Newton–okounkov Polytopes of Flag Varieties

A Newton-Okounkov body is a convex body constructed from a polarized variety with a valuation on its function field. Kaveh (resp., the first author and Naito) proved that the Newton-Okounkov body of a Schubert variety associated with a specific valuation is identical to the Littelmann string polytope (resp., the Nakashima-Zelevinsky polyhedral realization) of a Demazure crystal. These specific valuations are defined algebraically to be the highest term valuations with respect to certain local coordinate systems on a Bott-Samelson variety. Another class of valuations, which is geometrically natural, arises from some sequence of subvarieties of a polarized variety. In this paper, we show that the highest term valuation used by Kaveh (resp., by the first author and Naito) and the valuation coming from a sequence of specific subvarieties of the Schubert variety are identical on a perfect basis with some positivity properties. The existence of such a perfect basis follows from a categorification of the negative part of the quantized enveloping algebra. As a corollary, we prove that the associated Newton-Okounkov bodies coincide through an explicit affine transformation.Comment: 21 pages, to appear in J. London Math. Soc. (2

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“…In the paper [26], Kiritchenko considered the valuation associated with the sequence of translated Schubert varieties:…”

Section: Introductionmentioning

confidence: 99%

A comparison of Newton–Okounkov polytopes of Schubert varieties

Fujita

Oya

2017

Journal of London Math Soc

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“…The construction relies on the associated string parametrization and the multiplicative properties of the dual canonical bases. These results have been generalized to spherical varieties by Alexeev and Brion [1], see also the articles by Kaveh and Kiritchenko [52,55] for another approach via the framework of Newton-Okounkov bodies [54].…”

Section: Introductionmentioning

confidence: 89%

On toric degenerations of flag varieties

Fang

Fourier

Littelmann

2017

EMS Series of Congress Reports

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“…In types A and C, this fact has both representation theoretic [FFL11I,FFL11II] and combinatorial proofs [ABS]. In type A, there is also a convex geometric proof [Ki17,Section 4]. It would be interesting to check whether this proof extends to type C.…”

Section: Valuations On Flag Varietiesmentioning

confidence: 99%

“…To check whether the polytope P λ := m i P i coincides with the convex body ∆ v (G/B, L λ ) for λ = m i ω i we have to compare their volumes. For instance, one could try to construct a volume preserving piecewise linear map between P λ and the corresponding GZ-polytope in type B extending the construction of [Ki17,Section 4.2].…”

Section: Valuations On Flag Varietiesmentioning

confidence: 99%

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