Gelfand–Tsetlin Polytopes and Feigin–Fourier–Littelmann 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 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.
questiontheorem \aliascntresetthequestion \newaliascntpropositiontheorem \aliascntresettheproposition \newaliascntlemmatheorem \aliascntresetthelemma \newaliascntcorollarytheorem \aliascntresetthecorollary \newaliascntdefinitiontheorem \aliascntresetthedefinition \newaliascntexampletheorem \aliascntresettheexample \newaliascntremarktheorem \aliascntresettheremark \newaliascntconjecturetheorem \aliascntresettheconjecture
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 [5] 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 [3] 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 \thequestion.
[4] 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 [6] 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 \thedefinition.
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 [7]). 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 [1] 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 4 discusses the application of the combinatorial results of 3 to the representation theoretic polytopes that interest us.
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 [5] polytope is defined as follows: Consider the board given in Figure 1.
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 [3] 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 [3, 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 [6] 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
(1) |
for , . Then:
Theorem 3.1 ([6, 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 \thedefinition.
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 .
Definition \thedefinition.
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 .
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 (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.
Proof.
We start by showing that . Let and . Let , and be such that . The definition of implies that for all and . Thus,
Hence, .
To show that is bijective, we construct its inverse . We first define a map , where we define recursively by going up the poset according to the rule:
Since all the elements of height are in , is well-defined. We then define by applying and then forgetting the -coordinates. We will prove that, when restricted to , the map is the inverse of .
First we show that is the identity on . We begin by showing that ; i.e., that if and then . We prove by induction on . The claim certainly holds for . Suppose that we have proved it for all elements of height at most , and let have height . If , then
by definition. Otherwise, if , we have
by the inductive hypothesis. As
we conclude that , as desired.
We have shown that . By composing with the projection which forgets the coordinates, we obtain that is the identity on . Hence is injective.
To prove surjectivity, let and define . We start by showing that . Let . By definition,
As , this implies for all such that . If , this says that . If , this says that . As is arbitrary, it follows that .
Finally, we claim that . Once again, we prove that for all by induction on the height of . For height this statement is vacuous. Suppose that it holds for all elements of height at most , and consider with . Then
as desired. We have shown that is the identity on , hence is surjective.
We conclude that and are inverse functions, and therefore bijective, as we wished to show. The fact that they are continuous and piecewise affine follows directly from the definitions. ∎
We conclude this section with the generalization of the second part of Theorem 3.1, the compatibility of the transfer map with the integral lattice.
Theorem 3.3.
Let be a marked poset with . Then restricts to a bijection between and . Therefore and have the same Ehrhart polynomial.
Proof.
This follows immediately from the proof of Theorem 3.2, as both and preserve integrality. ∎
It is worth noting that Theorem 3.3 does not hold for general .
4 Applications
We now show how marked poset polytopes occur “in nature” in the representation theory of semisimple Lie algebras. More concretely, marked order polytopes occur as Gelfand–Tsetlin polytopes in type , , and , and marked chain polytopes occur as Feigin–Fourier–Littelmann polytopes in type .
4.1 Type .
Let be a dominant integral weight for . Let and be the marked order and chain polytopes determined by the marked Hasse diagram given in Figure 4.
Note that Figure 4 is obtained from Figure 1 by a clockwise rotation by . Hence from the definitions it is immediate that . Similarly, it follows immediately from the definitions that . Hence the equation
is implied by Theorem 3.3.
It would be interesting to see whether the explicit bijection of Theorem 3.3 gives interesting information about the transition matrix between the Gelfand–Tsetlin basis and the Feigin–Fourier–Littelmann basis of .
4.2 Type .
Now consider the symplectic Lie algebra . Here the role of Gelfand–Tsetlin patterns is played by the generalized Gelfand–Tsetlin patterns defined by Berenstein and Zelevinsky [1]. Fix a Cartan subalgebra . Choose simple roots such that for and is the long root. Let be the basis of such that for and . The corresponding fundamental weights are . This is the setting as used by Bourbaki [2]. We identify a weight with the -tuple of its coefficients with respect to the basis . Then dominant integral weights correspond to weakly decreasing -tuples of nonnegative integers. Given a dominant integral weight , Berenstein and Zelevinsky define an -pattern of highest weight to be a filling of the board in Figure 5 with nonnegative integers, such that every number is bounded from above by its upper left neighbor and bounded from below by its upper right neighbor (if any).
They show that is the number of such patterns [1, Th. 4.2].
Let and be the marked order and chain polytopes determined by the marked Hasse diagram given in Figure 6.
Note that Figure 6 is obtained from Figure 5 by a clockwise rotation by and apposition of the zeroes. From the definitions it is immediate that the -patterns of highest weight are the integral points in . This suggests the following:
Conjecture \theconjecture.
Indeed, this conjecture is proved in an article in preparation by Feigin, Fourier, and Littelmann. [4]
4.3 Type .
For the odd orthogonal Lie algebra , the situation is a bit more complicated. Fix a Cartan subalgebra . Choose simple roots such that for and is the short root. Let be the basis of such that for and . The corresponding fundamental weights are for and . This is the setting as used by Bourbaki [2]. We identify a weight with the -tuple of its coefficients with respect to the basis . Then dominant integral weights correspond to weakly decreasing -tuples in such that either all or none of the components are integers. Given a dominant integral weight , Berenstein and Zelevinsky [1] define an -pattern of highest weight to be a filling of the board in Figure 5 with elements of such that every number is bounded from above by its upper left neighbor and bounded from below by its upper right neighbor (if any), and such that all numbers which possess an upper right neighbor are congruent to modulo . Let be the set of -patterns of highest weight .
As in type , let be the marked order polytope defined by the marked Hasse diagram in Figure 6. Then , but does not consist of the integral points, but of the points determined by more complicated congruence conditions. Namely, decompose
where , , and consist of all elements in of height , , and , respectively, that are not contained in . Then consists of all such that for all . Hence consists of all
such that
for all . From the point of view taken in this article, appears to be the most natural candidate to parametrize a PBW basis of [3] in type . Note that the elements of can not appear directly as exponent vectors of a PBW basis, as their components are not necessarily integral, so we are missing at least a change of coordinates in this case.
4.4 Type .
The generalized Gelfand–Tsetlin polytopes [1] for the even orthogonal Lie algebras are not marked order polytopes, so our methods do not apply here. It would be interesting to find a suitable modification of our results to this case.
References
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