Newton-Okounkov polytopes of flag varieties

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.