A comparison of Newton-Okounkov polytopes of Schubert varieties

Abstract

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.

21 pages, to appear in J. London Math. Soc. (2)

References in corpus (6)

• 2-Kac-Moody algebras
• Convex bodies and algebraic equations on affine varieties
• Newton-Okounkov bodies of Bott-Samelson varieties and Grossberg-Karshon twisted cubes
• Newton-Okounkov polytopes of flag varieties
• Newton-Okounkov bodies for Bott-Samelson varieties and string polytopes for generalized Demazure modules
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Cited by in corpus (8)

• Quantum twist maps and dual canonical bases
• Newton-Okounkov bodies of flag varieties and combinatorial mutations
• Polyhedral realizations of crystal bases and convex-geometric Demazure operators
• Newton-Okounkov polytopes of Bott-Samelson varieties as Minkowski sums
• Newton-Okounkov polytopes of Schubert varieties arising from cluster structures
• Newton-Okounkov polytopes of flag varieties and marked chain-order polytopes
• Semi-toric degenerations of Richardson varieties arising from cluster structures on flag varieties
• Folding procedure for Newton-Okounkov polytopes of Schubert varieties