A comparison of Newton-Okounkov polytopes of Schubert varieties

TL;DR

This paper demonstrates that two different classes of valuations on Schubert varieties produce identical Newton-Okounkov bodies, linking algebraic and geometric perspectives through perfect bases and explicit transformations.

Contribution

It proves the equivalence of algebraic and geometric valuations on Schubert varieties and shows their Newton-Okounkov bodies coincide via an explicit affine transformation.

Findings

Algebraic and geometric valuations are identical on a perfect basis.

Newton-Okounkov bodies from these valuations coincide.

Explicit affine transformation relates the bodies.

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.