On the f-vectors of Gelfand-Cetlin polytopes

TL;DR

This paper provides a new combinatorial description of Gelfand-Cetlin polytopes' face structure, derives a differential equation for their f-vector generating function, and addresses an open problem in the field.

Contribution

It introduces an equivalent face structure description via ladder diagrams and solves an open problem related to the generating function of f-vectors.

Findings

Face structure described by ladder diagrams

Derived differential equation for f-vector generating function

Solved open problem from Gusev, Kritchenko, and Timorin

Abstract

A Gelfand-Cetlin polytope is a convex polytope obtained as an image of certain completely integrable system on a partial flag variety. In this paper, we give an equivalent description of the face structure of a GC-polytope in terms of so called the face structure of a ladder diagram. Using our description, we obtain a partial differential equation whose solution is the exponential generating function of f-vectors of GC-polytopes. This solves the open problem (2) posed by Gusev, Kritchenko, and Timorin in [GKT].