(right) drawn in $\mathbb{R}^2$ rather than in a $2$-dimensional subspace of $\mathbb{R}^3$. Both of these polytopes can be shown to be reflexive if we translate them so that their unique interior point is $(0,0)$.[]{data-label="fig:schur"}](SageImage-schur_21_3-reflexive-hexagon.png "fig:"){height="3.5cm"}

In Section \[sec:Reflexive\_Schur\], we characterize which Newton polytopes arising from Schur polynomials are reflexive. Figure \[fig:Reflexive-Nonreflexive-BG\] illustrates examples of reflexive and nonreflexive ${\mathrm{Newt}}(s_\lambda(\x))$, generated using Normaliz [@Normaliz] and SageMath [@sagemath].

![The polytope ${\mathrm{Newt}}(s_{(2,1,1,0)}(\x))$, on the left, is reflexive and the polytope ${\mathrm{Newt}}(s_{(2,1,0,0)}(\x))$, on the right, is not reflexive.[]{data-label="fig:Reflexive-Nonreflexive-BG"}](SageImage_Schur_211_4_shift-Reflexive.png "fig:"){height="5.5cm"} ![The polytope ${\mathrm{Newt}}(s_{(2,1,1,0)}(\x))$, on the left, is reflexive and the polytope ${\mathrm{Newt}}(s_{(2,1,0,0)}(\x))$, on the right, is not reflexive.[]{data-label="fig:Reflexive-Nonreflexive-BG"}](SageImage_Schur_21_4-Non-reflexive.png "fig:"){height="5.5cm"}

The Newton polytope ${\mathrm{Newt}}(s_\lambda(\x))$ of a Schur polynomial can be realized as the $(m-1)$-dimensional [**$\lambda$-permutohedron**]{} $\mathcal{P}_\lambda^m$ in $\mathbb{R}^m$, which is the convex hull of the $S_m$-orbit of $(\lambda_1,\ldots,\lambda_m)\in \mathbb{R}^m$.

Symmetric Grothendieck polynomials can be thought of as an inhomogeneous analogue of Schur polynomials. The following definition is due to Lenart [@Lenart Theorem 2.2].

Let $\x=(x_1,\ldots, x_m)$ and let $\lambda$ be a partition with at most $m$ parts. For any partition $\mu\supseteq \lambda$ with at most $m$ rows, let $a_{\lambda\mu}$ be the number of fillings of the skew shape $\mu/ \lambda$ such that the filling increases strictly along 