Gelfand-Zetlin Polytopes and Flag Varieties

Abstract

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 motivated by the geometric presentation of Schubert cells using Demazure modules due to Bernstein–Gelfand–Gelfand [3]. 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.