Newton-Okounkov polytopes of flag varieties

TL;DR

This paper computes Newton-Okounkov bodies for line bundles on complete flag varieties of GL_n, revealing they match Feigin-Fourier-Littelmann-Vinberg polytopes, thus connecting geometric valuations with known combinatorial polytopes.

Contribution

It introduces a specific geometric valuation from translated Schubert subvarieties and shows the resulting Newton-Okounkov bodies coincide with established polytopes in type A.

Findings

Newton-Okounkov bodies match Feigin-Fourier-Littelmann-Vinberg polytopes

Valuation from translated Schubert subvarieties is effective

Connects geometric valuations with combinatorial polytopes

Abstract

We compute the Newton--Okounkov bodies of line bundles on the complete flag variety of GL_n for a geometric valuation coming from a flag of translated Schubert subvarieties. The Schubert subvarieties correspond to the terminal subwords in the decomposition (s_1)(s_2s_1)(s_3s_2s_1)(...)(s_{n-1}...s_1) of the longest element in the Weyl group. The resulting Newton--Okounkov bodies coincide with the Feigin--Fourier--Littelmann--Vinberg polytopes in type A.