Lattice Polytopes Associated to Certain Demazure Modules of sl n + 1

°

Lattice Polytopes Associated to Certain Demazure Modules of sl _{n + 1}

RAIKA DEHY dehyr@ictp.trieste.it

Abdus Salam Institute, Mathematics Section, P.O. Box 586-34100 Trieste, Italy

RUPERT W.T. YU yuyu@mathlabo.univ-poitiers.fr

CNRS ESA 6086, Universit´e de Poitiers, Boulevard 3, Teleport 2, BP179, 86960 Futurescope cedex, France Received December 16, 1997; Revised June 19, 1998

Abstract. Letwbe an element of the Weyl group of sln+1. We prove that for a certain class of elementsw (which includes the longest elementw0of the Weyl group), there exist a lattice polytope1^wi ⊂R^`(w), for each fundamental weightωiof sln+1, such that for any dominant weightλ=Pn

i=1aiωi, the number of lattice points in the Minkowski sum1^w_λ =Pn

i=1ai1^wi is equal to the dimension of the Demazure module E_w(λ). We also define a linear map A^w:R^`(w)−→P⊗ZRwhere P denotes the weight lattice, such that char E_w(λ)=e^λP

e^−Aw(x) where the sum runs through the lattice points x of1^w_λ.

Keywords: lattice polytope, Demazure module, Minkowski sum, character formula

1. Introduction

In this paper, we present some results concerning the first of a two-part programme to prove the existence of degenerations of Schubert varieties of S L(n)into toric varieties (by degeneration of a Schubert variety into a toric variety, we mean a flat deformation where the generic fibre is a Schubert variety and the special fibre is a toric variety). This involves the construction of the lattice polytope which in turn, in the second part of the programme, will provide the toric variety into which the corresponding Schubert variety degenerates.

In this direction, Gonciulea and Lakshmibai [10] recently proved such degenerations for Schubert varieties in an arbitrary miniscule G/P, as well as the class of Kempf varieties in the flag variety S L(n)/B. For an arbitrary G of rank two, this has been proved by one of the authors [4].

Let us describe our results more precisely. Fix n ∈ N^∗ and K an algebraically closed field of characteristic 0. Letbbe a Borel subalgebra of sln+1(K)andh ⊂ ba Cartan subalgebra. Let αi, i = 1, . . . ,n, be the corresponding set of positive simple roots so that hαi, α^∨ji = ai j where(ai j)i,j is the Cartan matrix, and letωi be the corresponding fundamental weights. Denote by P, P⁺, W ,`(−)and¹respectively the weight lattice, the set of dominant weights, the Weyl group which is just the symmetric group of n+1 letters, the length function and the Bruhat order on W . Letλ ∈ P⁺andw ∈ W . Set V_λto be the finite-dimensional irreducible representation of highest weightλ,vwλto be a non-zero

weight vector of weight wλand E_w(λ)to be theb-module U(b)vwλwhich is called the Demazure module [5] associated tow. Set Wⁱ to be the stabilizer ofωi in W and Wi the quotient W/Wⁱ. Endow W_i with the induced Bruhat order that we shall denote equally by¹and ifσ ∈ W_i, then we shall denote by`(σ)the induced length ofσ, which is the minimum of the lengths of representatives ofσ.

The representation theory of a semisimple algebraic group G is closely related to the geometry of Schubert varieties (in particular G/B) since the Demazure modules can be realized as the global sections of line bundles over Schubert varieties. Degenerations of Schubert varieties into toric varieties will allow us to study the geometry of the former via toric varieties which are combinatorial.

Letλ=P

iaiωi be a dominant weight, then the dimension of E_w(λ)is a polynomial in the variables aiof degree`(w)because the dimension of its dual E_w(λ)^∗can be described as the Euler characteristic of the ample line bundle N

iL^⊗_ωi^ai over the Schubert variety associated towin G/P_λ([7, 18.3.6] or [2, 2.3]). Whereas, given convex lattice polytopes 1i inR^`(w), a theorem of Ehrhart [6] implies that under the condition that a lattice point in the Minkowski sum1:=P

iai1i = {P

iaivi wherevi ∈ 1i}is the sum over i of ai

lattice points of1i, the number of lattice points in1is a polynomial of degree`(w)in the variables a_i. On the other hand, suppose that we have a degeneration of the Schubert variety S_wequipped with line bundlesL_ω1, . . . ,L_ωninto the toric variety X equipped with line bundlesL1, . . . ,Ln. Then dim H⁰(S_w,N

iL^⊗_ωi^ai)=dim H⁰(X,N

iL^⊗_i^ai). But to say that X is equipped with line bundlesL1, . . . ,Lnis equivalent to having n lattice polytopes 1^w₁, . . . , 1^wn inR^`(w)such that dim H⁰(X,N

iL^⊗_i^ai)is the number of lattice points in the Minkowski sumPn

i=1ai1^wi (for example, see properties B3, B4 of Section 2.3 in [19]).

These facts lead us to construct a polytope1^wi for each fundamental weightωiand then we form the appropriate Minkowski sum.

We prove first in this paper the case where w=w0, the longest element of the Weyl group W .

Theorem 1.1 There exist lattice polytopes1i ⊂R^`(w0),i =1, . . . ,n,such that for any λ=Pn

i=1a_iωi ∈ P⁺,the number of lattice points in the Minkowski sum1_λ:=Pn i=1a_i1i

is the dimension of the irreducible representation V_λ.

Polytopes satisfying Theorem 1.1 (although there was no mention of the Minkowski sum decomposition, they do have a Minkowski sum decomposition) have been constructed using Gelfand-Tsetlin patterns in [9, 12], by Berenstein and Zelevinsky [1] and by Littelmann [16] via the combinatorics of Lakshmibai-Seshadri paths.

Our polytope 1λ is different and it turns out that the toric variety associated to this polytope is the same as the one constructed by Gonciulea and Lakshmibai in [10]. In fact the Minkowski sum decomposition gives a direct link between lattice points and the standard monomial basis (see [14, 17]) of the irreducible representation since we can prove that a lattice point of1_λcan be written as a sum over i of a_i lattice points of1i.

Furthermore, since standard monomial theory exists also for Demazure modules (and for other simple algebraic groups), we believe that our construction can be generalized to any simple algebraic group G as follows.

Conjecture 1.2 Letw ∈ W . There exist lattice polytopes1^wi ⊂R^`(w),i =1, . . . ,n, such that for anyλ=Pn

i=1aiωi ∈ P⁺,the number of lattice points in the Minkowski sum 1^w_λ :=P_n

i=1ai1i^wis the dimension of the Demazure module E_w(λ).

As a matter of fact, the polytopes1iconstructed in Theorem 1.1 are such that the vertices {v_τ}_τ∈W_i are indexed by Wi. We believe that1^wi of the conjecture can be chosen as the convex hull of{v_τ}_τ¹wembedded (by a permutation of coordinates) inR^`(w).

Indeed, we prove that this is true whenwcan be written in a certain way (see Section 7 for details). Unfortunately, this does not cover all the elements of the Weyl group except in the case where G = S L(2)or S L(3). By weakening to a notion called polytopes with integral structure, one of the authors proved in [3] that one can construct a polytope with integral structure for anyw∈ W such that the number of lattice points in the polytope is the dimension of the associated Demazure module. However, there is no Minkowski sum decomposition and these polytopes do not provide directly toric varieties.

This paper is organised as follows. In Section 2, we construct for each fundamental weight ωi a lattice convex polytope 1i whose vertices are indexed by the set W_i. We shall prove later in Section 5 that1i is triangulable by primitive simplices parametrized by maximal chains. We then present an example in Section 3. In Sections 4–6, we show how, in the case where w = w0, a lattice point in the Minkowski sumPn

i=1a_i1i can be written as a sum over i of ai lattice points of1i, and that these points exhaust the dimension of the irreducible representation V_λ whereλ = Pn

i=1aiωi. Sections 7 and 8 contain a discussion of the case of Demazure modules where we specify and prove the cases where the conjecture is true. We give another example in Section 9 and finally, in Section 10, we present applications of our results concerning combinatorial descriptions of weight multiplicities as lattice points of a polytope with rational vertices.

We shall use the above notations throughout this paper. Furthermore, let s1, . . . ,sn be the reflections associated to the positive simple roots. For any N ∈N, we shall endowR^N with the following partial-ordering: let X , Y ∈R^N be such that X 6=Y , then

X <Y if and only if Y−X ∈R₊^N

2. Construction of the polytope∆ifor each fundamental weight wi

Let 1≤i ≤n be fixed in this section. Recall that Wi can be identified with the subset of W consisting of elementswsuch thatws_j ºwfor all j6=i . It is also well known that W_i is in bijection with the set of i -tuples(r₁, . . . ,r_i)such that 0≤r₁ <r₂ <· · · <r_i ≤n.

Namely, we can think of W =S_n+1as the group of permutations on the set{0,1, . . . ,n}. Then the bijectionw7→(r₁, . . . ,r_i)is given by{r₁, . . . ,r_i} =w({0,1, . . . ,i−1}).

The induced Bruhat order on W_i is then given by:

(r₁, . . . ,r_i)≺(s₁, . . . ,s_i)⇔(r₁, . . . ,r_i) < (s₁, . . . ,s_i)

where on the right hand side, the i -tuples are considered as elements ofRⁱ.

Note that in this notation, the smallest element is (0,1,2, . . . ,i −1) that we shall denote sometimes simply by 1 when there is no confusion, and the biggest element is (n−i +1,n−i+2, . . . ,n), and that the length of the latter is(n−i+1)i . In fact, the minimal representative of(r₁, . . . ,r_i)is

s_r1s_r1−1· · ·s₁s_r2s_r2−1· · ·s₂s_r3· · ·s_ris_ri−1· · ·s_i

where s_rj· · ·s_j =1 if r_j < j and its length is the sum over j of r_j−j+1.

We shall fix a particular reduced decomposition of w0. Namely, we use the lexico- graphic minimal expressionw0 =s₁s₂s₁s₃s₂s₁· · ·s_ns_n−1· · ·s₁. Notice that each minimal representative of W_i can be written as a subexpression of this reduced decomposition.

Remark 2.1 We shall think of this as n blocks where block 1 is s₁, block 2 is s₂s₁, . . . , block n is s_ns_n−1· · ·s₁.

Let us write the standard basis vectors inR^`(w0) as e_pq with 1 ≤ q ≤ p ≤ n. Let 1≤i ≤n, and(r₁, . . . ,r_i)be an element of W_i, we then define

ϕ(r1, . . . ,ri)= Xn p=n−i+1

rX_p+i−n

q=p+i−n

epq∈R^`(w0)

Definition 2.2 Let c:τ1 Â · · · Âτmbe a chain in Wi. We define Scto be the convex hull of the points{ϕ(τj)}^mj=1and we define1i to be the convex hull of the points{ϕ(τ)}τ∈W_i. Lemma 2.3

(a) The vertices of1iare the only lattice points in1iand they are indexed by the elements of Wi.

(b) The mapϕis order-preserving.

(c) Let c:τ1Â · · · Âτ₍n−i+1)i Â1 be a maximal chain in Wi. The polytope Scis a simplex of dimension(n−i+1)i and its volume is 1/((n−i+1)i)!.

Proof: The first two assertions are direct consequences of the definition ofϕi. For part (c), notice that the pointsϕ(τ1), . . . , ϕ(τ₍n−i+1)i)are linearly independent andϕ(1)is zero in R^`(w0). So Sc is a simplex. Since ϕ(τ1), . . . , ϕ(τ(n−i+1)i)can be obtained from the canonical basis via a matrix (with integer entries) of determinant 1 or−1, the volume of Sc

is 1/((n−i+1)i)!. 2

We deduce from our definition the following properties between the polytopes1i. Proposition 2.4

(a) The intersection of1iand1jis{0}whenever i 6= j . (b) Let x=P

p,qxpqepq∈1i,then xpq =0 if p<n−i+1.

(c) If x = P

p,qxpqepq ∈ 1i is such that xst 6= 0,then xst⁰ 6= 0 for all t⁰ = t,t + 1, . . . ,rs+i−n.

Proof: Assertions (b) and (c) are straightforward. So let us prove (a). We can assume that i < j . Notice that the coefficient of eni for any non-zero element of1i is non-zero while it is zero for any element of1j. Thus (a) follows. 2

Letλ=Pn

i=1aiωi be a dominant weight (thus each ai ∈N) and V_λbe the irreducible sln+1-module of highest weightλ.

Definition 2.5 We define the polytope1λto be the Minkowski sumPn i=1ai1i. Since the 1i’s are lattice convex polytopes, the polytope 1_λ is also a lattice convex polytope. We can now state our theorem in the case wherew=w0.

Theorem 2.6 The number of lattice points in1λis equal to the dimension of V_λ. 3. Example

The first interesting example is sl4. We writew0 =s1s2s1s3s2s1 and we have, in terms of minimal representatives,

W1= {1,s1,s2s1,s3s2s1}, W2 = {1,s2,s3s2,s1s2,s1s3s2,s2s1s3s2} W3= {1,s3,s2s3,s1s2s3}

We then obtain viaϕthe following table where each row contains the coefficients of aϕ(τ):

s1 s2 s1 s3 s2 s1

e11 e22 e21 e33 e32 e31

s1 0 0 0 0 0 1 (1)

s2s1 0 0 0 0 1 1 (2)

s3s2s1 0 0 0 1 1 1 (3)

s2 0 0 0 0 1 0 (0, 2)

s3s2 0 0 0 1 1 0 (0, 3)

s1s2 0 0 1 0 1 0 (1, 2)

s1s3s2 0 0 1 1 1 0 (1, 3)

s2s1s3s2 0 1 1 1 1 0 (2, 3)

s3 0 0 0 1 0 0 (0, 1, 3)

s2s3 0 1 0 1 0 0 (0, 2, 3)

s1s2s3 1 1 0 1 0 0 (1, 2, 3)

The images of(0),(0,1),(0,1,2)are all(0,0,0,0,0,0).

Let us now consider the adjoint representation. The highest weight isω1+ω3. One then verifies easily by hand that a lattice point of11+13is the sum of a lattice point of11and

a lattice point of13. Hence a quick computation shows that the lattice points are the ones in11and13together with 8 other points:

e31+e33, e31+e33+e22, e31+e33+e22+e11

e32+e31+e33+e22, e32+e31+e33+e22+e11

e31+e32+2e33, e31+e32+2e33+e22, e31+e32+2e33+e22+e11

Thus there are 15 lattice points in11+13which is the dimension of sl₄.

Remark thatϕ(2)+ϕ(0,1,3)=ϕ(3)+ϕ(0,1,2)is the only sum repeated here. This can be seen to correspond to the tensor product decomposition

V_ω1⊗V_ω3∼=V_ω1⊗¡

V_ω1¢_∗ ∼=gl₄=sl4⊕V0

4. Correspondence with semi-standard Young tableaux Let λ = Pn

i=1aiωi be a dominant weight. SetW be the disjoint union of the Wi and W(λ)=Q_n

i=1

Q_ai

j=1Wi. We can associate to an element ofW(λ)viaϕa lattice point of 1λ. Namely, an element(wi j)i,j ofW(λ)is sent toP

i,jϕ(wi j)in1λ.

However this association is not necessarily injective (that is, a lattice point can be the image of another element inW(λ)). We claim that with respect to a certain partial ordering ofW, there is a unique such element which is decreasing. At the end of this section, we shall show that the set of lattice points corresponding to the elements inW(λ)is in bijection with the set of semi-standard Young tableaux of typeλ.

Let us first define our partial order inW, denoted by≺, which extends the induced Bruhat ordering in Wi. Let(r1, . . . ,ri)and(s1, . . . ,sj)be two elements ofW, then

(r1, . . . ,ri)≺(s1, . . . ,sj)⇔(−|1, . . . ,{z −1}

n−i

,r1, . . . ,ri) < (−|1, . . . ,{z −1}

n−j

,s1, . . . ,sj)

where the elements on the right hand side are inRⁿ.

Remark 4.1 Using the notations above, if we have r ≺s then i ≤ j . Furthermore, there is a unique maximal element(1,2, . . . ,n)and a unique minimal element(0).

Lemma 4.2

(a) The setWis a lattice,that is,every pair of elements ofWhave a well defined max and min.

(b) If r ∈Wi,s∈Wjand i ≤ j,then we have min(r,s)∈Wi and max(r,s)∈Wj. (c) (^MAX–MIN)Let r,s∈W,thenϕ(r)+ϕ(s)=ϕ(max(r,s))+ϕ(min(r,s)).

Proof: Let r=(r₁, . . . ,r_i)be an element ofW. By adding−1’s on the left as above, we can associate to r , an element R=(R1, . . . ,Rn)ofRⁿ.

Definition 4.3 Let r , s be elements ofWand R, S the corresponding associated elements inRⁿ. We define min(R,S) = (T1, . . . ,Tn)where Ti = min(Ri,Si)and min(r,s)the element ofWassociated to min(R,S)by taking away all the−1’s.

We define max(r,s)similarly.

One verifies easily assertions (a) and (b) from this definition. It suffices therefore to check max–min.

Let r =(r₁, . . . ,r_i)and s=(s₁, . . . ,s_j)where i ≤ j . Then ϕ(r)+ϕ(s)=

Xn p=n−i+1

rX_p+i−n

q=p+i−n

e_pq+ Xn p=n−j+1

s_p+j−nX

q=p+j−n

e_pq

= Xn p=n−i+1

rX_p+i−n

q=p+i−n

epq+

n−i

X

p=n−j+1 s_p+j−nX

q=p+j−n

epq+ Xn p=n−i+1

sX_p+j−n

q=p+j−n

epq

= Xn p=n−i+1

à _r

p+i−n

X

q=p+i−n

epq+

sX_p+j−n

q=p+j−n

epq

! +

n−i

X

p=n−j+1 sX_p+j−n

q=p+j−n

epq

= Xn p=n−i+1

min(r_p+i−nX,s_p+j−n) q=p+i−n

e_pq+ Xn p=n−j+1

max(r_p+i−nX,s_p+j−n) q=p+j−n

e_pq

= ϕ(min(r,s))+ϕ(max(r,s)) 2 We shall now state and prove our claim.

Proposition 4.4 Letθ= {θi j}i=1,...,n;j=1,...,ai be an element ofW(λ). Then there exists a unique elementψ= {ψi j}ofW(λ)such that

(i) ψi j ¹ψk`if i <k or if i =k and j≤`;

(ii) P

i,jϕ(θi j)=P

i,jϕ(ψi j).

Before proving this proposition, let us remark that condition (i) says that

ψ11¹ψ12¹ · · · ¹ψ1a1¹ψ21¹ · · · ¹ψ2a2¹ · · · ¹ψ₍n−1)a_n−1 ¹ψn1¹ · · · ¹ψnan

This is similar to the definition for a Young tableaux of Lakshmibai and Seshadri of typeλ modulo liftings to the Weyl group W , see [14]. As we shall see, our theorem says that this is exactly the same definition.

Proof: We shall prove the existence by induction on a = Pn

i=1ai. It is clear that the induction hypothesis holds for a=1. (In fact, by max–min of Lemma 4.2, it holds equally for a=2).

Let us now suppose that the induction hypothesis holds for a−1. Let r be maximal such that a_r 6=0. By the induction hypothesis, we can suppose thatθ⁰=θ\ {θr ar}satisfies the conditions (i) and (ii) of the proposition.

We shall now divideθ⁰into three disjoint totally-ordered sets. Let E_≺= {θi j|θi j≺θr a_r}, E_º= {θi j|θi j ºθr a_r}

and

E0= {θi j|θi j and θr a_rare not comparable}.

Note that the elements of E_ºare all in Wr.

If E₀is empty, then we can insertθr arin the sequence to obtain a totally-ordered sequence and hence by rearranging the subscripts, we obtain an element of W(λ) satisfying the required conditions.

Suppose now that E₀is not empty. Thenθr ar is in neither E_≺, E_ºnor E₀. Letφbe the maximal element in E0. By max–min of Lemma 4.2, replacingφandθr a_r by max(φ, θr a_r) and min(φ, θr a_r)does not alter the sum viaϕ. Furthermore, if we let E⁰_≺, E⁰_ºand E⁰₀be the new partition as defined above relative toθr a⁰ _r =max(φ, θr a_r), then the cardinal of E₀⁰ is strictly less than E0since min(φ, θr a_r)will belong to E_≺⁰.

Now repeat the same procedure until E0is empty and we have the existence since E0is a finite set.

Let us turn to the uniqueness which is a consequence of the following lemma.

Lemma 4.5 Let r and s be two distinct elements of Wi. Then there exist k,mksuch that one of the following conditions is satisfied:

(i) the ekmk-coordinate is 1 forϕ(r)and the ek`-coordinate forϕ(s)is 0 for all mk≤`≤k.

(ii) the ekm_k-coordinate is 1 forϕ(s)and the ek`-coordinate forϕ(r)is 0 for all mk≤`≤k.

Furthermore let us suppose that (i) is satisfied(we have obviously the same statement with the roles of r and s exchanged when (ii) is satisfied). Then we can choose k and mk such that for all t ∈ Wj satisfying t ¹ s,the ekm_k-coordinate is 1 forϕ(r)and the ek`-coordinate forϕ(t)is 0 for all mk≤`≤k.

Proof: Let us denote r =(r1, . . . ,ri)and s =(s1, . . . ,si). Since r and s are distinct, there exists k such that either rk>skor sk>rk. Suppose we have rk>sk(resp. sk>rk).

Since rk >sk ≥ k−1 (resp. sk >rk ≥ k−1), r (resp. s) has non-zero entries in the (n−i +k)^thblock. By the definition of our embedding, it is clear that if we put m_k=r_k (resp. m_k=s_k), then the conditions of (i) (resp. (ii)) are satisfied.

To prove the last statement, let us suppose that (i) is satisfied. Then, there exists k such that r_k>s_k. Now let t∈W_jbe such that t ¹s. By Remark 4.1, we must have i≥ j and hence we can write t=(t1, . . . ,ti)by adding−1’s on the left. Since t¹s, we have tk≤sk<rk. It follows again from our embedding that we have our result by letting mk=rk. 2 We can now finish our proof. Letθ andθ⁰ = {θi j⁰}be two elements inW(λ)satisfying the conditions of the proposition. Let r be maximal such that ar 6=0. Thenθr a_r andθr a⁰ _rare maximal inθandθ⁰respectively. Ifθr a_r 6=θr a⁰ _r, then by applying the previous lemma, we can suppose that there exists k, mksuch that the entry ekm_k is 1 forϕ(θr a_r)and the entries ek`

forϕ(θr a⁰ r)is 0 for all mk ≤`≤k. Hence by the same lemma, the same entries forϕ(θi j⁰)

are 0 for all i , j sinceθr a⁰_r is maximal inθ⁰. It follows thatP

i,jϕ(θi j)−P

i,jϕ(θ_{i j}⁰)6=0 which contradicts the fact thatθandθ⁰satisfy the second condition of the proposition.

Thusθr ar =θr a⁰ _r. Now by induction on a, the sum of the a_i’s, the elementsθ,θ⁰must be the same (the case a=1 is equivalent to the fact thatϕis an embedding). 2 Thus we have proved what we claimed at the start of this section. Let us denote byW(λ)d

the set of elements inW(λ)satisfying property (i) of the proposition. Now given an element θ inW(λ)d, using the notations(r1, . . . ,ri)for elements in Wi, we can arrange eachθi j

as a row of numbers flushright, and stack them in order with the largest row on top, the smallest row on the bottom, what we obtain then is a semi-standard Young tableau of type λ. For example, the sequence(1),(0,1),(0,2),(0,2,3)corresponds to the semi-standard Young tableau

0 2 3

0 2

0 1

1

By the uniqueness proved in the proposition, we obtain a well-defined map fromW(λ)d

to the set of semi-standard Young tableaux of typeλ, which is obviously injective. On the other hand, given a semi-standard Young tableau of typeλ, we obtain an element ofW(λ)d

by reading off the rows. It is clear that this is the inverse of the former map. Now by Lemma 2.3, lattice points in1i are in bijection with elements of Wi, thus Proposition 4.4 says thatW(λ)dis in bijection with the set of lattice points in1λwhich can be written as a sum over i of ailattice points of1i, we can hence state

Theorem 4.6 The set of lattice points of1_λwhich can be written as a sum over i of ai

lattice points of1iis in bijection with the set of semi-standard Young tableaux of typeλ. Remark 4.7 In fact, the existence part of Proposition 4.4 can be proved with semi- standard Young tableaux since it involves only max–min of Lemma 4.2 and not the explicit embedding. The idea is to put the maximal entry of each column at the top row and then use induction which is roughly what we have done.

5. Characterization of lattice points in∆λ

Let λ = Pn

i=1aiωi be a dominant weight. Recall from Definition 3.2 that 1λ is the Minkowski sumPn

i=1ai1i, where11, . . . , 1nare the polytopes associated to the funda- mental weightsω1, . . . , ωnwhich were defined in Section 2. In this section we shall prove the following theorem:

Theorem 5.1 A lattice point in the Minkowski sumPn

i=1ai1ican be written as the sum of a1lattice points in11,a2lattice points of12and so on.

As in the previous section, denote byW the union over all i of Wi equipped with the partial order defined in the same section, and for any dominant weightµ, denote byW(µ)d

the set of elements inW(µ)satisfying property (i) of Proposition 4.4. Letθ = {θi j}i,j be an element ofW(µ)d, we shall denote by C_µ(θ)the convex cone generated by{ϕ(θi j)}i,j. Theorem 5.2 Let x ∈1λ,then there exist a dominant weightµand aθ ∈W(µ)d such that x ∈C_µ(θ).

This theorem is a direct consequence of the following technical lemma.

Lemma 5.3 Let{σi j}i=1,...,n;j=1,...,ai be a sequence of elements ofWsuch thatσi j ∈W_i. Let p_{i j} ∈ R₊. Then there exists{σ_{i j}⁰}i=1,...,n;j=1,...,a_i⁰ a sequence of elements ofW and

p_{i j}⁰ ∈R₊such that (i) σi j⁰ ∈Wi.

(ii) σi j⁰ ≺σkl⁰ if i <k or if i =k and j<l.

(iii) P_ai

j=1pi j =Pa_i⁰ j=1p⁰_{i j}. (iv) Pn

i=1

Pai

j=1pi jϕ(σi j)=Pn i=1

Pa_i⁰

j=1p_{i j}⁰ ϕ(σi j⁰). Proof: We shall prove the lemma by induction on thePn

i=1ai.

The assertion is obvious when the sum is 1. So let us suppose that the sum is strictly bigger than one. Let l be maximal such that al >0. By the induction hypothesis, we can assume that{σi j}i=1,...,l;j=1,...,a_i\{σla_l}satisfies (i) and (ii) of the lemma. For simplicity we shall denoteσla_l byσand q= pla_l.

Ifσ ºσi jfor all i=1, . . . ,l, j =1, . . . ,aior ifσ =σl jfor some j ≤al−1, then we are done.

So let us suppose the contrary. Then there existsτ =σcdminimal such thatσ 6Âτ. Let κ =σr sbe maximal such that P :=P

τ¹σi j≺κ p_{i j}≤q. Denote by m_{i j}=min(σ, σi j)∈W_i and by M_l^{i j}=max(σ, σi j)∈Wl. Note that we have

M_l^{i j}º M_l^i,j−1º · · · ºM_l^cd Âσ Âmi j º · · · ºmcd

Now using repeatedly max–min of Lemma 4.2, we obtain:

Xl i=1

a_i

X

j=1

pi jϕ(σi j)= X

σi j≺τ

pi jϕ(σi j)+X

σi jºκ

pi jϕ(σi j)

+ X

τ¹σi j≺κ

pi j(ϕ(σi j)+ϕ(σ ))+(q−P)ϕ(σ)

= X

σi j≺τ

pi jϕ(σi j)+X

σi jÂκ

pi jϕ(σi j)

+ X

τ¹σi j≺κ

pi j

¡ϕ(mi j)+ϕ¡ M_l^{i j}¢¢

+pr sϕ(κ)+(q−P)ϕ(σ)

Now if pr s≤q−P, then we must haveκ =σl,a_l−1. Consequently, we have Xl

i=1 ai

X

j=1

p_{i j}ϕ(σi j)= X

σi j≺τ

p_{i j}ϕ(σi j)+ X

τ¹σi j¹κ

p_{i j}¡

ϕ(m_{i j})+ϕ¡ M_l^{i j}¢¢

+(q−P−p_{r s})ϕ(σ)

Thus we obtain a chain

M_l^{r s}º · · · º M_l^cdÂσ Âmr sºmr,s−1º · · · ºmc,d Âσc,d−1Â · · · Âσ11 Â1 from which we can compress into a chain{σ_{i j}⁰}where i = 1, . . . ,l and j = 1, . . . ,a⁰_i satisfying the required properties of the lemma.

If p_{r s} ≥q−P, then:

Xl i=1

ai

X

j=1

pi jϕ(σi j)= X

σi j≺τ

pi jϕ(σi j)+X

σi jÂκ

pi jϕ(σi j)

+ X

τ¹σi j≺κ

pi j

¡ϕ(mi j)+ϕ¡ M_l^{i j}¢¢

+(q−P)(ϕ(κ)+ϕ(σ))+(pr s−(q−P))ϕ(κ)

= X

σi j≺τ

p_{i j}ϕ(σi j)+X

σi jÂκ

p_{i j}ϕ(σi j)

+ X

τ¹σi j≺κ

p_{i j}¡

ϕ(m_{i j})+ϕ¡ M_l^{i j}¢¢

+(q−P)¡

ϕ(m_{r s})+ϕ¡ M_l^{r s}¢¢

+(p_{r s}−(q−P))ϕ(κ)

Thus we obtain a chain

σl,al−1Â · · · Âκ Âm_{r s}ºm_r,s−1º · · · ºm_cd Âσc,d−1· · · Âσ11Â1

from which we can compress into a chain{σi j⁰}where i = 1, . . . ,l and j = 1, . . . ,a⁰_i satisfying (i), (ii) and (iii) of the lemma (look at the coefficents). Therefore we have

Xl i=1

ai

X

j=1

p_{i j}ϕ(σi j)= Xl

i=1 a_i⁰

X

j=1

p⁰_{i j}ϕ(σi j⁰)+ X

τ¹σi j≺κ

p_{i j}ϕ¡ M_l^{i j}¢

+(q−P)ϕ¡ M_i^{r s}¢

We now observe that the length of the remaining elements M_l^cd, . . . ,M_l^{r s}are strictly greater than that ofσ. Thus we can repeat the same reasoning and the lemma is proved because

there is a maximal element in Wl. 2

Corollary 5.4 The polytope1i is triangulable by primitive simplices of dimension(n− i+1)i .

Proof: Recall from [11] that a simplex is called primitive of dimension d if its vertices are lattice points and its volume is 1/d!.

It is clear from the proof of the preceding lemma applied to the sequence{σi j}j=1,...,ai

that1i is the union of all the S_cwhere c is a chain in W_i(see Definition 2.2). Moreover, if c⁰is a (strict) subchain of c, then S_c0 lies in the boundary of S_c. Since by Lemma 2.3, S_cis a primitive simplex of dimension(n−i+1)i when c is a maximal chain, to show that1i

is triangulable by primitve simplices, it suffices to show that the interior of any two distinct simplices Scand Sc⁰do not meet.

Consider two chains c:σ1Â · · · Âσ_`Â1 and c⁰:τ1Â · · · ÂτmÂ1. Suppose that the intersection of the interiors of Scand Sc⁰is not empty and that Q belongs to this intersection.

We can therefore write Q as (recall thatϕ(1)=(0, . . . ,0)∈R<sup>`(w0)</sup>) X`

j=1

pjϕ(σj)=Q= Xm k=1

qkϕ(τk) (∗)

where p_j,q_k∈]0,1[ and p₁+ · · · +p_`≤1, q₁+ · · · +q_m≤1.

Assume thatσ1 6=τ1. Writingσ1=(s1, . . . ,si)andτ1 =(t1, . . . ,ti). By Lemma 4.5, there exists a coordinate epqwhich is non-zero on the left hand side of(∗), whereas it is zero on the right hand side (becauseτ1is maximal in the chain c⁰). So we have a contradiction and thereforeσ1=τ1.

Without loss of generality, we can suppose p1≥q1. We can then rewrite(∗)as follows:

(p₁−q₁)ϕ(σ1)+X^`

j=2

p_jϕ(σj)=Q⁰= Xm k=2

q_kϕ(τk) (∗∗)

Consequently, we must have p1 = q1. Now by repeating the same argument (or use induction on `+m) on (∗∗), we conclude that ` = m, pj = qj andσj = τj for all j=1, . . . , `. That is c=c⁰. Thus the corollary is proved. 2 Proof of Theorem 5.1: Suppose that x is a lattice point ofPl

i=1ai1i. Without loss of generality we can assume that al6=0. We can write x=x1+ · · · +xlwhere

x_i= p_{i 1}ϕ(σi 1)+ · · · +p_iriϕ¡ σiri

¢ with p_{i j} >0 and

ri

X

j=1

p_{i j} =a_i

whereσi j∈Wi. By Theorem 5.2, we can assume thatσi jis a strictly increasing sequence of elements ofW.

If rl =1, then plrl =aland so xl =alϕ(σlrl)which implies that x−xlis a lattice point ofPl−1

i=1ai1i.

If rl >1, then by Lemma 4.5, there exists a coordinate e_αβsuch that the e_αβcoordinate of x is equal to plr_l. So plr_l is a positive integer and x−plr_lϕ(σlr_l)is a lattice point of P_l−1

i=1ai1i+(al−plr_l)1l.

Thus, in both cases the assertion follows by induction onPl

i=1ai. 2