An iterated residue perspective on stable Grothendieck polynomials

TL;DR

This paper introduces the iterated residue technique as a new tool to study stable Grothendieck polynomials, providing new formulas, proofs, and insights into their algebraic and combinatorial properties in $K$-theory.

Contribution

The paper develops the iterated residue method and applies it to derive formulas, straightening laws, and proofs of key properties of stable Grothendieck polynomials, offering a novel approach in the field.

Findings

New formulas for iterated residues and Grothendieck polynomials

Proofs of straightening laws and multiplication formulas

Alternative proofs of the $K$-Pieri rule and sign patterns

Abstract

Grothendieck polynomials are important objects in the study of the $K$-theory of flag varieties. Their many remarkable properties have been studied in the context of algebraic geometry and tableaux combinatorics. We explore a new tool, similar to generating sequences, which we call the iterated residue technique. We prove new formulas on the calculus of iterated residues and use them to prove straightening laws and multiplication formulas for stable Grothendieck polynomials. As a further application of our method, we give new proofs that the $K$-Pieri rule and the expansions of Grothendieck polynomials in the Schur basis both exhibit alternating signs. As a consequence, we observe that our method implies a new combinatorial statement of the $K$-Pieri rule. Our results indicate that the iterated residue technique should be further explored as a new line of attack on open conjectures regarding positivity and stability, for example of quiver polynomials and Thom polynomials, in $K$-theory.