  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 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 A. Okounkov for the Gelfand-Cetlin polytopes of
the symplectic group. As a corollary we recover 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 spherical varieties, as well as their toric
degenerations follow, recovering previous results of Caldero, Alexeev-Brion and
the author.
