Newton–Okounkov Polytopes of Flag Varieties for Classical Groups

Abstract

For classical groups $$SL_n(\mathbb {C})$$, $$SO_n(\mathbb {C})$$ and $$Sp_{2n}(\mathbb {C})$$, 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.