Newton-Okounkov Bodies for Minuscule Homogeneous Spaces and Beyond

Given a triple (X,π,s) consisting of a homogeneous space X=G/P, a dominant weight π giving a projective embedding of X, and a reduced expression s for the minimal coset representative of w_0 in the parabolic quotient W/W_P, we construct a polytope which we call a Chevalley polytope. When π is a minuscule weight, this construction admits a combinatorial interpretation in terms of the minuscule poset associated to π and the Chevalley polytopes in this case are Newton-Okounkov bodies for X satisfying the integer decomposition property. We conjecture similar properties outside of the minuscule case (this is ongoing joint work with Peter Spacek) and illustrate our conjectures with many examples.