.1090/S0894-0347-97-00237-3
[12] Fomin, S., Greene, C.: Noncommutative Schur functions and their applications. Discret. Math. 193(1–3), 179–200 (1998) Selected papers in honor of Adriano Garsia (Taormina, 1994) · Zbl 1011.05062
[13] Fomin, S., Kirillov, A.N.: Grothendieck polynomials and the Yang-Baxter equation. Proc. Formal Power Series and Alg. Comb. 183–190 (1994)
[14] Fomin, S., Kirillov, A.N.: The Yang-Baxter equation, symmetric functions, and Schubert polynomials. In: Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993), vol. 153, pp. 123–143 (1996) · Zbl 0852.05078
[15] Fulton W. (1999). Universal Schubert polynomials. Duke Math. J. 96(3): 575–594 · Zbl 0981.14022 · doi:10.1215/S0012-7094-99-09618-7
[16] Knutson A., Miller E and Shimozono M. (2006). Four positive formulae for type A quiver polynomials. Invent. Math. 166: 229–325 · Zbl 1107.14046 · doi:10.1007/s00222-006-0505-0
[17] Lascoux, A.: Transition on Grothendieck Polynomials. Physics and Combinatorics, 2000 (Nagoya), pp. 164–179. World Scientific Publishing, River Edge (2001) · Zbl 1052.14059
[18] Lascoux A. and Schützenberger M.-P. (1982). Polynômes de Schubert. Acad C.R. Sci. Paris Ser. I Math. 294(13): 447–450 · Zbl 0495.14031
[19] Lascoux A. and Schützenberger M.-P. (1982). Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux. C. R. Acad. Sci. Paris Ser. I Math. 295(11): 629–633 · Zbl 0542.14030
[20] Miller E. (2005). Alternating formulae for K-theoretic quiver polynomials. Duke Math. J. 128(1): 1–17 · Zbl 1099.05079 · doi:10.1215/S0012-7094-04-12811-8
[21] Robinson G. de B. (1938). On the representations of the symmetric group. Am. J. Math. 60: 745–760 · Zbl 0019.25102 · doi:10.2307/2371609
[22] Schensted C. (1961). Longest increasing and decreasing subsequences. Can. J. Math. 13: 179–191 · Zbl 0097.25202 · doi:10.4153/CJM-1961-015-3
[23] Stanley R.P. (1984). On the number of reduced decompositions of elements of Coxeter groups. Eur. J. Comb. 5: 359–372 · Zbl 0587.20002
[24] Zelevinskiĭ A.V. (1985). Two remarks on graded nilpotent classes. Uspekhi Mat. Nauk 40(1(241)): 199–200
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.