# Stable Grothendieck polynomials and K-theoretic factor sequences - Zurich Open Repository and Archive

Buch, A S; Kresch, A; Shimozono, M; Tamvakis, H; Yong, A (2008). Stable Grothendieck polynomials and K-theoretic factor sequences. Mathematische Annalen, 340(2):359-382.

## Abstract

We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of [Fomin-Kirillov '94] in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as new proofs of the rule of [Fomin-Greene '98] for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood-Richardson rule of [Buch '02]. 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-Schutzenberger '82]. In particular, we provide the first $K$-theoretic analogue of the factor sequence formula of [Buch-Fulton '99] for the cohomological quiver polynomials.

## Abstract

We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of [Fomin-Kirillov '94] in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as new proofs of the rule of [Fomin-Greene '98] for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood-Richardson rule of [Buch '02]. 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-Schutzenberger '82]. In particular, we provide the first $K$-theoretic analogue of the factor sequence formula of [Buch-Fulton '99] for the cohomological quiver polynomials.

## Citations

9 citations in Web of Science®
11 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2008 13 Jan 2009 15:38 05 Apr 2016 12:39 Springer 0025-5831 The original publication is available at www.springerlink.com https://doi.org/10.1007/s00208-007-0155-6 http://arxiv.org/abs/math/0601514http://www.ams.org/mathscinet-getitem?mr=2368984