2 &*(yellow) 3 \\
   *(yellow) 3 
\end{ytableau}\\ \\
(2,4,3) &= (1,2,0) + (1,1,1) + (0,1,2) \end{aligned}$$

The Newton polytope of a symmetric Grothendieck polynomial
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A notable difference between the Newton polytope of Schur polynomials versus symmetric Grothendieck polynomials is that unlike the case of Schur polynomials, $t{\mathrm{Newt}}(G_\lambda(\x)) \neq {\mathrm{Newt}}(G_{t\lambda}(\x))$. Motivated by our study of the integer decomposition property of the Newton polytope of symmetric Grothendieck polynomials, we make the following definition.

Let $h$ be a positive integer. Let $\x=(x_1,\ldots, x_m)$ and let $\lambda \vdash n$ be a partition with at most $m$ parts. For any partition $\mu \supseteq \lambda$ with at most $m$ rows, let $b_{h,\lambda\mu}$ be the number of fillings of the skew shape $\mu/\lambda$ such that the filling increases strictly along each row and each column, and the filling in the $r$-th row is from $\{1,\ldots, h(r-1)\}$. Let $$A(h,\lambda) = \{ \mu \mid b_{h,\lambda\mu}\neq0 \}.$$ The [**inflated symmetric Grothendieck polynomial**]{} indexed by $\lambda$ and $h$ is $$G_{h,\lambda}(\x) = \sum_{\mu\in A(h,\lambda)} (-1)^{|\mu/\lambda|} b_{h,\lambda\mu}s_\mu(\x).$$

Let $\lambda =(2,1,0) \vdash 3$, $m=3$, $h=2$, and $\x=(x_1,x_2,x_3)$. Then $$G_{2,(2,1,0)}(\x) = s_{(2,1,0)}(\x) - (2s_{(2,2,0)}(\x) + 4s_{(2,1,1)}(\x))
   + 8s_{(2,2,1)}(\x) - 11 s_{(2,2,2)}(\x).$$ Compare with Example \[210example\].

Note that $G_{1,\lambda}(\x)=G_\lambda(\x)$ is the usual symmetric Grothendieck polynomial.

