Newton-Okounkov polytopes of flag varieties for classical groups

Abstract: For classical groups SL(n), SO(n) and Sp(2n), we define uniformly geometric valuations on the corresponding complete flag varieties. The valuation in every type comes from a natural coordinate system on the open Schubert cell and is combinatorially related to the Gelfand-Zetlin pattern in the same type. In types A and C, we identify the corresponding Newton-Okounkov polytopes with the Feigin-Fourier-Littelmann-Vinberg polytopes. In types B and D, we compute low-dimensional examples and formulate open questions.