On the support of Grothendieck polynomials

Abstract

Grothendieck polynomials ${\mathfrak{G}}_w$ of permutations $w\in S_n$ were introduced by Lascoux and Schützenberger 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 Möbius function values of the aforementioned poset with $\hat{0}$ appended. We prove special cases of our conjectures for Grassmannian and fireworks permutations.

15 pages, 7 figures