Newton–Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases

A Newton–Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is deeply connected with representation theory. For instance, the Littelmann string polytopes and the Feigin–Fourier–Littelmann–Vinberg polytopes are examples of Newton–Okounkov convex bodies. In this paper, we prove that the Newton–Okounkov convex body of a Schubert variety with respect to a specific valuation is identical to the Nakashima–Zelevinsky polyhedral realization of a Demazure crystal. As an application of this result, we show that Kashiwara’s involution ( ∗ -operation) corresponds to a change of valuations on the rational function field.