of $\lambda^{(N)}$.

\[proposition\_layers\_of\_polytope\] Let $h$ be a positive integer, and let $\lambda$ be a partition with at most $m$ parts. Suppose $\deg G_{h,\lambda}(\x) = |\lambda|+N$, and let $\lambda^{(0)},\ldots, \lambda^{(N)}$ be the sequence of dominating partitions for $G_{h,\lambda}(\x)$. Further, let $H_k$ be the hyperplane in $\mathbb{R}^m$ defined by $\sum_{i=1}^m x_i = |\lambda|+k$. Then $${\mathrm{Newt}}(G_{h,\lambda}(\x)) \cap H_k = {\mathrm{Newt}}(s_{\lambda^{(k)}}(\x)).$$

For $k\geq0$, if $(p_1,\ldots, p_m) \in H_k$, then $\sum_{i=1}^m p_i = |\lambda|+k$, thus ${\mathrm{Newt}}(G_{h,\lambda}(\x))\cap H_k$ is the convex hull of the content vectors of the partitions $\mu\in A(h,\lambda)$ such that $\mu \vdash |\lambda|+k$.

A result of Rado [@Rado Proposition 2.5] states that $${\mathrm{Newt}}(s_\alpha(\x))\subseteq {\mathrm{Newt}}(s_\beta(\x)) \hbox{ if and only if } \alpha \,\unlhd\, \beta$$ for any two partitions $\alpha, \beta$. By Lemma \[lem.dominance\](a), since $\lambda^{(k)}$ dominates all partitions $\mu\in A(h,\lambda)$ such that $\mu \vdash |\lambda|+k$, then ${\mathrm{Newt}}(s_\mu(\x)) \leq {\mathrm{Newt}}(s_{\lambda^{(k)}}(\x))$, and we conclude that $H_k \cap {\mathrm{Newt}}(G_{h,\lambda}(\x)) = {\mathrm{Newt}}(s_{\lambda^{(k)}}(\x))$.

The intersection of ${\mathrm{Newt}}(G_{h,\lambda}(\x))$ with the hyperplane $H_k$ corresponds to the homogeneous component of $G_{h,\lambda}(\x)$ of degree $|\lambda|+k$.

\[proposition\_snp\_of\_sgp\] The inflated symmetric Grothendieck polynomial $G_{h,\lambda}(\x)$ has SNP.

In [@Escobar-Yong], the proof that $G_{1,\lambda}(\x)=G_\lambda(\x)$ has SNP does not depend on the inflation parameter $h$ other than in [@Escobar-Yong Claim A], which describes the structure of ${\mathrm{Newt}}(G_\lambda(\x))$ arising 