Chevalley Polytopes and Newton-Okounkov Bodies

We construct a family of polytopes, which we call Chevalley polytopes, associated to homogeneous spaces $X=G\mathrm{/}P$ in their projective embeddings $X\hookrightarrow \mathbb{P}(V_{\varpi })$ together with a choice of reduced expression for the minimal coset representative $w^P$ of $w_0$ in $W\mathrm{/}{W}_P$. When $X$ is minuscule in its minimal embedding, we describe our construction in terms of order polytopes of minuscule posets and use the associated combinatorics to show that minuscule Chevalley polytopes are Newton-Okounkov bodies for $X$ and that the Plücker coordinates on $X$ form a Khovanskii basis for $\mathbb{C}[X]$. We conjecture similar properties for general $X$ and general embeddings $X\hookrightarrow \mathbb{P}(V_{\varpi })$, along with a remarkable decomposition property which we consider as a polytopal shadow of the Littlewood-Richardson rule. We highlight a connection between Chevalley polytopes and string polytopes and give examples where Chevalley polytopes possess better combinatorial properties than string polytopes. We conclude with several examples further illustrating and supporting our conjectures.