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. Stanley (1986) a montré que chaque ensemble fini partiellement ordonné permet de définir deux polyèdres, le polyèdre de l'ordre et le polyèdre des cha\^ınes. Ces polyèdres ont le même polynôme de Ehrhart, bien qu'ils soient tout à fait distincts du point de vue combinatoire. On généralise ce résultat à une famille plus générale de polyèdres, construits à partir d'un ensemble partiellement ordonné ayant des entiers attachés à certains de ses éléments. Par cette construction, on explique en termes combinatoires la relation entre les polyèdres de Gelfand-Tsetlin (1950) et ceux de Feigin-Fourier-Littelmann-Vinberg (2010, 2005), qui apparaissent dans la théorie des représentations des algèbres de Lie linéaires spéciales. On utilise les polyèdres de Gelfand-Tsetlin généralisés par Berenstein et Zelevinsky (1989) afin d'obtenir des analogues (conjecturés) des polytopes de Feigin-Fourier-Littelmann-Vinberg pour les algèbres de Lie symplectiques et orthogonales impaires.
Highlights
Feigin, Fourier, and Littelmann [3] constructed a different basis of V (λ ), conjecturally announced by Vinberg [8]
The basis elements are parametrized by the integral points in a certain polytope FFLV(λ ) ⊂ Rn(n−1)/2
Fourier, and Littelmann [3] associate a different polytope with a dominant integral weight λ as follows: The positive roots of sln are Φ+ = {αi, j | 0 ≤ i < j ≤ n}, where αi, j = εi − ε j
Summary
Consider the simple complex Lie algebra sln. The irreducible representations of sln are parametrized up to isomorphism by dominant integral weights, i.e., weakly decreasing n-tuples of integers determined up to adding multiples of (1, . . . , 1). Fourier, and Littelmann used two subtle algebraic arguments to prove that their basis spans V (λ ) and is linearly independent When they had only produced the first half of the proof, they asked the second author of this paper: Question 1.1. For any polytope with integer coordinates Q there exists a polynomial EQ(t), the Ehrhart polynomial of Q, with the following property: for every positive integer n, the n-th dilate nQ of Q contains exactly EQ(n) lattice points (see [7]). With this notion, our answer to Question 1.1 is given by the following two results. A possible way to read this article is to skip §2 and continue there directly
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