Séminaire Lotharingien de Combinatoire 49 (2003), Article B49d

LITTELMANN’S REFINED DEMAZURE CHARACTER FORMULA REVISITED

STEEN RYOM-HANSEN

Abstract. We give a purely combinatorial derivation of Littelmann’s refined Demazure character formula.

1. Introduction

The Demazure character formula is a generalization of Weyl’s character formula. It was first stated by Demazure in [D], who showed that it would follow from a certain string property. However, it turned out that this property did not hold in the original setting. The first correct proofs of the formula were therefore given by Andersen, [A] and Ramanan-Ramanathan [RR], using methods closely related to Frobenius splitting.

This work is concerned with the crystal basis approach to the Demazure character formula. In that setting the string property indeed does hold as demonstrated by Kashiwara in [K1]. We briefly review the deduction of the character formula from it.

We then go on to show that the string property can be obtained using only combinatorial properties of the crystals: the Kashiwara operators ẽ_{i}, f̃_{i}, together with the ∗-operation. This is different from the previous deductions of the formula, which use either a representation theoretical interpretation of the formula or appeal to Littelmann’s path models. Our deduction should be contrasted with the remarks following Proposition 6.3.10 in Joseph’s book, [J1].

I would like to thank the referee for many useful suggestions.

2. The refined Demazure character formula

2.1. Let us briefly recall the notion of crystal as introduced by Kashiwara. We refer to [K1,K2,J] for all unexplained notation. Let C := (c_{i,j})_{i,j∈I} be a generalized Cartan matrix. Crystals are certain combinatorial objects associated to C. They consist of a set B with maps ẽ_{i}, f̃_{i} : B → B ∪ {0} and maps ϵ_{i}, φ_{i} : B → ℤ ∪ {−∞}, wt_{i} : B → P, ∀i ∈ I, that satisfy certain conditions.

There is a crystal B(λ) associated to the Weyl module V (λ) of the quantized universal algebra U_{q}(𝔤). The limit crystal is called B(∞).

Given two crystals B_{1} and B_{2} one can make B_{1} × B_{2} into a crystal, which is called the tensor product B_{1} ⊗ B_{2}. For example we have that

f̃_{i}(b_{1} ⊗ b_{2}) = \{ \begin{matrix}f̃_{i}b_{1} ⊗ b_{2} & if φ_{i}(b_{1}) > ϵ_{i}(b_{2}), \\ b_{1} ⊗ f̃_{i}b_{2} & otherwise. \end{matrix}

There is also a sum construction. (But notice that not all crystals arise from the representation theoretical crystals using such constructions).

We shall mainly view crystals as combinatorial objects in the above sense, but shall also appeal to Kashiwara’s ∗-operation on B(∞) (see [K1]). We first of all need the following property: for all i ∈ I there is an injective morphism of crystals Ψ_{i} : B(∞) → B(∞) ⊗ B_{i} where B_{i} is the crystal defined in example 1.2.6. of [K1]. It satisfies the following conditions

Ψ_{i} : u_{∞} ↦ u_{∞} ⊗ b_{i}, (2.1.1)

Ψ_{i}(f̃_{i}^{∗}b) = b′ ⊗ f̃_{i}b′′ where Ψ_{i}(b) = b′ ⊗ b′′, (2.1.2)

f̃_{i}Ψ_{i}(b) = Ψ_{i}(f̃_{i}b) and ẽ_{i}Ψ_{i}(b) = Ψ_{i}(ẽ_{i}b), (2.1.3)

where u_{∞} is the unique element of B_{∞} of weight 0, and where B(∞)⊗B_{i} has the above structure of a tensor product. Joseph has given a purely combinatorial proof of the existence of Ψ_{i}, [J2].

Now, for a reduced expression s_{i_{n}}s_{i_{n−1}} … s_{i_{1}} of the Weyl group element w, we define B_{w}(∞) ⊂ B(∞) and B_{w}(λ) ⊂ B(λ) in the following recursive way

B_{w}(∞) := ⋃_{k} f̃_{i_{n}}^{k}B_{s_{in}w}(∞), B_{1}(∞) := {u_{∞}},

B_{w}(λ) := ⋃_{k} f̃_{i_{n}}^{k}B_{s_{in}w}(λ), B_{1}(λ) := {u_{λ}}.

A priori, these definitions might depend on the choice of reduced expression s_{i_{n}}s_{i_{n−1}} … s_{i_{1}} of w. We shall later show that in fact B_{w}(∞) and B_{w}(λ) are independent of this choice.

2.2. Let D_{i} be the additive operator on ℤ[B(λ)] given by

D_{i}b = \{ \begin{matrix}∑_{0≤k≤wt_{i}(b)} f̃_{i}^{k}b & if & wt_{i}(b) ≥ 0, \\ − ∑_{1≤k≤−wt_{i}(b)−1} ẽ_{i}^{k}b & if & wt_{i}(b) < 0. \end{matrix}

Then the refined Demazure character formula, [K1,L1], is the following equality in ℤ[B(λ)]

(2.2.1)  ∑_{b∈B_{w}(λ)} b = D_{i_{n}}D_{i_{n−1}} … D_{i_{1}}u_{λ}.

The D_{i}’s induce the usual Demazure operators on the group ring of the weight lattice ℤ[P] under the weight map w : ℤ[B(λ)] → ℤ[P]. Thus, if W is finite and we take w = w_{0} the longest element of the Weyl group, (2.2.1) generalizes the original Demazure expression of the Weyl character, see e.g. [A].

2.3. In the rest of this section we shall review Kashiwara’s proof of (2.2.1). The idea is to reduce to the verification of the following three properties of B_{w}(λ):

(1) B_{w}^{∗}(∞) = B_{w^{−1}}(∞),

(2) ẽ_{i}B_{w}(∞) ⊂ B_{w}(∞) ∪ {0} ∀i ∈ I,

(3) f̃_{j}b ∈ B_{w}(∞) ⇒ f̃_{j}^{k}b ∈ B_{w}(∞) ∀b ∈ B_{w}(∞), ∀k ∈ ℕ, ∀j ∈ I .

Let us denote any subset of B(∞) or of B(λ) an i-string if it is of the form

(2.3.1)  S = {f̃_{i}^{k}b | k ≥ 0, where b ∈ B(λ) satisfies ẽ_{i}b = 0}.

We call b the highest weight vector of S. The key “Demazure string property” of these i-strings is then the following: for any i-string S ⊂ B(∞) we have that

(2.3.2)  B_{w}(∞) ∩ S is either S or {b} or the empty set .

This is seen by combining (2) and (3).

2.4. The string property is also valid for B(λ): to see this one defines for λ ∈ P the crystal on one element T_{λ} := {t_{λ}} as follows:

wt_{i}(t_{λ}) = ⟨λ, α_{i}⟩ ε_{i}(t_{λ}) = −∞, φ_{i}(t_{λ}) = −∞,
ẽ_{i}(t_{λ}) = 0 = f̃_{i}(t_{λ}).

Let λ ∈ P^{+}. Then u_{λ} ↦ u_{∞} ⊗ t_{λ} defines an embedding of crystals ι_{λ} : B(λ) ↦ B(∞) ⊗ T_{λ} that commutes with the ẽ_{i}’s.

Now, B_{w}(λ) is the inverse image of B_{w}(∞) ⊗ T_{λ} under ι_{λ}. Furthermore, the inverse image under ι_{λ} of an i-string for B(∞) is an i-string for B(λ). Thus (2.3.2) implies the string property for B(λ).

2.5. For completeness we now include Kashiwara’s proof of the following lemma.

Lemma 2.1. The refined Demazure formula (2.2.1) follows from the string property for B(λ).

Proof. If ẽ_{i}b = 0 for b ∈ B(λ) then clearly D_{i}b is an i-string having b as its highest weight vector. Moreover, an easy calculation shows that D_{i}S = S for S any i-string. Now Theorem 2 of [K2] says that

(2.5.1)  B(λ) = ⋃_{k_{i}≥0,j_{i}∈I,m≥0} f̃_{j_{m}}^{k_{m}} f̃_{j_{m−1}}^{k_{m−1}} … f̃_{j_{1}}^{k_{1}}u_{λ}.

Hence, B(λ) is the disjoint union of i-strings for any i ∈ I, since istrings are either disjoint or coincide.

We now prove (2.2.1) by induction on l(w). We thus assume the formula for s_{i_{n}}w = s_{i_{n−1}}s_{i_{n−2}} … s_{i_{1}} and need to check the equality

(2.5.2)  ∑_{b∈B_{w}(λ)} b = D_{i_{n}}( ∑_{b∈B_{s_{in} w}(λ)} b ).

As D_{i} leaves any i-string invariant it is enough to verify the following equality

(2.5.3)  ∑_{b∈B_{w}(λ)∩S} b = D_{i_{n}}( ∑_{b∈B_{s_{in} w}(λ)∩S} b ).

Now (2.3.2) severely restricts the shape of these intersections, and even further restrictions are imposed by the condition

B_{w}(λ) ∩ S = ⋃_{k} f̃_{i_{n}}^{k}(B_{s_{in}w}(λ) ∩ S),

which is a consequence of the definitions. All together, we are left with only three possibilities, namely

(1) B_{w}(λ) ∩ S = B_{s_{in}w}(λ) ∩ S = ∅,

(2) B_{w}(λ) ∩ S = B_{s_{in}w}(λ) ∩ S = S,

(3) B_{w}(λ) ∩ S = S and B_{s_{in}w}(λ) ∩ S = {b} where ẽ_{i}b = 0. In all three cases it is straightforward to check that (2.5.3) holds true.

□

We have thus reduced ourselves to the verification of (1), (2) and (3) of 2.3. Kashiwara proves (1) and (2) by realizing the B_{w}(λ)’s as crystals of the Demazure modules whereas the proof of the string property (3) relies on the combinatorial properties of the operators ẽ_{i}^{∗} and f̃_{i}^{∗} together with (1) and (2).

Here we shall demonstrate that (1) and (2) can be obtained in the same combinatorial spirit that is employed for (3), that is without relying on an interpretation of B_{w}(λ)’s as crystals for any modules. Now it is known that Littelmanns’s Path model is equivalent to the crystal combinatorics, see eg. [J1] and references therein, and that (2) and (3) (which suffice to obtain the string property (2.3.2)) can be obtained in that setting, [L2]. Still, Joseph remarks on page 181 in [J1] that it seems extremely difficult to establish (2) purely combinatorially.

3. Properties of B_{w}(∞)

3.1. Recall the injective morphism Ψ_{i} : B(∞) → B(∞) ⊗ B_{i} from the previous section. Using its properties (2.1.1), (2.1.2) and (2.1.3) one can obtain information about the commutation of f̃_{i} and ẽ_{i}; this is illustrated by the following lemma.

Lemma 3.1. For any i, j ∈ I and b ∈ B(∞) we have

⋃_{k,n} f̃_{i}^{n}f̃_{j}^{∗k}b = ⋃_{k,n} f̃_{j}^{∗k}f̃_{i}^{n}b.

Proof. If i ≠ j then by Corollary 2.2.2 of [K1] f̃_{i} and f̃_{j}^{∗} commute and there is nothing to prove. So we assume i = j. Write

Ψ_{i}(b) = b_{0} ⊗ f̃_{i}^{m}b_{i},

and let φ := φ_{i}(b_{0}) and ε := m. Now, Ψ_{i} is an embedding so to show the equality of the lemma it is enough to see that both sides have the same image under Ψ_{i}. So we replace b by b_{0} ⊗ f̃_{i}^{m}b_{i} and keep in mind that the action of f̃_{j}^{∗k} is on the right factor while f̃_{i} acts as on a tensor product.

Let now Ψ_{i}(b) = b_{0} ⊗ f̃_{i}^{m}b_{i} be represented as a point in the crystal graph associated to B(∞) ⊗ B_{i}. The crystal graph is a representation of the action of f̃_{i} on B(∞) ⊗ B_{i}, so there is an arrow between two points in the graph if f̃_{i} carries the corresponding crystal elements to each other.

If φ ≤ m the action of f̃_{i} is on the second factor and there is a horizontal arrow leaving b_{0} ⊗ f̃_{i}^{m}b_{i} and if φ > m there is a vertical arrow leaving b_{0} ⊗ f̃_{i}^{m}b_{i}

One typically gets a picture as the following one.

B_{i} B(∞)  \begin{aligned}∗ & & ∗ & & ∗ & & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ \\ ↓ & & ↓ & & ↓ \\ ∗ & & ∗ & & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ \\ ↓ & & ↓ \\ ∗ & & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ \\ ↓ \\ ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ \end{aligned}

The subset of B(λ) ,

⋃_{k} f̃_{i}^{k}(b_{0} ⊗ f̃_{i}^{m}b_{i}),

is represented by the points of the graph that can be hit by a sequence of arrows starting in b_{0} ⊗ f̃_{i}^{m}b_{i}.

On the other hand the action of f̃_{i}^{∗} is always on the second factor of the tensor product, so f̃_{i}^{∗} always takes a point in the graph to its right neighbour. Using this information one can now calculate the two sides of the lemma; in both cases one gets the infinite rectangle whose upper left corner is Ψ_{i}(b) = b_{0} ⊗ f̃_{i}^{m}b_{i} and whose lower left corner is the point below b_{0} ⊗ f̃_{i}^{m}b_{i} in which the arrows change direction. The lemma is proved. □

3.2. We can use the above to show the following result.

Theorem 3.2. B_{w}(∞) = ⋃_{k_{1},...k_{n}} f̃_{i_{1}}^{∗k_{1}} … f̃_{i_{n}}^{∗k_{n}}u_{∞}.

Proof. By definition f̃_{i}^{∗k}u_{∞} = f̃_{i}^{k}u_{∞} for all k and all i. So we get that

B_{w}(∞) = ⋃_{k_{1},...k_{n}} f̃_{i_{n}}^{k_{n}} … f̃_{i_{2}}^{k_{2}} f̃_{i_{1}}^{∗k_{1}}u_{∞}.

Using Lemma 2.1 we can move f̃_{i_{1}}^{∗k_{1}} to the front position. We then proceed with f̃_{i_{2}}^{k_{2}} etc. The theorem is proved. □

3.3. We can now deduce the property (1) of B_{w}(∞):

Corollary 3.3. B_{w}^{∗}(∞) = B_{w^{−1}}(∞).

Proof. Let b ∈ B_{w}(∞), i.e. b = f̃_{i_{n}}^{k_{n}} … f̃_{i_{1}}^{k_{1}}u_{∞} for some k_{1}, … k_{n}. The definition of f̃_{i}^{∗} then gives that

b^{∗} = f̃_{i_{n}}^{∗k_{n}} f̃_{i_{n−1}}^{∗k_{n−1}} … f̃_{i_{1}}^{∗k_{1}}u_{∞}

But from Theorem 3.2 we see that b^{∗} ∈ B_{w^{−1}}(∞) and the corollary is proved. □

3.4. We shall now consider the property (2). To that end we prove the following lemma

Lemma 3.4. For all i, j ∈ I and for all b ∈ B(∞) we have that

ẽ_{i} ⋃_{k} f̃_{j}^{∗k}b ⊂ ⋃_{k} f̃_{j}^{∗k}ẽ_{i}b ∪ ⋃_{k} f̃_{j}^{∗k}b ∪ {0}

Proof. Again only the case i = j is nontrivial; otherwise ẽ_{i} and f̃_{j}^{∗} commute. We apply the morphism Ψ_{i} to both sides of the lemma and can then check the inclusion in the crystal graph:

B_{i} B(∞)  \begin{aligned}∗ & & ∗ & & ∗ & & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ \\ ↓ & & ↓ & & ↓ \\ ∗ & & ∗ & & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ \\ ↓ & & ↓ \\ ∗ & & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ \\ ↓ \\ ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ & → & ∗ \end{aligned}

The graph is infinite to the right. We understand that ẽ_{i}b = 0 if there is no arrow ending at the point corresponding to b. Again, f̃_{i}^{∗} acts by shifting a b to the right while f̃_{i} follows the arrows (and hence ẽ_{i} follows the arrows in negative direction).

Let us start out by verifying that there are no points missing in the above picture. So we must check that if the arrow leaving b is vertical and there is no arrow ending at b then neither should there be any arrow ending at b’s right neighbour.

Let thus b be as indicated and write Ψ_{i}(b) = b_{0}⊗f̃_{i}^{m}b_{i}. Then φ(b_{0}) > ε(f̃_{i}^{m}b_{i}) = m because the arrow leaving b is vertical. Now ẽ_{i}(b) = 0 implies that ẽ_{i}(b_{0}) = 0 because Ψ_{i} commutes with ẽ_{i} and no element of B_{i} is mapped to 0 under ẽ_{i}. Since φ(b_{0}) ≥ ε(f̃_{i}^{m+1}b_{i}) we indeed get that

ẽ_{i}(Ψ_{i}(f̃_{i}^{∗}b)) = ẽ_{i}(b_{0} ⊗ f̃_{i}^{m+1}b_{i}) = ẽ_{i}b_{0} ⊗ f̃_{i}^{m+1}b_{i} = 0

We now split the verification of the lemma into several cases. Firstly we consider the case of a b with ẽ_{i}(b) = 0. Then the left hand side of the lemma consists of those points in the row of b from which a horizontal arrow is leaving. But this is contained in the right hand side of the lemma.

Then we consider the case of a vertical arrow entering and a vertical arrow leaving b. In that case the left hand side of the lemma consists of all the points that are positioned to the right of b (including b itself) together with the points in the row above b that have an arrow leading into one of the first points. In addition, the right hand side consists of the first points together with their upper neighbours. Thus the inclusion also holds in this case.

We then consider the case of a vertical arrow entering and a horizontal arrow leaving b. Then the left hand side of the lemma consists of the points positioned to the right of b together with b itself and its immediate predecessor. This is contained in the right hand side of the lemma (only the k = 0 part of the first union is needed).

Finally we consider the case of horizontal arrows entering as well as leaving b. In that case the left hand side consists of all points to the right of b together with b’s immediate predecessor, which is included in the right hand side (only the first union is needed). □

3.5. We can now show the property (2) of B_{w}(∞):

Theorem 3.5. For i ∈ I we have ẽ_{i}B_{w}(∞) ⊂ B_{w}(∞) ∪ {0}

Proof. We argue by induction on l(w) and thus assume the theorem for l(w) − 1. By Theorem 3.2 B_{w}(∞) satisfies the equality

B_{w}(∞) = ⋃_{k_{1}} f̃_{i_{1}}^{∗k_{i}}B_{ws_{i_{1}}}(∞)

By induction hypothesis ẽ_{i}B_{ws_{i_{1}}}(∞) ⊂ B_{ws_{i_{1}}}(∞) ∪ {0}. Combining this with lemma 3.4 we obtain the induction step. The theorem is proved. □

4. The Braid Relations

In this section we verify that the crystal Demazure operators D_{i} satisfy the braid relations on dominant weights. From this it follows that B_{w}(λ) is independent of the choice of reduced expression for w. Note that Kashiwara has observed that the D_{i} do not satisfy the braid relations in general.

4.1. Since W is a Weyl group, it is enough to check the braid relations for W of type A_{2}, B_{2} or G_{2}. Indeed, for

w = w_{1}s_{i_{k}}s_{i_{k−1}}s_{i_{k}}w_{2} = w_{1}s_{i_{k−1}}s_{i_{k}}s_{i_{k−1}}w_{2}

a braid relation of type A_{2} it is enough to check the case w_{1} = 1. By the refined sum formula (2.2.1) applied to w_{2} one should then show that

(4.1.1) D_{i_{k}}D_{i_{k−1}}D_{i_{k}} ( ∑_{b∈B_{w_{2}}(λ)} b) = D_{i_{k−1}}D_{i_{k}}D_{i_{k−1}} ( ∑_{b∈B_{w_{2}}(λ)} b) .

Using (2.2.1) once more, the left hand side of this is the sum over all elements of B_{s_{k}s_{k−1}s_{k}w_{2}}(λ) while the right hand side is the sum over the elements of B_{s_{k−1}s_{k}s_{k−1}w_{2}}(λ). We write w_{2} = s_{i_{l}}s_{i_{l−1}} … s_{i_{1}} and get then by repeated use of Lemma 3.1 like in Theorem 3.2 that

B_{s_{k}s_{k−1}s_{k}w_{2}}(λ) = ⋃_{k_{1},...k_{l}} f̃_{i_{1}}^{∗k_{1}} … f̃_{i_{l−1}}^{∗k_{l−1}} f̃_{i_{l}}^{∗k_{l}}B_{s_{k}s_{k−1}s_{k}}(λ).

Similarly, the right hand side of (4.1.1) is the sum over

⋃_{k_{1},...k_{l}} f̃_{i_{1}}^{∗k_{1}} … f̃_{i_{l−1}}^{∗k_{l−1}} f̃_{i_{l}}^{∗k_{l}}B_{s_{k}s_{k−1}s_{k}}(λ).

The A_{2}-case of the braid relations then implies (4.1.1). Similarly, one reduces the other braid relations to rank 2 cases.

4.2. To check the A_{2}, B_{2} or G_{2} cases, we appeal to the representation theoretical interpretation of B(λ) as basis at q = 0 of the irreducible highest weight module V (λ) for the quantum group U_{q}(𝔤).

Let us consider the A_{2}-case and write λ = (λ_{1}, λ_{2}) in terms of the fundamental weights (ω_{1}, ω_{2}). Then f̃_{1}^{λ_{2}}u_{λ} is nonzero, since it is the lowest element of the 2-string with highest element u_{λ}. But by weight considerations f̃_{1}^{λ_{2}}u_{λ} must be mapped to 0 under ẽ_{1} and therefore it is the highest element of the 1-string, whose lowest element is f̃_{2}^{λ_{1}+λ_{2}} f̃_{1}^{λ_{1}}u_{λ} and especially nonzero. Continuing, we find that

f̃_{1}^{λ_{2}} f̃_{2}^{λ_{1}+λ_{2}} f̃_{1}^{λ_{1}}u_{λ} ∈ B_{s_{1}s_{2}s_{1}}(λ) ⊂ B(λ)

is nonzero. The lowest weight vector space of V (λ) is one dimensional and so this element is the unique lowest element if B(λ) .

Now, by (2) of (2.3), B_{s_{1}s_{2}s_{1}}(λ) is invariant under all the ẽ_{i} operators. Since it moreover contains the lowest element, it must be equal to all of B(λ). The same conclusion holds for B_{s_{2}s_{1}s_{2}}(λ) and then B_{s_{2}s_{1}s_{2}}(λ) = B_{s_{1}s_{2}s_{1}}(λ) as promised.

References

[A] H. H. Andersen, Schubert varieties and Demazure’s character formula, Invent. Math., 79 (1985) 611-618

[D] M. Demazure, Une nouvelle formule des caractères. Bull. Sc. Math. 98 (1974) 163-172

[J1] A. Joseph, Quantum Groups and Their Primitive Ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Band 29, Springer-Verlag

[J2] A. Joseph, Combinatoire de Crystaux, Cours de troisième cycle. Université P. et M. Curie, Année 2001-2002

[K1] M. Kashiwara, Crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993) 839-858

[K2] M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J., 63 (1991), 465-516

[L1] P. Littelmann, Crystal graphs and Young tableaux, Journal of Algebra 175 1995 no. 1, 65-87

[L2] P. Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebra, Invent. Math. 116 (1994) 329-346

[R] S. Ramanan & A. Ramanathan, Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), 217-224

Matematisk Afdeling, Universitetsparken 5, DK-2100 København Ø, Danmark, steen@math.ku.dk