Newton polytopes of fireworks Grothendieck polynomials

TL;DR

This paper investigates the structure of fireworks Grothendieck polynomials, revealing their support properties and convexity, and establishing relationships with layered permutations to understand their combinatorial and geometric features.

Contribution

It characterizes the support of fireworks Grothendieck polynomials, proves M-convexity of their homogenization, and relates their support to layered permutations, advancing understanding of their combinatorial structure.

Findings

Support of $rak G_w$ is as large as possible, determined by divisibility conditions.

Homogenization of $rak G_w$ has M-convex support.

Existence of layered permutation $ ilde{w}$ with support containing that of $w$.

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