Proof of a conjectured Möbius inversion formula for Grothendieck polynomials

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. Mészáros, L. Setiabrata, and A. St. Dizier conjectured that ${\mathfrak{G}}_w$ can be easily computed from ${\mathfrak{S}}_w$ via Möbius inversion on a certain poset. We prove this conjecture.

4 pages