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Crystal bases and Newton–Okounkov bodies

Let G be a connected reductive algebraic group. We prove that the string parametrization of a crystal basis for a finite dimensional irreducible representation of G coincides with a natural valuation on the field of rational functions on the flag variety G/B, constructed out of a sequence of (translated) Schubert varieties, or equivalently a coordi- nate system on a Bott-Samelson variety. This shows that the string polytopes associated to irreducible representations, can be realized as Newton-Okounkov bodies for the flag variety. This fully generalizes an earlier result of A. Okounkov for the Gelfand-Cetlin polytopes of the symplectic group. As another corollary we deduce a multiplicativity property of the canonical basis due to P. Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spher- ical varieties, as well as toric degenerations for them follow.