from the homogeneous components of $G_\lambda(\x)$. Using the description of the homogeneous components of $G_{h,\lambda}(\x)$ from Proposition \[proposition\_layers\_of\_polytope\], the rest of the proof in [@Escobar-Yong] applies to arbitrary $h\in \mathbb{Z}_{\geq1}$ and shows that $G_{h,\lambda}(\x)$ has SNP.

We revisit Example \[210example\] from the viewpoint of dominating partitions. (Here $h=1$.)

\[eg.sgp21\] Let $\lambda =(2,1,0)\vdash 3$, $m=3$, and $\x=(x_1,x_2,x_3)$. Then $$G_{(2,1,0)}(\x) = \textcolor{blue}{s_{(2,1,0)}(\x)} - \textcolor{green}{\left(s_{(2,2,0)}(\x) + 2s_{(2,1,1)}(\x)\right)} + \textcolor{purple}{2s_{(2,2,1)}(\x)} - \textcolor{orange}{s_{(2,2,2)}(\x)}.$$ The sequence of dominating partitions for $G_{(2,1,0)}(\x)$ is $$\textcolor{blue}{\lambda^{(0)}=\ydiagram{2,1}}\quad 
\textcolor{green}{\lambda^{(1)}=\ydiagram{2,2}}\quad 
\textcolor{purple}{\lambda^{(2)}=\ydiagram{2,2,1}}\quad 
\textcolor{orange}{\lambda^{(3)}=\ydiagram{2,2,2}}$$

[SageImage\_sym\_groth\_21\_slices\_crop.jpg]{} (100,3)[![The lattice points of ${\mathrm{Newt}}(G_{(2,1,0)}(\mathbf{x}))$ are color-coded to reflect the structure arising from the four homogeneous components of $G_{(2,1,0)}(\mathbf{x})$; see Example \[eg.sgp21\]. The intersection of ${\mathrm{Newt}}(G_{(2,1,0)}(\mathbf{x}))$ with the hyperplane $x_1+x_2+x_3 =4$ is shown on the right. The extreme points are given by the $S_3$-orbit of the dominating partition $\lambda^{(1)}=(2,2,0)$. []{data-label="fig.Grothendieck"}](SageImage_sym_groth_21_slice_crop.png "fig:"){height="5.2cm"}]{}

### Inflated symmetric Grothendieck polynomials and IDP

We now show that the Newton polytopes of inflated symmetric Grothendieck polynomials have IDP. As a corollary, symmetric Grothendieck polynomials have IDP.

By Proposition \[proposition\_layers\_of\_polytope\], we know that ${\mathrm{Newt}}(G_{h,\lambda}(\x))$ is the convex hull of the $S_m$-orbits of the sequence of dominating partitions $\lambda^{(0)}, \ldots, \lambda^{(N)}$, but the next result shows that it suffices to take a certain subset of these partitions.

