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 $\mathrm{supp}(\mathfrak G_{\pi(w)})\supseteq \mathrm{supp}(\mathfrak G_w)$.