Stable Grothendieck polynomials and K-theoretic factor sequences

TL;DR

This paper introduces a new combinatorial rule for expanding stable Grothendieck polynomials, generalizing previous results and providing new proofs and formulas in K-theory related to quiver and Grothendieck polynomials.

Contribution

It presents a nonrecursive combinatorial rule for stable Grothendieck polynomial expansion, generalizing prior rules and introducing new K-theoretic formulas and properties.

Findings

New combinatorial rule for stable Grothendieck polynomial expansion

Generalization of insertion algorithms for proofs

First K-theoretic factor sequence formula for 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.