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.

The Minkowski Property and Reflexivity of Marked Poset Polytopes

“…♦ Remark 3.8. Let G = SL n+1 or Sp 2n , B be a Borel subgroup in G, and G/B be the complete flag variety embedded in P(V (2ρ)) (embedding using the anti-canonical bundle on G/B) where 2ρ is the sum of positive roots in G. As shown in [14,6] and [10], there exist flat toric degenerations of G/B to the toric varieties associated to the marked order polytopes and marked chain polytopes associated to Gelfand-Tsetlin posets and marking is given by 2ρ (see [1,13] for the definition of the posets). By Theorem 3.4, these toric varieties are Gorenstein and Fano.…”

A Continuous Family of Marked Poset Polytopes

“…This conjecture was proved by Feigin, Littelmann and the second author in 2010 [6]. As observed by Ardila, Bliem and Salazar [1] shortly after, these two families of polytopes are related in a similar way as Stanley's order and chain polytopes. They introduced marked order and marked chain polytopes, i.e., one fixes a subset of poset elements consisting of at least all extremal elements, provides an integeral marking for them and obtains defining inequalities from covering relations between any elements and chains between marked elements for marked order and marked chain polytopes, respectively.…”

“…In contrast to the transfer maps defined in [16,Theorem 3.2] and [1,Theorem 3.4], the inverse transfer map ψ t in Theorem 2.1 is given using a recursion. Unfolding the recursion, we might as well express the inverse transfer map for p ∈P in the closed form ψ t (y) p = max c (t p 1 · · · t pr y p 0 + t p 2 · · · t pr y p 1 + · · · + y pr ) , where the maximum ranges over all saturated chains c : p 0 ≺ p 1 ≺ · · · ≺ p r with p 0 ∈ P * , p i ∈P for i ≥ 1 and r ≥ 0 ending in p r = p.…”

“…Before coming back to marked poset polyhedra, we finish with a result on the f -vectors of continuous degenerations. 1] be a continuous degeneration of polyhedra. We have…”

Degenerate Schubert Varieties in Type A

“…It is easy to see that M a is a cyclic S(a)-module generated by m a , the image of m in When a = n − and M = V (λ) for λ ∈ P + , the S(n − )-module V a (λ) is studied in [14]. A monomial basis of V a (λ), parametrized by the Feigin-Fourier-Littelmann-Vinberg polytope, is constructed in [1]; and the defining ideal of the S(n − )-module V a (λ) is made explicit.…”