We prove that for every complex classical group G the string polytope associated to a special reduced decomposition and any dominant integral weight λ will be a lattice polytope if and only if the highest weight representation of the Lie algebra of G with highest weight λ integrates to a representation of G itself. This affirms an earlier conjecture and shows that every partial flag variety of a complex classical group admits a flat projective degeneration to a Gorenstein Fano toric variety.

Assume that the valuation semigroup Γ ( λ ) \Gamma (\lambda ) of an arbitrary partial flag variety corresponding to the line bundle L λ \mathcal {L_\lambda } constructed via a full-rank valuation is finitely generated and saturated. We use Ehrhart theory to prove that the associated Newton–Okounkov body — which happens to be a rational, convex polytope — contains exactly one lattice point in its interior if and only if L λ \mathcal {L}_\lambda is the anticanonical line bundle. Furthermore, we use this unique lattice point to construct the dual polytope of the Newton–Okounkov body and prove that this dual is a lattice polytope using a result by Hibi. This leads to an unexpected, necessary and sufficient condition for the Newton–Okounkov body to be reflexive.