Gelfand-Zetlin polytopes and flag varieties

TL;DR

This paper establishes a geometric correspondence between Schubert cycles in flag varieties and faces of Gelfand-Zetlin polytopes, providing a polyhedral interpretation of Schubert calculus.

Contribution

It introduces a novel geometric construction linking Schubert cycles to Gelfand-Zetlin polytope faces, extending the combinatorial understanding of flag varieties.

Findings

Faces of Gelfand-Zetlin polytopes correspond to Schubert cycles.

Classical Chevalley formula interpreted via Gelfand-Zetlin polytopes.

Resembles toric variety polytope structures.

Abstract

I construct a correspondence between the Schubert cycles on the variety of complete flags in C^n and some faces of the Gelfand-Zetlin polytope associated with the irreducible representation of SL_n(C) with a strictly dominant highest weight. The construction is based on a geometric presentation of Schubert cells by Bernstein-Gelfand-Gelfand 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.