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A Newton–Okounkov body is a convex body constructed from a polarized variety with a valuation on its function field. Kaveh (respectively, 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 (respectively, 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 (respectively, 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.

A comparison of Newton–Okounkov polytopes of Schubert varieties