1 October 2015 Crystal bases and Newton–Okounkov bodies
Kiumars Kaveh
Duke Math. J. 164(13): 2461-2506 (1 October 2015). DOI: 10.1215/00127094-3146389


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.


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Kiumars Kaveh. "Crystal bases and Newton–Okounkov bodies." Duke Math. J. 164 (13) 2461 - 2506, 1 October 2015. https://doi.org/10.1215/00127094-3146389


Received: 1 October 2013; Revised: 28 October 2014; Published: 1 October 2015
First available in Project Euclid: 5 October 2015

zbMATH: 06527348
MathSciNet: MR3405591
Digital Object Identifier: 10.1215/00127094-3146389

Primary: 14M15
Secondary: 05E10 , 14M27

Keywords: Bott–Samelson variety , crystal basis , flag variety , Gelfand–Cetlin polytope , Newton–Okounkov body , SAGBI basis , spherical variety , string parameterization , string polytope , Toric Degeneration

Rights: Copyright © 2015 Duke University Press


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Vol.164 • No. 13 • 1 October 2015
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