On the support of Grothendieck polynomials

TL;DR

This paper explores the structure of Grothendieck polynomials, conjecturing a poset organization of their nonzero terms and linking coefficients to M"obius functions, with proofs for specific permutation classes.

Contribution

It introduces conjectures about the poset structure of Grothendieck polynomial terms and their coefficients, providing proofs for Grassmannian and fireworks permutations.

Findings

Conjecture: exponents of nonzero terms form a poset isomorphic to a subposet of a7^n.

Conjecture: coefficients relate to Mb6bius function values of the poset.

Proved special cases for Grassmannian and fireworks permutations.

Abstract

Grothendieck polynomials $\mathfrak{G}_w$ of permutations $w\in S_n$ were introduced by Lascoux and Sch\"utzenberger in 1982 as a set of distinguished representatives for the K-theoretic classes of Schubert cycles in the K-theory of the flag variety of $\mathbb{C}^n$. We conjecture that the exponents of nonzero terms of the Grothendieck polynomial $\mathfrak{G}_w$ form a poset under componentwise comparison that is isomorphic to an induced subposet of $\mathbb{Z}^n$. When $w\in S_n$ avoids a certain set of patterns, we conjecturally connect the coefficients of $\mathfrak{G}_w$ with the M\"obius function values of the aforementioned poset with $\hat{0}$ appended. We prove special cases of our conjectures for Grassmannian and fireworks permutations.