On the Support of Grothendieck Polynomials

Grothendieck polynomials \(\mathfrak {G}_w\) of permutations \(w\in S_n\) were introduced by Lascoux and Schützenberger (C R Acad Sci Paris Sér I Math 295(11):629–633, 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 Möbius function values of the aforementioned poset with \(\hat{0}\) appended. We prove special cases of our conjectures for Grassmannian and fireworks permutations