Decompositions of Grothendieck Polynomials

TL;DR

This paper introduces a new combinatorial basis and positive formulas for Grothendieck polynomials, facilitating the understanding of their multiplication in K-theory of flag varieties and extending to connective K-theory.

Contribution

It provides a positive combinatorial Littlewood-Richardson rule for Grothendieck polynomials and introduces glide polynomials as a new basis with broad applications.

Findings

Derived a positive combinatorial formula for Grothendieck polynomial expansion in glide basis.

Established a Littlewood-Richardson rule for multiplying Grothendieck polynomials.

Extended techniques to $eta$-Grothendieck polynomials and related bases.

Abstract

We investigate the longstanding problem of finding a combinatorial rule for the Schubert structure constants in the $K$-theory of flag varieties (in type $A$). The Grothendieck polynomials of A. Lascoux-M.-P. Sch\"{u}tzenberger (1982) serve as polynomial representatives for $K$-theoretic Schubert classes; however no positive rule for their multiplication is known outside the Grassmannian case. We contribute a new basis for polynomials, give a positive combinatorial formula for the expansion of Grothendieck polynomials in these glide polynomials, and provide a positive combinatorial Littlewood-Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the $\beta$-Grothendieck polynomials of S. Fomin-A. Kirillov (1994), representing classes in connective $K$-theory, and we state our results in this more general context. A specialization of the glide basis recovers the fundamental slide polynomials of S. Assaf-D. Searles (2016), which play an analogous role with respect to the Chow ring of flag varieties. Additionally, the stable limits of another specialization of glide polynomials are T. Lam-P. Pylyavskyy's (2007) basis of multi-fundamental quasisymmetric functions, $K$-theoretic analogues of I. Gessel's (1984) fundamental quasisymmetric functions. Those glide polynomials that are themselves quasisymmetric are truncations of multi-fundamental quasisymmetric functions and form a basis of quasisymmetric polynomials.