For minuscule homogeneous spaces, Chevalley polytopes coincide with order polytopes of minuscule posets, producing Newton-Okounkov bodies and Khovanskii bases; the general case is conjectural.
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Chevalley Polytopes and Newton-Okounkov Bodies
For minuscule homogeneous spaces, Chevalley polytopes coincide with order polytopes of minuscule posets, producing Newton-Okounkov bodies and Khovanskii bases; the general case is conjectural.