Abstract
We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of Fomin and Kirillov (Proc Formal Power Series Alg Comb, 1994) in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as new proofs of the rule of Fomin and Greene (Discret Math 193:565–596, 1998) for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood–Richardson rule of Buch (Acta Math 189(1):37–78, 2002). The proof is based on a generalization of the Robinson–Schensted and Edelman–Greene insertion algorithms. Our results are applied to prove a number of new formulas and properties for K-theoretic quiver polynomials, and the Grothendieck polynomials of Lascoux and Schützenberger (C R Acad Sci Paris Ser I Math 294(13):447–450, 1982). In particular, we provide the first K-theoretic analogue of the factor sequence formula of Buch and Fulton (Invent Math 135(3):665–687, 1999) for the cohomological quiver polynomials.
Similar content being viewed by others
References
Billey S., Jockusch W and Stanley R.P. (1993). Some combinatorial properties of Schubert polynomials. J. Algebraic Comb. 2(4): 345–374
Buch A.S. (2002). Grothendieck classes of quiver varieties. Duke Math. J. 115(1): 75–103
Buch A.S. (2002). A Littlewood–Richardson rule for the K-theory of Grassmannians. Acta Math. 189(1): 37–78
Buch A.S. (2005). Alternating signs of quiver coefficients. J. Am. Math. Soc. 18(1): 217–237
Buch A.S., Fehér L.M. and Rimányi R. (2005). Positivity of quiver coefficients through Thom polynomials. Adv. Math. 197(1): 306–320
Buch A.S. and Fulton W. (1999). Chern class formulas for quiver varieties. Invent. Math. 135(3): 665–687
Buch A.S., Kresch A., Tamvakis H and Yong A. (2005). Grothendieck polynomials and quiver formulas. Am. J. Math. 127(3): 551–567
Buch A.S., Kresch A., Tamvakis H and Yong A. (2004). Schubert polynomials and quiver formulas. Duke Math. J. 122: 125–143
Buch A.S., Sottile F and Yong A. (2005). Quiver coefficients are Schubert structure constants. Math. Res. Lett. 12(4): 567–574
Edelman P. and Greene C. (1987). Balanced tableaux. Adv. Math. 63(1): 42–99
Fomin S., Gelfand S. and Postnikov A. (1997). Quantum Schubert polynomials. J. Am. Math. Soc 10: 565–596
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)
Fomin, S., Kirillov, A.N.: Grothendieck polynomials and the Yang-Baxter equation. Proc. Formal Power Series and Alg. Comb. 183–190 (1994)
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)
Fulton W. (1999). Universal Schubert polynomials. Duke Math. J. 96(3): 575–594
Knutson A., Miller E and Shimozono M. (2006). Four positive formulae for type A quiver polynomials. Invent. Math. 166: 229–325
Lascoux, A.: Transition on Grothendieck Polynomials. Physics and Combinatorics, 2000 (Nagoya), pp. 164–179. World Scientific Publishing, River Edge (2001)
Lascoux A. and Schützenberger M.-P. (1982). Polynômes de Schubert. Acad C.R. Sci. Paris Ser. I Math. 294(13): 447–450
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
Miller E. (2005). Alternating formulae for K-theoretic quiver polynomials. Duke Math. J. 128(1): 1–17
Robinson G. de B. (1938). On the representations of the symmetric group. Am. J. Math. 60: 745–760
Schensted C. (1961). Longest increasing and decreasing subsequences. Can. J. Math. 13: 179–191
Stanley R.P. (1984). On the number of reduced decompositions of elements of Coxeter groups. Eur. J. Comb. 5: 359–372
Zelevinskiĭ A.V. (1985). Two remarks on graded nilpotent classes. Uspekhi Mat. Nauk 40(1(241)): 199–200
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Buch, A.S., Kresch, A., Shimozono, M. et al. Stable Grothendieck polynomials and K-theoretic factor sequences. Math. Ann. 340, 359–382 (2008). https://doi.org/10.1007/s00208-007-0155-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-007-0155-6