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Polyhedral realizations of crystal bases and convex-geometric Demazure operators

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Abstract

The main object in this paper is a family of rational convex polytopes whose lattice points give a polyhedral realization of a highest weight crystal basis. Every polytope in this family is identical to a Newton–Okounkov body of a flag variety, and it gives a toric degeneration. In this paper, we prove that a specific class of polytopes in this family is given by Kiritchenko’s Demazure operators on polytopes. This implies that polytopes in this class are all lattice polytopes. As an application, we give a sufficient condition for the corresponding toric variety to be Gorenstein Fano.

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Acknowledgements

The author is greatly indebted to Satoshi Naito for numerous helpful suggestions and fruitful discussions. The author would also like to express his gratitude to Dave Anderson and Valentina Kiritchenko for useful comments and suggestions. At the conference “Algebraic Analysis and Representation Theory” in June 2017, the author gave a poster presentation on the result of this paper. But there was a gap in the proof at that time, and the condition of the main result has been corrected from the one at the conference.

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  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Naoki Fujita

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Correspondence to Naoki Fujita.

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The work was partially supported by Grant-in-Aid for JSPS Fellows (No. 16J00420).

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Fujita, N. Polyhedral realizations of crystal bases and convex-geometric Demazure operators. Sel. Math. New Ser. 25, 74 (2019). https://doi.org/10.1007/s00029-019-0522-7

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