Gelfand–Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes as marked poset polytopes

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–Vinberg polytopes (2010, 2005), 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–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras.