Associated to each polynomial $f$ is the Newton polytope ${\mathrm{Newt}}(f)$, which is the convex hull of the exponent vectors occurring in the monomials in $f$. A polynomial has [**saturated Newton polytope**]{} if every lattice point appearing in the Newton polytope corresponds to the exponent vector of a monomial in $f$ with nonzero coefficient [@Monical-Tokcan-Yong].

If a polynomial $f$ has saturated Newton polytope, then checking if a monomial has nonzero coefficient is equivalent to checking if the corresponding integer lattice point is in the Newton polytope. Adve, et al. [@Adve-Robichaux-Yong] use this perspective to study the computational complexity of the “nonvanishing problem” for polynomials, with a focus on Schubert polynomials.

In this paper, we study Newton polytopes arising from Schur polynomials and a generalization of symmetric Grothendieck polynomials, which we call inflated symmetric Grothendieck polynomials. We denote these polytopes by ${\mathrm{Newt}}(s_\lambda)$ and ${\mathrm{Newt}}(G_{h,\lambda})$, respectively. The polytopes ${\mathrm{Newt}}(s_\lambda)$ and ${\mathrm{Newt}}(G_{1,\lambda})$ have previously been studied by Monical, Tokcan, and Yong [@Monical-Tokcan-Yong] and Escobar and Yong [@Escobar-Yong], but many open questions remain. We are particularly interested in determining when ${\mathrm{Newt}}(s_\lambda)$ and ${\mathrm{Newt}}(G_{h,\lambda})$ are reflexive and when they have the integer decomposition property (IDP), both of which we define in Section \[Section:Background\].

The Ehrhart series 