Proof of a conjectured M\"obius inversion formula for Grothendieck polynomials

TL;DR

This paper proves a conjecture that relates Grothendieck polynomials to Schubert polynomials through M"obius inversion on a poset, providing a new computational approach in algebraic geometry.

Contribution

We prove a conjecture that expresses Grothendieck polynomials in terms of Schubert polynomials using M"obius inversion, establishing a new link between these polynomials.

Findings

Confirmed the conjectured M"obius inversion formula for Grothendieck polynomials.

Provided a method to compute Grothendieck polynomials from Schubert polynomials.

Strengthened the theoretical understanding of the relationship between K-theory and cohomology classes.

Abstract

Schubert polynomials $\mathfrak{S}_w$ are polynomial representatives for cohomology classes of Schubert varieties in a complete flag variety, while Grothendieck polynomials $\mathfrak{G}_w$ are analogous representatives for the $K$-theory classes of the structure sheaves of Schubert varieties. In the special case that $\mathfrak{S}_w$ is a multiplicity-free sum of monomials, K. M\'esz\'aros, L. Setiabrata, and A. St. Dizier conjectured that $\mathfrak{G}_w$ can be easily computed from $\mathfrak{S}_w$ via M\"obius inversion on a certain poset. We prove this conjecture.