Crystal bases and Newton–Okounkov bodies

Abstract

Let $G$ be a connected reductive algebraic group. We prove that the string parameterization of a crystal basis for a finite-dimensional irreducible representation of $G$ extends to a natural valuation on the field of rational functions on the flag variety $G/B$, which is a highest-term valuation corresponding to a coordinate 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 is closely related to an earlier result of Okounkov for the Gelfand–Cetlin polytopes of the symplectic group. As a corollary, we recover a multiplicativity property of the canonical basis due to Caldero. We generalize the results to spherical varieties. From these the existence of SAGBI bases for the homogeneous coordinate rings of flag and spherical varieties, as well as their toric degenerations, follow recovering results by Alexeev and Brion, Caldero, and the author.