Crystal bases and Newton-Okounkov bodies

TL;DR

This paper demonstrates that string parametrizations of crystal bases for reductive groups can be viewed as Newton-Okounkov bodies on flag varieties, linking representation theory with algebraic geometry and toric degenerations.

Contribution

It extends the string parametrization to a valuation on the field of rational functions, connecting crystal bases with Newton-Okounkov bodies and generalizing to spherical varieties.

Findings

String polytopes are realized as Newton-Okounkov bodies.

Establishes a valuation extending crystal basis parametrization.

Recovers multiplicativity of the canonical basis.

Abstract

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.