shall show that the partitions $\theta(1),\ldots, \theta(t)$ are in $A(h,\lambda)$. By Lemma \[lem.dominance\](c), it suffices to show that $\lambda \subseteq \theta(i) \subseteq \lambda^{(N)}$, where $\deg G_{h,\lambda}(\x) = |\lambda|+N$.

The tableau $T_i$ is comprised of every $t$-th column of $T$, so its shape is $$\theta(i)=(\lambda_1 + \ell_{i,1}, \ldots, \lambda_m + \ell_{i,m})$$ for some nonnegative integers $\ell_{i,1},\ldots, \ell_{i,m}$. Thus $\theta(i)\supseteq \lambda$.

Now, $\nu^{(k)}\in A(th,t\lambda)$, so Proposition \[prop.dilate\] gives $$0\leq \nu_r^{(k)} - t\lambda_r  \leq ta_r,$$ for each $r=1,\ldots, m$, where $\lambda^{(N)} = (\lambda_1 +a_1,\ldots, \lambda_m+a_m)$. Again, since the partition $\theta(i)$ is comprised of every $t$-th column $\nu^{(k)}$, then this implies $$0 \leq \theta(i)_r - \lambda_r \leq a_r$$ for each $r=1,\ldots, m$. Therefore, $\theta(i)\subseteq \lambda^{(N)}$. So by Lemma \[lem.dominance\](c), $\theta(i)\in A(h,\lambda)$.

Finally, let $k_i = \ell_{i,1}+\cdots+\ell_{i,m}$. Since $\theta(i)\vdash |\lambda|+k_i$ is a partition in $A(h,\lambda)$, then $\lambda^{(k_i)} \unrhd \theta(i)$. If $\bv_i$ denotes the content vector of $T_i$, then $\bv_i$ is a lattice point in ${\mathrm{Newt}}(s_{\lambda^{(k_i)}}(\x))$, since ${\mathrm{Newt}}(s_{\lambda^{(k_i)}}(\x))$ has IDP. As $\p = \bv_1 + \cdots + \bv_t$, then ${\mathrm{Newt}}(G_{h,\lambda}(\x))$ has IDP.

The Newton polytope ${\mathrm{Newt}}(G_\lambda(\x))$ has the integer decomposition property.

Reflexivity
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Reflexive and Gorenstein Newton polytopes of Schur polynomials {#sec:Reflexive_Schur}
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To characterize which Newton polytopes arising from Schur polynomials are reflexive, we first 