Summary: GelfandTsetlin Polytopes and
FeiginFourierLittelmannVinberg Polytopes as
Marked Poset Polytopes
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 GelfandTsetlin polytopes (1950) and the FeiginFourier
LittelmannVinberg polytopes (2010, 2005), which arise in the representa-
tion theory of the special linear Lie algebra. We then use the generalized
GelfandTsetlin polytopes of Berenstein and Zelevinsky (1989) to propose
conjectural analogues of the FeiginFourierLittelmannVinberg polytopes
corresponding to the symplectic and odd orthogonal Lie algebras.