Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann polytopes as marked poset polytopes

TL;DR

This paper extends Stanley's poset polytope framework to connect Gelfand-Tsetlin and Feigin-Fourier-Littelmann polytopes, providing combinatorial insights and proposing conjectures for other Lie algebra types.

Contribution

It generalizes the relationship between order and chain polytopes to include Gelfand-Tsetlin and Feigin-Fourier-Littelmann polytopes, linking combinatorics and representation theory.

Findings

Unified combinatorial explanation for Gelfand-Tsetlin and Feigin-Fourier-Littelmann polytopes

Identification of polytopes with the same Ehrhart polynomial despite different structures

Proposal of conjectural polytopes for symplectic and orthogonal Lie algebras

Abstract

Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand-Tsetlin polytopes (1950) and the Feigin-Fourier-Littelmann polytopes (2010), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand-Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin-Fourier-Littelmann polytopes corresponding to the symplectic and odd orthogonal Lie algebras.