Mathematics – Algebraic Geometry
Scientific paper
2011-01-09
Mathematics
Algebraic Geometry
27 pages
Scientific paper
We prove that the Littelmann-Berenstein-Zelevinsky "string parametrization" of a "crystal basis", for an irreducible representation of a connected reductive group G, coincides with a natural geometric valuation on the field of rational functions on the flag variety G/B. This valuation is constructed out of a sequence of (translated) Schubert varieties, or equivalently a coordinate system on a Bott-Samelson variety. It follows that the "string polytopes" associated to irreducible representations, can be realized as Newton-Okounkov bodies for flag variety, fully generalizing 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 dual 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, and hence toric degenerations for them, follow.
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