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arXiv · 1603.01346

Newton-Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases

Naoki Fujita·Satoshi Naito
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Abstract

A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is deeply connected with representation theory. For instance, the Littelmann string polytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes are examples of Newton-Okounkov convex bodies. In this paper, we prove that the Newton-Okounkov convex body of a Schubert variety with respect to a specific valuation is identical to the Nakashima-Zelevinsky polyhedral realization of a Demazure crystal. As an application of this result, we show that Kashiwara's involution (*-operation) corresponds to a change of valuations on the rational function field.

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Naoki Fujita, Satoshi Naito. 2016-03-04. Newton-Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases. https://arxiv.org/abs/1603.01346

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