Gelfand–Tsetlin Polytopes and Feigin–Fourier–Littelmann Polytopes asMarked Poset Polytopes

Gelfand–Tsetlin Polytopes and Feigin–Fourier–Littelmann Polytopes as
Marked Poset Polytopes

Federico Ardila Supported in part by the National Science Foundation CAREER Award DMS-0956178 and the National Science Foundation Grant DMS-0801075.    Thomas Bliem Supported by the Deutsche Forschungsgemeinschaft Grant SPP 1388.    Dido Salazar Supported in part by the National Science Foundation Grant DGE-0841164.
E-mail addresses:\emailfederico@math.sfsu.edu, \emailbliem@math.sfsu.edu, \emaildusalaza@sfsu.edu.
Postal address: Department of Mathematics, San Francisco State University, 1600 Holloway Ave, San Francisco CA 94132, USA.
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 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.

1 Introduction

Consider the simple complex Lie algebra . The irreducible representations of are parametrized up to isomorphism by dominant integral weights, i.e., weakly decreasing -tuples of integers determined up to adding multiples of . Given a dominant integral weight , let denote the corresponding irreducible -module. The module has a distinguished basis, the Gelfand–Tsetlin [gelfand1950] basis, parametrized by the points with integral coordinates (“integral points” or “lattice points” for short) in the Gelfand–Tsetlin polytope .

Recently, Feigin, Fourier, and Littelmann [feigin2010] constructed a different basis of , related to the Poincaré–Birkhoff–Witt basis of the universal enveloping algebra , where is the span of the negative root spaces. Again, the basis elements are parametrized by the integral points in a certain polytope .

Feigin, Fourier, and Littelmann used two subtle algebraic arguments to prove that their basis indeed spans and is linearly independent. When they had only produced the first half of the proof, they asked the second author of this paper:

Question 1.0.

[fourier2010] Is there a combinatorial explanation for the fact that and contain the same number of lattice points?

This question provided the motivation for this paper. We answer it by generalizing a result of Stanley [Stanley86] on poset polytopes, as we now describe. Let be a finite poset. Let be a subset of which contains all minimal and maximal elements of . Let be a vector in , which we think of as a marking of the elements of with real numbers. We call such a triple a marked poset.

Definition 1.0.

The marked order polytope of is

where and represent elements of , and represents an element of . The marked chain polytope of is

where represent elements of , and represent elements of .

For any polytope with integer coordinates there exists a polynomial , the Ehrhart polynomial of , with the following property: for every positive integer , the -th dilate of contains exactly lattice points (see [Stanleybook]). With this notion, our answer to § 1 is given by the following two results.

Theorem 1.1.

For any marked poset with , the marked order polytope and the marked chain polytope have the same Ehrhart polynomial.

Theorem 1.2.

For every partition there exists a marked poset such that and .

We also consider the extension of these constructions to other Lie algebras. Berenstein and Zelevinsky proposed a construction of generalized Gelfand–Tsetlin polytopes [berenstein1989] for other semisimple Lie algebras. For the symplectic and odd orthogonal Lie algebras, their polytopes are also in the family of marked order polytopes. Therefore Theorem 1.1 yields candidates for the Feigin–Fourier–Littelmann polytopes in types and .

The paper is organized as follows. In § 2 we discuss the relevant aspects of the representation theory of the simple complex Lie algebras . Section 3 treats marked order and chain polytopes, and gives a bijection between their lattice points. Section LABEL:sec:applications discusses the application of the combinatorial results of § 3 to the representation theoretic polytopes that interest us.

We note that the combinatorial § 3 is self-contained, and may be of independent interest beyond the representation theoretic application. A possible way to read this article is to skip § 2 and continue there directly.

2 Preliminaries

Consider the simple complex Lie algebra . Let be the Cartan subalgebra consisting of its diagonal matrices. For , let denote the projection onto the -th diagonal component. As , the coefficient vector of an integral weight is only determined as an element of . We identify an integral weight with the corresponding equivalence class of coefficient vectors. If is a weight and we use the symbol in a context where it has to be interpreted as an -tuple , we use the convention that a representative has been chosen implicitly. Fix simple roots for . The corresponding fundamental weights are . Hence dominant integral weights correspond to weakly decreasing -tuples of integers, or partitions.

Given a dominant integral weight , the associated Gelfand–Tsetlin [gelfand1950] polytope is defined as follows: Consider the board given in Figure 1.

Figure 1: Board defining Gelfand–Tsetlin patterns.

Each one of the empty boxes stands for a real variable. The polytope is given by the fillings of the board with real numbers with the following property: each number is less than or equal to its upper left neighbor and greater than or equal to its upper right neighbor. Note that the ambiguity in choosing an -tuple for the weight amounts to an integral translation of , and hence does not affect its number of integral points. In fact, the integral points in parametrize the Gelfand–Tsetlin basis of , hence .

Feigin, Fourier, and Littelmann [feigin2010] associate a different polytope with a dominant integral weight as follows: The positive roots of are , where . A Dyck path is by definition a sequence in such that and are simple, and if , then either or . Denote the coordinates on by for . Let . Then the polytope is given by the inequalities

for all and

for all Dyck paths such that and .

For all , let be a nonzero element of the root space . Let be a highest weight vector of . Fix any total order on . As ranges over the lattice points of , the elements form a basis of [feigin2010, Th. 3.11]. Hence .

The previous discussion shows that . In the sequel, we give a combinatorial explanation and an extension of this fact.

3 Marked poset polytopes

To any finite poset , Stanley [Stanley86] associated two polytopes in : the order polytope and the chain polytope. He showed that there is a continuous, piecewise linear bijection between them, which restricts to a bijection between their sets of integral points. In this section we construct a generalization of the order and chain polytopes, and prove the analogous result. We begin with a review of Stanley’s work.

3.1 Stanley’s order and chain polytopes

Let be a finite poset. For we say that covers , and write , when and there is no with . We identify with its Hasse diagram: the graph with vertex set , having an edge going down from to whenever covers .

The order polytope and chain polytope of are,

respectively.

Stanley proved that, even though and can have quite different combinatorial structures, they have the same Ehrhart polynomial. He did this as follows. Define the transfer map by

(0)

for , . Then:

Theorem 3.1 ([Stanley86, Theorem 3.2]).

The transfer map restricts to a continuous, piecewise linear bijection from onto . For any , restricts to a bijection from onto .

3.2 Marked poset polytopes

We now recall the definition of marked order and chain polytopes, and prove that they satisfy a generalization of Theorem 3.1.

An element of a poset is called extremal if it is maximal or minimal.

Definition 3.1.

A marked poset consists of a finite poset , a subset containing all its extremal elements, and a vector . We identify it with the marked Hasse diagram, where we label the elements with in the Hasse diagram of .

Figure 2: A marked Hasse diagram defining a partial order on the set with and .
Definition 3.1.

The marked order polytope of is

where and represent elements of , and represents an element of . The marked chain polytope of is

where represent elements of , and represent elements of .

Figure 3: The marked order polytope of the marked poset in Figure 2 is given by the inequalities and . The marked chain polytope is given by the inequalities , , , , and . Note that they are not combinatorially isomorphic.

Stanley’s construction is a special case of ours as follows: Given any finite poset , add a new smallest and largest element to obtain for . Let and . Then

The following definitions will be needed in the proof of Theorem 3.2: The length of a chain is . The height of is the length of the longest chain ending at . If is graded, the height of an element is just its rank.

Theorem 3.2.

Let be a marked poset. The map defined by

for each restricts to a continuous, piecewise affine bijection from onto .

The following alternative description of may be useful. Let be Stanley’s transfer map as defined in (3.1). Let be the canonical projection which forgets the coordinates in , and let be the canonical inclusion into the fiber over , which adds a coordinate to each . Then .

These maps (and some more to be defined in the proof) are illustrated in the following diagram.

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