Crystal bases and Newton–Okounkov bodies

Kaveh, Kiumars

doi:10.1215/00127094-3146389

Cited by 91 publications

(143 citation statements)

References 41 publications

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“…This definition generalizes the notion of a SAGBI (Subalgebra Analogue of Gröbner Basis for Ideals) basis for a subalgebra of a polynomial ring, and was introduced in [Ka11] and [Ma11]. In general a Khovanskii basis may or may not be finite.…”

Section: A Toric Degeneration Associated To a Valuationmentioning

confidence: 99%

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Integrable systems, toric degenerations and Okounkov bodies

2015

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Abstract. Let X be a smooth projective variety of dimension n over C equipped with a very ample Hermitian line bundle L. In the first part of the paper, we show that if there exists a toric degeneration of X satisfying some natural hypotheses (which are satisfied in many settings), then there exists a surjective continuous map from X to the special fiber X 0 which is a symplectomorphism on an open dense subset U . From this we are then able to construct a completely integrable system on X in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions {H 1 , . . . , Hn} on X which are continuous on all of X, smooth on an open dense subset U of X, and pairwise Poisson-commute on U . Moreover, our integrable system in fact generates a Hamiltonian torus action on U . In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the 'moment map' µ = (H 1 , . . . , Hn) : X → R n is precisely the Newton-Okounkov body ∆ = ∆(R, v) associated to the homogeneous coordinate ring R of X, and an appropriate choice of a valuation v on R. Our main technical tools come from algebraic geometry, differential (Kähler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Lojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties X, this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.

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Section: A Toric Degeneration Associated To a Valuationmentioning

confidence: 99%

“…The key ingredient in his construction is a multiplicativity property of the (dual) canonical basis with respect to the string parametrization. In [Ka11] it is shown that Anderson's toric degeneration recounted in Section 8 is a generalization of Caldero's construction.…”

Section: Examplesmentioning

confidence: 99%

Integrable systems, toric degenerations and Okounkov bodies

2015

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“…Proposition 5.7 (see [Kav,Theorem 4.1, Corollary 4.2, and Remark 4.6]). Let i ∈ I r be a reduced word for w ∈ W , and λ a dominant integral weight.…”

Section: 2mentioning

confidence: 99%

“…The following is a fundamental property of valuations. Proposition 3.2 (see, for instance, [Kav,Proposition 1.8]). Let v be a valuation on R. Assume that σ 1 , .…”

Section: Let Us Write Zmentioning

confidence: 99%

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

Fujita

Naito

2016

Math. Z.

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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 NewtonOkounkov 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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Schubert Calculus on Newton–Okounkov Polytopes

Kiritchenko

Padalko²

2022

Springer Proceedings in Mathematics &Amp; Statistics

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Crystal bases and Newton–Okounkov bodies

Cited by 91 publications

References 41 publications

Integrable systems, toric degenerations and Okounkov bodies

Integrable systems, toric degenerations and Okounkov bodies

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

Schubert Calculus on Newton–Okounkov Polytopes

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