Newton polytopes of fireworks Grothendieck polynomials

Abstract

We show that the support of the Grothendieck polynomial ${\mathfrak{G}}_w$ of any fireworks permutation is as large as possible: a monomial appears in ${\mathfrak{G}}_w$ if and only if it divides ${\mathbf{x}}^{\mathrm{wt}(\overline{D(w)})}$ and is divisible by some monomial appearing in the Schubert polynomial ${\mathfrak{S}}_w$. Our formula implies that the homogenization of ${\mathfrak{G}}_w$ has M-convex support. We also show that for any fireworks permutation $w\in S_n$, there exists a layered permutation $\pi (w)\in S_n$ so that $\mathrm{supp}({\mathfrak{G}}_{\pi (w)})\supseteq \mathrm{supp}({\mathfrak{G}}_w)$.

17 pages, 8 figures. v3: final version, to appear at Comb. Theory