Gelfand-zetlin Polytopes and Flag Varieties

I construct a correspondence between the Schubert cycles on the variety of complete flags in Cn and some faces of the Gelfand-Zetlin polytope associated with the irreducible representation of SLn(C) with a strictly dominant highest weight. The construction is based on a geometric presentation of Schubert cells by Bernstein–Gelfand– Gelfand [2] using Demazure modules. The correspondence between the Schubert cycles and faces is then used to interpret the classical Chevalley formula in Schubert calculus in terms of the Gelfand-Zetlin polytopes. The whole picture resembles the picture for toric varieties and their polytopes.