Journal of Algebraic Combinatorics 10 (1999), 149–172
c 1999 Kluwer Academic Publishers. Manufactured in The Netherlands.
°
Lattice Polytopes Associated to Certain Demazure
Modules of sln+1
RAIKA DEHY
Abdus Salam Institute, Mathematics Section, P.O. Box 586-34100 Trieste, Italy
dehyr@ictp.trieste.it
RUPERT W.T. YU
yuyu@mathlabo.univ-poitiers.fr
CNRS ESA 6086, Université de Poitiers, Boulevard 3, Teleport 2, BP179, 86960 Futurescope cedex, France
Received December 16, 1997; Revised June 19, 1998
Abstract. Let w be an element of the Weyl group of sln+1 . We prove that for a certain class of elements w
(which includes the longest element w0 of the Weyl group), there existP
a lattice polytope 1iw ⊂ R`(w) , for each
n
,
such
that
for
any
dominant
weight
λ
=
fundamental weight ωi of slP
n+1
i=1 ai ωi , the number of lattice points in
n
w
w
the Minkowski sum 1λ = i=1 ai 1i is equal to the dimension of the Demazure module E w (λ). We also define
P
w
a linear map Aw : R`(w) −→ P ⊗Z R where P denotes the weight lattice, such that char E w (λ) = eλ e−A (x)
w
where the sum runs through the lattice points x of 1λ .
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. Let b be a Borel subalgebra of sln+1 (K ) and h ⊂ b a Cartan
subalgebra. Let αi , i = 1, . . . , n, be the corresponding set of positive simple roots so
that hαi , α ∨j i = ai j where (ai j )i, j is the Cartan matrix, and let ωi be the corresponding
fundamental weights. Denote by P, P + , W , `(−) and ¹ respectively 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 . Let λ ∈ P + and w ∈ W . Set Vλ to be
the finite-dimensional irreducible representation of highest weight λ, vwλ to be a non-zero
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DEHY AND YU
weight vector of weight wλ and E w (λ) to be the b-module U (b)vwλ which is called the
Demazure module [5] associated to w. Set W i to be the stabilizer of ωi in W and Wi the
quotient W/W i . Endow Wi with the induced Bruhat order that we shall denote equally
by ¹ and if σ ∈ Wi , then we shall denote by `(σ ) the induced length of σ , which is the
minimum of the lengths of representatives of σ .
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
P which are combinatorial.
Let λ = i ai ωi be a dominant weight, then the dimension of E w (λ) is a polynomial in
of its dual E w (λ)∗ can be described
the variables ai of degree `(w) because the dimension N
i
as the Euler characteristic of the ample line bundle i L⊗a
ωi over the Schubert variety
associated to w in G/Pλ ([7, 18.3.6] or [2, 2.3]). Whereas, given convex lattice polytopes
1i in R`(w) , a theorem of Ehrhart
P that under the condition that a lattice point
P [6] implies
in the Minkowski sum 1 := i ai 1i = { i ai vi where vi ∈ 1i } is the sum over i of ai
lattice points of 1i , the number of lattice points in 1 is a polynomial of degree `(w) in
the variables ai . On the other hand, suppose that we have a degeneration of the Schubert
. . , Lωn into the toric variety
with
variety Sw equipped with line bundles Lω1 , .N
N X⊗aequipped
i
0
i
line bundles L1 , . . . , Ln . Then dim H 0 (Sw , i L⊗a
)
=
dim
H
(X,
L
).
But
to
say
ωi
i i
, Ln is equivalent to having n lattice polytopes
that X is equipped with line bundles L1 , . . .N
⊗ai
w
`(w)
0
1w
1 , . . . , 1n in R
Pn such wthat dim H (X, i Li ) is the number of lattice points in the
Minkowski sum i=1 ai 1i (for example, see properties B3, B4 of Section 2.3 in [19]).
These facts lead us to construct a polytope 1iw for each fundamental weight ωi and 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
exist lattice polytopes 1i ⊂ R`(w0 ) , i = 1, . . . , n, such that
Pn 1.1 There
Pn for any
+
λ = i=1 ai ωi ∈ P , the number of lattice points in the Minkowski sum 1λ := i=1
ai 1i
is the dimension of the irreducible representation Vλ .
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 of 1λ can be written as a sum over i of ai lattice points of 1i .
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.
LATTICE POLYTOPES FOR DEMAZURE MODULES
151
Conjecture 1.2 Let P
w ∈ W . There exist lattice polytopes 1iw ⊂ R`(w) , i = 1, . . . , n,
n
ai ωi ∈ P + , the number of lattice points in the Minkowski sum
such that
any λ = i=1
Pfor
n
w
w
1λ := i=1 ai 1i is the dimension of the Demazure module E w (λ).
As a matter of fact, the polytopes 1i constructed in Theorem 1.1 are such that the vertices
{vτ }τ ∈Wi are indexed by Wi . We believe that 1iw of the conjecture can be chosen as the
convex hull of {vτ }τ ¹w embedded (by a permutation of coordinates) in R`(w) .
Indeed, we prove that this is true when w can 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 any w ∈ 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 Wi . We
shall prove later in Section 5 that 1i is triangulable by primitive simplices parametrized
by maximal chains. We then present an example in Section 3. In SectionsP
4–6, we show
n
ai 1i can
how, in the case where w = w0 , a lattice point in the Minkowski sum i=1
be written as a sum over i of ai lattice points of 1i , and P
that these points exhaust the
n
ai ωi . Sections 7 and 8
dimension of the irreducible representation Vλ where λ = i=1
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 endow R 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 ∆i for 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 elements w such that ws j º w for all j 6= i. It is also well known that Wi
is in bijection with the set of i-tuples (r1 , . . . , ri ) such that 0 ≤ r1 < r2 < · · · < ri ≤ n.
Namely, we can think of W = Sn+1 as the group of permutations on the set {0, 1, . . . , n}.
Then the bijection w 7→ (r1 , . . . , ri ) is given by {r1 , . . . , ri } = w({0, 1, . . . , i − 1}).
The induced Bruhat order on Wi is then given by:
(r1 , . . . , ri ) ≺ (s1 , . . . , si ) ⇔ (r1 , . . . , ri ) < (s1 , . . . , si )
where on the right hand side, the i-tuples are considered as elements of Ri .
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DEHY AND YU
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 , . . . , ri ) is
sr1 sr1 −1 · · · s1 sr2 sr2 −1 · · · s2 sr3 · · · sri sri −1 · · · si
where sr j · · · 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 lexicographic minimal expression w0 = s1 s2 s1 s3 s2 s1 · · · sn sn−1 · · · s1 . Notice that each minimal
representative of Wi 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 s2 s1 , . . . ,
block n is sn sn−1 · · · s1 .
Let us write the standard basis vectors in R`(w0 ) as e pq with 1 ≤ q ≤ p ≤ n. Let
1 ≤ i ≤ n, and (r1 , . . . , ri ) be an element of Wi , we then define
ϕ(r1 , . . . , ri ) =
n
X
rX
p+i−n
e pq ∈ R`(w0 )
p=n−i+1 q= p+i−n
Definition 2.2 Let c: τ1 Â · · · Â τm be a chain in Wi . We define Sc to be the convex hull
of the points {ϕ(τ j )}mj=1 and we define 1i to be the convex hull of the points {ϕ(τ )}τ ∈Wi .
Lemma 2.3
(a) The vertices of 1i are the only lattice points in 1i and they are indexed by the elements
of Wi .
(b) The map ϕ is order-preserving.
(c) Let c: τ1  · · ·  τ(n−i+1)i  1 be a maximal chain in Wi . The polytope Sc is 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 of ϕi . For part
(c), notice that the points ϕ(τ1 ), . . . , ϕ(τ(n−i+1)i ) are linearly independent and ϕ(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 polytopes 1i .
Proposition 2.4
(a) The intersection
of 1i and 1 j is {0} whenever i 6= j.
P
(b) Let x = P p,q x pq e pq ∈ 1i , then x pq = 0 if p < n − i + 1.
0
(c) If x =
p,q x pq e pq ∈ 1i is such that x st 6= 0, then x st 0 6= 0 for all t = t, t +
1, . . . , rs+i−n .
153
LATTICE POLYTOPES FOR DEMAZURE MODULES
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 of 1i is non-zero
2
while it is zero for any element of 1 j . Thus (a) follows.
Pn
ai ωi be a dominant weight (thus each ai ∈ N) and Vλ be the irreducible
Let λ = i=1
sln+1 -module of highest weight λ.
Definition 2.5
We define the polytope 1λ to be the Minkowski sum
Pn
i=1
ai 1i .
Since the 1i ’s are lattice convex polytopes, the polytope 1λ is also a lattice convex
polytope. We can now state our theorem in the case where w = w0 .
Theorem 2.6 The number of lattice points in 1λ is equal to the dimension of Vλ .
3.
Example
The first interesting example is sl4 . We write w0 = s1 s2 s1 s3 s2 s1 and we have, in terms of
minimal representatives,
W1 = {1, s1 , s2 s1 , s3 s2 s1 },
W3 = {1, s3 , s2 s3 , s1 s2 s3 }
W2 = {1, s2 , s3 s2 , s1 s2 , s1 s3 s2 , s2 s1 s3 s2 }
We then obtain via ϕ the following table where each row contains the coefficients of a ϕ(τ ):
s1
s2
s1
s3
s2
s1
e11
e22
e21
e33
e32
e31
s1
0
0
0
0
0
1
(1)
s2 s1
0
0
0
0
1
1
(2)
s3 s2 s1
0
0
0
1
1
1
(3)
s2
0
0
0
0
1
0
(0, 2)
s3 s2
0
0
0
1
1
0
(0, 3)
s1 s2
0
0
1
0
1
0
(1, 2)
s1 s3 s2
0
0
1
1
1
0
(1, 3)
s2 s1 s3 s2
0
1
1
1
1
0
(2, 3)
s3
0
0
0
1
0
0
(0, 1, 3)
s2 s3
0
1
0
1
0
0
(0, 2, 3)
s1 s2 s3
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 is ω1 + ω3 . One then
verifies easily by hand that a lattice point of 11 + 13 is the sum of a lattice point of 11 and
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DEHY AND YU
a lattice point of 13 . Hence a quick computation shows that the lattice points are the ones
in 11 and 13 together 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 in 11 + 13 which is the dimension of sl4 .
Remark that ϕ(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
¡ ¢∗
Vω1 ⊗ Vω3 ∼
= Vω1 ⊗ Vω1 ∼
= gl4 = sl4 ⊕ V0
4.
Correspondence with semi-standard Young tableaux
Pn
ai ωi be a dominant weight. Set W be the disjoint union of the Wi and
Let λ = Q i=1
n Qai
W(λ) = i=1
j=1 Wi . We can associate to an element
P of W(λ) via ϕ a lattice point of
1λ . Namely, an element (wi j )i, j of W(λ) is sent to i, j ϕ(wi j ) in 1λ .
However this association is not necessarily injective (that is, a lattice point can be the
image of another element in W(λ)). We claim that with respect to a certain partial ordering
of W, 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 in W(λ) is in bijection
with the set of semi-standard Young tableaux of type λ.
Let us first define our partial order in W, denoted by ≺, which extends the induced Bruhat
ordering in Wi . Let (r1 , . . . , ri ) and (s1 , . . . , s j ) be two elements of W, then
(r1 , . . . , ri ) ≺ (s1 , . . . , s j ) ⇔ (−1, . . . , −1, r1 , . . . , ri ) < (−1, . . . , −1, s1 , . . . , s j )
|
|
{z
}
{z
}
n−i
n− j
where the elements on the right hand side are in Rn .
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 set W is a lattice, that is, every pair of elements of W have a well defined max and
min.
(b) If r ∈ Wi , s ∈ W j and i ≤ j, then we have min(r, s) ∈ Wi and max(r, s) ∈ W j .
(c) (MAX–MIN) Let r, s ∈ W, then ϕ(r ) + ϕ(s) = ϕ(max(r, s)) + ϕ(min(r, s)).
Proof: Let r = (r1 , . . . , ri ) be an element of W. By adding −1’s on the left as above, we
can associate to r , an element R = (R1 , . . . , Rn ) of Rn .
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LATTICE POLYTOPES FOR DEMAZURE MODULES
Definition 4.3 Let r , s be elements of W and R, S the corresponding associated elements
in Rn . We define min(R, S) = (T1 , . . . , Tn ) where Ti = min(Ri , Si ) and min(r, s) the
element of W associated 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 , . . . , ri ) and s = (s1 , . . . , s j ) where i ≤ j. Then
ϕ(r ) + ϕ(s) =
n
X
rX
p+i−n
p=n−i+1 q= p+i−n
=
n
X
rX
p+i−n
=
Ã
n−i
X
e pq +
e pq
rX
p+i−n
sX
p+ j−n
e pq +
n
X
min(r p+i−n
X,s p+ j−n )
p=n−i+1
q= p+i−n
!
sX
p+ j−n
e pq
q= p+ j−n
e pq +
n
X
e pq +
p=n− j+1 q= p+ j−n
p=n−i+1 q= p+i−n
=
sX
p+ j−n
p=n− j+1 q= p+ j−n
p=n−i+1 q= p+i−n
n
X
n
X
e pq +
+
sX
p+ j−n
e pq
p=n−i+1 q= p+ j−n
n−i
X
sX
p+ j−n
e pq
p=n− j+1 q= p+ j−n
n
X
max(r p+i−n
X,s p+ j−n )
p=n− j+1
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 Let θ = {θi j }i=1,...,n; j=1,...,ai be an element of W(λ). Then there exists a
unique element ψ = {ψi j } of W(λ) such that
(i) ψ
Pk or if i = k and j ≤ `;
Pi j ¹ ψk` if i <
ϕ(θ
)
=
(ii)
ij
i, j
i, j ϕ(ψi j ).
Before proving this proposition, let us remark that condition (i) says that
ψ11 ¹ ψ12 ¹ · · · ¹ ψ1a1 ¹ ψ21 ¹ · · · ¹ ψ2a2 ¹ · · · ¹ ψ(n−1)an−1 ¹ ψn1 ¹ · · · ¹ ψnan
This is similar to the definition for a Young tableaux of Lakshmibai and Seshadri of type λ
modulo liftings to the Weyl group W , see [14]. As we shall see, our theorem says that this
is exactly the same definition.
Pn
ai . It is clear that the
Proof: We shall prove the existence by induction on a = i=1
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 ar 6= 0. By the induction hypothesis, we can suppose that θ 0 = θ \ {θrar } satisfies the
conditions (i) and (ii) of the proposition.
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DEHY AND YU
We shall now divide θ 0 into three disjoint totally-ordered sets. Let
E ≺ = {θi j | θi j ≺ θrar },
E º = {θi j | θi j º θrar }
and
E 0 = {θi j | θi j and θrar are not comparable}.
Note that the elements of E º are all in Wr .
If E 0 is empty, then we can insert θrar in 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 E 0 is not empty. Then θrar is in neither E ≺ , E º nor E 0 . Let φ be the
maximal element in E 0 . By max–min of Lemma 4.2, replacing φ and θrar by max(φ, θrar )
and min(φ, θrar ) does not alter the sum via ϕ. Furthermore, if we let E ≺0 , E º0 and E 00 be
0
= max(φ, θrar ), then the cardinal of E 00
the new partition as defined above relative to θra
r
is strictly less than E 0 since min(φ, θrar ) will belong to E ≺0 .
Now repeat the same procedure until E 0 is empty and we have the existence since E 0 is
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, m k such that
one of the following conditions is satisfied:
(i) the ekm k -coordinate is 1 for ϕ(r ) and the ek` -coordinate for ϕ(s) is 0 for all m k ≤ ` ≤ k.
(ii) the ekm k -coordinate is 1 for ϕ(s) and the ek` -coordinate for ϕ(r ) is 0 for all m k ≤ ` ≤ 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
m k such that for all t ∈ W j satisfying t ¹ s, the ekm k -coordinate is 1 for ϕ(r ) and the
ek` -coordinate for ϕ(t) is 0 for all m k ≤ ` ≤ 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 > sk or 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)th block. By the definition of our embedding, it is clear that if we put m k = rk
(resp. m k = sk ), 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
rk > sk . Now let t ∈ W j be such that t ¹ s. 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 t ¹ s, we have tk ≤ sk < rk .
2
It follows again from our embedding that we have our result by letting m k = rk .
We can now finish our proof. Let θ and θ 0 = {θi0j } be two elements in W(λ) satisfying
0
the conditions of the proposition. Let r be maximal such that ar 6= 0. Then θrar and θra
are
r
0
0
maximal in θ and θ respectively. If θrar 6= θrar , then by applying the previous lemma, we
can suppose that there exists k, m k such that the entry ekm k is 1 for ϕ(θrar ) and the entries ek`
0
for ϕ(θra
) is 0 for all m k ≤ ` ≤ k. Hence by the same lemma, the same entries for ϕ(θi0j )
r
LATTICE POLYTOPES FOR DEMAZURE MODULES
157
P
P
0
is maximal in θ 0 . It follows that i, j ϕ(θi j ) − i, j ϕ(θi0j ) 6= 0
are 0 for all i, j since θra
r
which contradicts the fact that θ and θ 0 satisfy the second condition of the proposition.
0
. Now by induction on a, the sum of the ai ’s, the elements θ , θ 0 must be
Thus θrar = θra
r
the same (the case a = 1 is equivalent to the fact that ϕ is an embedding).
2
Thus we have proved what we claimed at the start of this section. Let us denote by W(λ)d
the set of elements in W(λ) satisfying property (i) of the proposition. Now given an element
θ in W(λ)d , using the notations (r1 , . . . , ri ) for elements in Wi , we can arrange each θi 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 from W(λ)d
to the set of semi-standard Young tableaux of type λ, which is obviously injective. On the
other hand, given a semi-standard Young tableau of type λ, we obtain an element of W(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 in 1i are in bijection with elements of Wi , thus Proposition 4.4
says that W(λ)d is in bijection with the set of lattice points in 1λ which can be written as
a sum over i of ai lattice points of 1i , we can hence state
Theorem 4.6 The set of lattice points of 1λ which can be written as a sum over i of ai
lattice points of 1i is in bijection with the set of semi-standard Young tableaux of type λ.
Remark 4.7 In fact, the existence part of Proposition 4.4 can be proved with semistandard 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
Pn
Let λ =
i ωi be a dominant weight. Recall from Definition 3.2 that 1λ is the
i=1 aP
n
ai 1i , where 11 , . . . , 1n are the polytopes associated to the fundaMinkowski sum i=1
mental weights ω1 , . . . , ωn which were defined in Section 2. In this section we shall prove
the following theorem:
Pn
Theorem 5.1 A lattice point in the Minkowski sum i=1
ai 1i can be written as the sum
of a1 lattice points in 11 , a2 lattice points of 12 and so on.
158
DEHY AND YU
As in the previous section, denote by W the union over all i of Wi equipped with the
partial order defined in the same section, and for any dominant weight µ, denote by W(µ)d
the set of elements in W(µ) satisfying property (i) of Proposition 4.4. Let θ = {θi j }i, j be
an element of W(µ)d , we shall denote by Cµ (θ ) the convex cone generated by {ϕ(θi j )}i, j .
Theorem 5.2 Let x ∈ 1λ , then there exist a dominant weight µ and a θ ∈ W(µ)d such
that x ∈ Cµ (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 of W such that σi j ∈ Wi .
Let pi j ∈ R+ . Then there exists {σi0j }i=1,...,n; j=1,...,ai0 a sequence of elements of W and
pi0 j ∈ R+ such that
(i) σi0j ∈ Wi .
(ii) σi0j ≺ σkl0 if i < k or if i = k and j < l.
Pai
Pai0
0
(iii)
j=1 pi j =
j=1 pi j .
Pn Pai
Pn Pai0
0
0
(iv)
i=1
i=1
j=1 pi j ϕ(σi j ) =
j=1 pi j ϕ(σi j ).
Pn
Proof: We shall prove the lemma by induction on the i=1
ai .
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,...,ai \{σlal } satisfies (i) and (ii) of the lemma. For simplicity we
shall denote σlal by σ and q = plal .
If σ º σi j for all i = 1, . . . , l, j = 1, . . . , ai or if σ = σl j for some j ≤ al − 1, then we
are done.
So let us suppose the contrary. Then
P there exists τ = σcd minimal such that σ 6Â τ . Let
κ = σr s be maximal such that P := τ ¹σi j ≺κ pi j ≤ q. Denote by m i j = min(σ, σi j ) ∈ Wi
ij
and by Ml = max(σ, σi j ) ∈ Wl . Note that we have
ij
i, j−1
Ml º Ml
º · · · º Mlcd  σ  m i j º · · · º m cd
Now using repeatedly max–min of Lemma 4.2, we obtain:
ai
l X
X
pi j ϕ(σi j ) =
X
pi j ϕ(σi j ) +
σi j ≺τ
i=1 j=1
+
X
X
pi j (ϕ(σi j ) + ϕ(σ )) + (q − P)ϕ(σ )
pi j ϕ(σi j ) +
σi j ≺τ
+
X
τ ¹σi j ≺κ
pi j ϕ(σi j )
σi j ºκ
τ ¹σi j ≺κ
=
X
X
σi j Âκ
pi j ϕ(σi j )
¡ i j ¢¢
¡
pi j ϕ(m i j ) + ϕ Ml
+ pr s ϕ(κ) + (q − P)ϕ(σ )
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LATTICE POLYTOPES FOR DEMAZURE MODULES
Now if pr s ≤ q − P, then we must have κ = σl,al −1 . Consequently, we have
ai
l X
X
pi j ϕ(σi j ) =
i=1 j=1
X
pi j ϕ(σi j ) +
σi j ≺τ
X
τ ¹σi j ¹κ
¡ i j ¢¢
¡
pi j ϕ(m i j ) + ϕ Ml
+ (q − P − pr s )ϕ(σ )
Thus we obtain a chain
Mlr s º · · · º Mlcd  σ  m r s º m r,s−1 º · · · º m c,d  σc,d−1  · · ·  σ11  1
from which we can compress into a chain {σi0j } where i = 1, . . . , l and j = 1, . . . , ai0
satisfying the required properties of the lemma.
If pr s ≥ q − P, then:
ai
l X
X
pi j ϕ(σi j ) =
X
pi j ϕ(σi j ) +
σi j ≺τ
i=1 j=1
+
X
pi j ϕ(σi j )
σi j Âκ
X
τ ¹σi j ≺κ
¡ i j ¢¢
¡
pi j ϕ(m i j ) + ϕ Ml
+ (q − P)(ϕ(κ) + ϕ(σ )) + ( pr s − (q − P))ϕ(κ)
X
X
pi j ϕ(σi j ) +
pi j ϕ(σi j )
=
σi j ≺τ
+
X
τ ¹σi j ≺κ
σi j Âκ
¡ i j ¢¢
pi j ϕ(m i j ) + ϕ Ml
¡
¡
¢¢
¡
+ (q − P) ϕ(m r s ) + ϕ Mlr s + ( pr s − (q − P))ϕ(κ)
Thus we obtain a chain
σl,al −1  · · ·  κ  m r s º m r,s−1 º · · · º m cd  σc,d−1 · · ·  σ11  1
from which we can compress into a chain {σi0j } where i = 1, . . . , l and j = 1, . . . , ai0
satisfying (i), (ii) and (iii) of the lemma (look at the coefficents). Therefore we have
ai
l X
X
i=1 j=1
0
pi j ϕ(σi j ) =
ai
l X
X
i=1 j=1
pi0 j ϕ(σi0j ) +
X
τ ¹σi j ≺κ
¡ ij¢
¡
¢
pi j ϕ Ml + (q − P)ϕ Mir s
We now observe that the length of the remaining elements Mlcd , . . . , Mlr s are strictly greater
than that of σ . Thus we can repeat the same reasoning and the lemma is proved because
2
there is a maximal element in Wl .
Corollary 5.4 The polytope 1i is triangulable by primitive simplices of dimension (n −
i + 1)i.
160
DEHY AND YU
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
that 1i is the union of all the Sc where c is a chain in Wi (see Definition 2.2). Moreover, if
c0 is a (strict) subchain of c, then Sc0 lies in the boundary of Sc . Since by Lemma 2.3, Sc is
a primitive simplex of dimension (n − i + 1)i when c is a maximal chain, to show that 1i
is triangulable by primitve simplices, it suffices to show that the interior of any two distinct
simplices Sc and Sc0 do not meet.
Consider two chains c: σ1  · · ·  σ`  1 and c0 : τ1  · · ·  τm  1. Suppose that the
intersection of the interiors of Sc and Sc0 is not empty and that Q belongs to this intersection.
We can therefore write Q as (recall that ϕ(1) = (0, . . . , 0) ∈ R`(w0 ) )
X̀
p j ϕ(σ j ) = Q =
j=1
m
X
qk ϕ(τk ) (∗)
k=1
where p j , qk ∈]0, 1[ and p1 + · · · + p` ≤ 1, q1 + · · · + qm ≤ 1.
Assume that σ1 6= τ1 . Writing σ1 = (s1 , . . . , si ) and τ1 = (t1 , . . . , ti ). By Lemma 4.5,
there exists a coordinate e pq which is non-zero on the left hand side of (∗), whereas it is zero
on the right hand side (because τ1 is maximal in the chain c0 ). So we have a contradiction
and therefore σ1 = τ1 .
Without loss of generality, we can suppose p1 ≥ q1 . We can then rewrite (∗) as follows:
( p1 − q1 )ϕ(σ1 ) +
X̀
p j ϕ(σ j ) = Q 0 =
j=2
m
X
qk ϕ(τk ) (∗∗)
k=2
Consequently, we must have p1 = q1 . Now by repeating the same argument (or use
induction on ` + m) on (∗∗), we conclude that ` = m, p j = q j and σ j = τ j for all
2
j = 1, . . . , `. That is c = c0 . Thus the corollary is proved.
P
Proof of Theorem 5.1: Suppose that x is a lattice point of li=1 ai 1i . Without loss of
generality we can assume that al 6= 0. We can write x = x1 + · · · + xl where
¡ ¢
xi = pi1 ϕ(σi1 ) + · · · + piri ϕ σiri
with pi j > 0 and
ri
X
pi j = ai
j=1
where σi j ∈ Wi . By Theorem 5.2, we can assume that σi j is a strictly increasing sequence
of elements of W.
If rl = 1, then plrl = al and so xl = al ϕ(σlrl ) which implies that x − xl is a lattice point
P
of l−1
i=1 ai 1i .
If rl > 1, then by Lemma 4.5, there exists a coordinate eαβ such that the eαβ coordinate
of x is equal to plrl . So plrl is a positive integer and x − plrl ϕ(σlrl ) is a lattice point of
Pl−1
i=1 ai 1i + (al − plrl )1l .
P
2
Thus, in both cases the assertion follows by induction on li=1 ai .
LATTICE POLYTOPES FOR DEMAZURE MODULES
6.
161
End of proof of Theorem 2.6
By Theorem 5.1, an integral point in 1λ is a sum over i of ai lattice points of 1i . Hence
by Theorem 4.6, the set of lattice points of 1λ is in bijection with the set of semi-standard
Young tableaux of type λ. Now by a classical result from the theory of invariants (see for
example [8] or [18]) that the number of semi-standard Young tableaux of type λ is exactly
the dimension of the sln+1 -module Vλ . Thus Theorem 2.6 is proved.
7.
The case of Demazure modules
In the previous sections, we explained how to construct for each fundamental weight a
polytope 1i whose vertices are indexed by the set Wi . Let Wiw be the set {σ ∈ Wi | σ ¹ w̄}
where w̄ is the class of w in Wi . Then we can define the polytope 1iw to be the convex hull
of the set of vertices of 1i corresponding to the set Wiw . It is clear that the vertices of 1iw
are indexed by the set Wiw .
WePwould like to embed 1iw in R`(w) in P
such a way that given a dominant weight
λ = i ci ωi , the number of lattice points in i ci 1iw is the dimension of the Demazure
module E w (λ). For some w we can construct such an embedding. In this section we shall
describe this embedding and explain why it works.
Recall that W is considered as the permutations of the set {0, 1, . . . , n}, with simple
transpositions si = (i − 1, i). Consider the unique factorization of a permutation w ∈ W
in the form
w = s(1, c1 )s(2, c2 ) · · · s(n, cn )
where we denote s(a, b) = sa sa−1 · · · sb and s(a, a + 1) = 1. Then c j is the cardinal
of the set {d such that d ≤ w −1 ( j), w(d) ≤ j}. It follows that there exist k ∈ N∗ ,
1 ≤ a1 < a2 < · · · < ak ≤ n and 1 ≤ b j ≤ a j for all j = 1, . . . , k such that
w = s(a1 , b1 )s(a2 , b2 ) · · · s(ak , bk )
Note that we have w0 = s(1, 1)s(2, 1) · · · s(n, 1). We shall use this notation in this section.
Definition 7.1 Let d ≤ e be positive integers. We shall call the subexpression s(ad , bd )
· · · s(ae , be ) of w, a part if the following conditions are satisfied:
(i) ae + 1 < bm for all m > e,
(ii) ad−1 + 1 < bm for all d ≤ m ≤ e.
A part of w is connected if it is not the product of two distinct parts of w.
It is clear that w is the product of connected parts, say w = P1 · · · Pl , and that when
1 ≤ i 6= j ≤ l, then Pi commutes with P j .
Theorem 7.2 Suppose that w ∈ W is either the identity or else each connected part
s(ad , bd ) · · · s(ae , be ) of w satisfies one of the following conditions:
162
DEHY AND YU
(i) bd ≥ bd+1 ≥ · · · ≥ be .
(ii) ad = bd < ad+1 = bd+1 < · · · < ae = be ,
any dominant
Then for each
ϕiw of 1iw in Rl(w) such that forP
P i, there exists an embedding
w
w
w
`(w)
=
weight λ = i ci ωi , we have Card(1λ ∩ Z ) = dim E w (λ) where 1w
λ
i ci ϕi (1i ).
Remark 7.3
(1) As remarked by one of the referees, there is no obvious relation between the set of
Kempf elements (for the definition of Kempf elements, see [13]) and the set of w
whose connected parts satisfy condition (i) or (ii) of the theorem. Note that in the case
of sl3 , all the elements of the Weyl group satisfy the conditions of the theorem, while
s2 s1 is not a Kempf element.
(2) In the case of sl4 , there are exactly 7 elements in W which satisfy neither of the 2 conditions. Namely they are s(1, 1)s(3, 2), s(2, 1)s(3, 3), s(2, 1)s(3, 2), s(1, 1)s(2, 2)s(3, 2),
s(1, 1)s(2, 1)s(3, 3), s(1, 1)s(2, 1)s(3, 2) and s(1, 1)s(2, 2)s(3, 1). However, a case
by case analysis shows that, by using the same construction, the theorem is true in these
cases.
(3) Let λ be a dominant weight. If w0 ≡ w modulo Wλ , the stabiliser of λ, then E w0 (λ) =
E w (λ). Therefore, if we can find a w satisfying the conditions of Theorem 7.2, then the
number of lattice points in m1w
λ is equal to the dimension of E w0 (mλ) for any m ∈ N.
In particular, such elements can always be found in the case of fundamental weights
(that is, when the stabiliser is Wi for some i).
Let us fix w = s(a1 , b1 )s(a2 , b2 ) · · · s(ak , bk ).
Let us make the idea behind our embedding more precise. Since 1iw is the convex hull of
its vertices and that the vertices are in one-to-one correspondence with the elements of Wiw ,
we simply need to specify the image of the vertex corresponding to an element σ ∈ Wiw ,
denoted by ϕiw (σ ). We have
σ = (r1 , . . . , ri ) = s(r1 , n − i + 1)s(r2 , n − i + 2) · · · s(ri , i).
The following description of ϕiw (σ ) may seem vague, but with the example that follows
it will become more transparent. We shall index the standard basis of R`(w) using the
expression of w, that is, we write the standard basis as e pq where p = 1, . . . , k and q =
b p , . . . , a p . Consider the rightmost subexpression of w identical to the above expression
of σ . Then we define the coefficient of e pq of ϕiw (σ ) to be 1 if the index belongs to this
subexpression, and zero otherwise.
Let us clarify all this with an example. Let w = s2 s1 s3 s2 s1 = s(2, 1)s(3, 1) be an element
of the Weyl group of sl4 . It satisfies condition (i) of Theorem 7.2. We have
W1w = {1, s1 , s2 s1 , s3 s2 s1 },
W2w = {1, s2 , s3 s2 , s1 s2 , s1 s3 s2 , s2 s1 s3 s2 },
W3w = {1, s3 , s2 s3 }
According to the discussion above, ϕiw (1) is the zero vector for any i. To specify ϕ1w (s1 ),
we “embed” s1 as right as possible in the expression s2 s1 s3 s2 s1 , so we get ϕ1w (s1 ) =
163
LATTICE POLYTOPES FOR DEMAZURE MODULES
(0, 0, 0, 0, 1). Similarly, we “embed” s2 s1 as right as possible in s2 s1 s3 s2 s1 , and we get
ϕ1w (s2 s1 ) = (0, 0, 0, 1, 1) and so forth. Hence, we obtain the following table.
s2
s1
s3
s2
s1
s1
0
0
0
0
1
s2 s1
0
0
0
1
1
s3 s2 s1
0
0
1
1
1
s2
0
0
0
1
0
s3 s2
0
0
1
1
0
s1 s2
0
1
0
1
0
s1 s3 s2
0
1
1
1
0
s2 s1 s3 s2
1
1
1
1
0
s3
0
0
1
0
0
s2 s3
1
0
1
0
0
Although it is easy to describe the image of σ in this way, this description needs to be
formalised so that we can prove that it works.
Definition 7.4 Let w = s(a1 , b1 ) · · · s(ak , bk ). Define pi = ( p1i , . . . , pii ) by reverse
i
= +∞, then
induction (i.e., starting from i and going down to 1). Set pi+1
(
pij
=
©
ª
max l | bl ≤ j ≤ al , l < pij+1
if it exists
−1
otherwise
In other words, if we write u j = s(a j , b j ), then pii is the biggest integer l such that si
i
occurs in u l (or in u l u l+1 · · · u k ). And pi−1
is the biggest integer l 0 such that si−1 si appears
as a subexpression of u l 0 u l 0 +1 · · · u k . And so on.
For instance, let us look at the example above where we let w = s2 s1 s3 s2 s1 = s(2, 1)s(3, 1).
Here we have a1 = 2, b1 = 1, a2 = 3, b2 = 1. Thus according to the definition p1 = (2),
p2 = (1, 2), p3 = (−1, 1, 2). Note that if w = w0 , then pi = (n − i + 1, . . . , n).
Remark 7.5 Note that the class of w in Wiw is w̄ = (0, 1, 2, . . . , l − 2, a pli , . . . , a pii ),
i
where l is maximal such that p1i = · · · = pl−1
= −1. Therefore an element (r1 , . . . , ri ) of
Wi is in Wiw if and only if rl−1 = l − 2 and r j ≤ a pij for l ≤ j ≤ i.
Now we can formalise the description of ϕiw given above. Let w = s(a1 , b1 ) · · · s(ak , bk )
and let us write the standard basis of R`(w) as e pq with p = 1, . . . , k and q = b p , b p +
1, . . . , a p . We define the map ϕiw : Wiw −→ R`(w) by sending
(r1 , . . . , ri ) 7−→
X X
p= pli l≤q≤rl
e pq
164
DEHY AND YU
Definition 7.6
w
the image of Wiw via ϕiw .
(i) We define,
Pnby abuse of notation, 1i to be the convex hull of
w
(ii) Let
Pn λ = wi=1 ci ωi be a dominant weight, then we define 1λ to be the Minkowski sum
i=1 ci 1i .
8.
Proof of Theorem 7.2
We shall first prove that the conditions (i) and (ii) of Theorem 7.2 give nice properties on
pi . Then we shall define a partial order on the union of Wiw similar to the one given in
Section 4. Finally, we prove Theorem 7.2 by showing that there is a one-to-one correspondence between lattice points in 1λ and the standard monomial basis of the Demazure
module E w (λ).
In this section, we shall fix an element w = s(a1 , b1 ) · · · s(ak , bk ) of W which satisfies
the conditions of Theorem 7.2. By definition, non-negative entries of pi are distinct. We
shall denote by B(pi ) the set of non-negative entries of pi .
Lemma 8.1 Let us suppose that B(pi ) is not empty and let u i (resp. vi ) be minimal (resp.
maximal ) in B(pi ).
(i) The element s(au i , bu i )s(au i +1 , bu i +1 ) · · · s(avi , bvi ) occurs (as a subexpression) in a
connected part of w.
(ii) The set B(pi ) is a set of consecutive integers, that is, B(pi ) = {m ∈ N∗ | u i ≤ m ≤ vi }
where u i and vi are as defined in (i).
(iii) If i < j and B(p j ) is non-empty, then vi ≤ v j .
Proof: Assertion (i) is a direct consequence of the definition of a connected part.
By definition, B(pi ) ⊂ {1, . . . , k}. Let us suppose that there exists r > 1 such that
r ∈ B(pi ) and r − 1 6∈ B(pi ). To prove (ii), il suffices to show that r is minimal in B(pi ).
There exists j such that r = pij . Hence, by the definition of pi , we have br ≤ j ≤ ar ,
and that either j − 1 < br −1 or ar −1 < j − 1.
If br − 1 ≤ j − 1 < br −1 , then br −1 ≥ br . We are therefore in a connected part of w
satisfying condition (i) of Theorem 7.2. It follows that pij−1 = −1.
If ar −1 < j −1, then j −1 > al for all l = 1, . . . , r −1. It follows again that pij−1 = −1.
Consequently, pij−1 = −1 in both cases and therefore r is minimal in B(pi ).
j
Finally, for (iii). Suppose that v j < vi , then av j < avi . Since vi = pii and v j = p j , we
would have
bvi ≤ i ≤ j ≤ av j < avi
j
Therefore v j = p j ≥ vi , contradiction.
2
In view of the lemma, we define s(pi ) to be s(au i , bu i )s(au i +1 , bu i +1 ) · · · s(avi , bvi ). By
part (i), s(pi ) occurs in a connected part of w.
LATTICE POLYTOPES FOR DEMAZURE MODULES
165
Corollary 8.2 Let i ≤ j be such that B(pi ) and B(p j ) are not empty. Let u i , vi , u j , v j
be as defined in Lemma 8.1. Then we have one of the following three possibilities:
(i) s(pi ) and s(p j ) both occur in a connected part of w which satisfies condition (i) of
Theorem 7.2. Furthermore vi = v j .
(ii) s(pi ) and s(p j ) both occur in a connected part of w which satisfies condition (ii) of
Theorem 7.2. Furthermore, vi + j − i = v j , u i = u j .
(iii) s(pi ) and s(p j ) occur in different connected parts of w. Furthermore, this implies that
i < j and avi + 1 < bl for all u j ≤ l ≤ v j .
Example 8.3 Let us look at some examples:
(1) Let w = w0 , then B(pi ) = {m ∈ Z | n − i + 1 ≤ m ≤ n} and so u i = n − i + 1,
vi = n.
(2) Let n = 3 and w = s(1, 1)s(3, 1), then B(p1 ) = {2}, u 1 = v1 = 2, B(p2 ) = {1, 2},
u 2 = 1, v2 = 2, and B(p3 ) = {2}, u 2 = v2 = 2.
`n
Wiw .
We shall now define a partial order on W w := i=1
w 0
0
0
w
Let r = (r1 , . . . , ri ) ∈ Wi , r = (r1 , . . . , r j ) ∈ W j with i ≤ j. And let u i , vi , u j , v j be
as in Lemma 8.1.
Lemma 8.4 Suppose that s(pi ) and s(p j ) both occur in a connected part of w which
satisfies condition (i) of Theorem 7.2, we define max(r, r0 ), min(r, r0 ) as in Section 4. Then
max(r, r0 ) ∈ W jw and min(r, r0 ) ∈ Wiw .
Proof: From Section 4, we know that max(r, r0 ) ∈ W j and min(r, r0 ) ∈ Wi . So we only
need to show that they are in W jw and Wiw respectively.
Let d(i) = vi − u i + 1. By Remark 7.5, we can write
r = (0, 1, . . . , i − d(i) − 1, ri−d(i)+1 , . . . , ri )
and
r0 = (0, 1, . . . , j − d( j) − 1, r 0j−d( j)+1 , . . . , r 0j ).
By Corollary 8.2, I := B(pi )∩ B(p j ) is the set of integers between max(u i , u j ) and vi = v j .
j
j
i
Let l ∈ I , then since pii = vi = v j = p j , there exists m such that l = pi−m
= p j−m . It
follows from Remark 7.5 that i −m −1 ≤ ri−m ≤ al and i −m −1 ≤ j −m −1 ≤ r 0j−m ≤ al .
0
To finish the proof, we shall show that rm+
j−i ≥ rm for all m ≤ max(i − d(i), j − d( j)).
If j − d( j) ≤ i − d(i), then I = B(pi ) ⊂ B(p j ). Since j ≥ i, we obtain by inspection
0
that rm+
j−i ≥ m + j − i − 1 ≥ m − 1 = r m for all m ≤ i − d(i).
If i − d(i) < j − d( j), then i < j and I = B(p j ) is a strict subset of B(pi ). Note that
j
p j−d( j) = −1. We claim that j − d( j) > au j −1 .
i
Let us prove our claim. If we have i − d(i) < j − d( j) < bu j −1 , then pi−d(i)+1
= −1
since we are working in a connected part satisfying condition (i) of Theorem 7.2. But this
will imply that B(pi ) has at most d(i) − 1 elements which is absurd.
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DEHY AND YU
So we have j − d( j) > au j −1 . It follows from the above expressions for r, and r0 that
rm0 = m − 1 ≥ au j −1 − ( j − d( j) − m) ≥ am+vi − j ≥ rm+i− j
for all j − i + u i ≤ m ≤ j − d( j).
Lastly, if j − i < m < j − i + u i , then rm+i− j = m + i − j − 1 < m − 1 = rm0 . Hence
our proof is complete.
2
Lemma 8.5 Suppose that s(pi ) and s(p j ) both occur in a connected part of w which satisfies condition (ii) of Theorem 7.2, we define min(r, r0 ) := (m 1 , . . . , m j ) and max(r, r0 ) :=
(M1 , . . . , Mi ) as follows:
(i) m l := min(rl , rl0 ) and Ml := max(rl , rl0 ) si u i ≤ l ≤ vi ;
(ii) m l := rl0 and Ml := rl otherwise.
Then max(r, r0 ) ∈ Wiw and min(r, r0 ) ∈ W jw .
Proof: By Corollary 8.2, I := B(pi ) ∩ B(p j ) is the set of integers between u i = u j and
vi = v j + i − j. Since we are working in a connected part satisfying condition (ii) of
Theorem 7.2, we can write as in the previous lemma:
r = (0, 1, . . . , i − d(i) − 1, ri−d(i)+1 , . . . , ri )
and
0
, . . . , ri0 , . . . , r 0j )
r0 = (0, 1, . . . , i − d(i) − 1, ri−d(i)+1
where m − 1 ≤ rm , rm0 ≤ m. It is now clear that max(r, r0 ) ∈ Wiw and min(r, r0 ) ∈ W jw .
2
Definition 8.6 If B(pi ) ∩ B(p j ) is not empty, then we define max(r, r0 ) and min(r, r0 )
according to the lemmas above. On the other hand, if i < j and B(pi ) ∩ B(p j ) is empty,
then we define max(r, r0 ) = r0 and min(r, r0 ) = r. Notice that if B(pi ) is empty, then Wiw
has only one element and therefore this definition is well-defined.
Moreover, we define a binary relation ¹w on W w by:
r ¹w r0 if and only if max(r, r0 ) = r0 and min(r, r0 ) = r
where here, we do not assume that i ≤ j.
Remark 8.7 When i < j and B(pi )∩ B(p j ) is empty, the binary relation r ¹w r0 coincides
with the partial ordering defined in Section 4.
Example 8.8
(1) Let w = w0 , then this definition is the same as the one defined in Section 4.
(2) Let n = 2 and w = s(1, 1)s(2, 2), then W1w = {1 = (0), s1 = (1)} and W2w = {1 =
(0, 1), s2 = (0, 2), s1 s2 = (1, 2)}. We have therefore,
(0, 1) ≺ (0, 2) ≺ (1, 2) ≺ (1), (0, 1) ≺ (0, 2) ≺ (0), (0) ≺ (1)
167
LATTICE POLYTOPES FOR DEMAZURE MODULES
and max((0), (1, 2)) = (1), min((0), (1, 2)) = (0, 2). Note that the maximal element
is (1).
Lemma 8.9 The binary relation ¹w defines a partial ordering on W w . Furthermore,
together with the operations max and min, it defines a lattice structure on W w (see
Lemma 4.2).
Proof: The only point which is unclear is transitivity. But by Corollary 8.2, if s(pi ), s(p j )
and s(pl ) are in the same connected part, then either vi = v j = vk or u i = u j = u k . The
former is just an analogue of the partial ordering in Section 4. The latter can be verified
easily using Lemma 8.4. Finally, the fact that the operations max and min induces a lattice
2
structure on W w is clear from the definition.
Pn
Theorem 8.10 Let λ = i=1
ci ωi be a dominant weight. Then a lattice point in 1λ can
w
be written as the sum of c1 lattice points in 1w
1 , c2 lattice points in 12 and so on.
Proof: This theorem is the analogue of Theorem 5.1, and it can be proved similarly.
The key point is that the proofs of Theorem 5.1 and Theorem 5.2 only require a partial
ordering equipped with a max–min operation satisfing Lemma 4.3 and 4.5. The analogues
of Lemma 4.3 and 4.5 can be easily shown.
2
Example 8.11 Let us first look at example 2 of 8.8. Consider w = s1 s2 = s(1, 1)s(2, 2).
It satisfies condition (i) of Theorem 7.2.
We obtain immediately that p1 = (1) and p2 = (1, 2). As above, we have in terms of
minimal representatives,
W1w = {1, s1 }
W2w = {1, s2 , s1 s2 }
We then obtain via ϕiw the following table:
s1
s2
e11
e22
s1
1
0
(1)
s2
0
1
(0, 2)
s1 s2
1
1
(1, 2)
The right most column corresponds to the notations of the elements in W w . The images
of (0, 1) and (0) are both 0.
If we consider the adjoint representation, then the highest weight is ω1 + ω2 and one
w
w
w
verifies easily by hand that the lattice points in 1w
1 + 12 are the ones in 11 and 12
168
DEHY AND YU
together with the point 2e11 + e22 . Thus the number of lattice points is 5 which is exactly
the dimension of the Demazure module E w (ω1 + ω2 ).
Note that ϕ1w (1) + ϕ2w (0, 2) = ϕ1w (0) + ϕ2w (1, 2). Again this can be seen to correspond
to the tensor product decomposition of b-modules.
We prove the following key lemmas. As before, we let r = (r1 , . . . , ri ) ∈ Wiw and r0 =
(r10 , . . . , r 0j ) ∈ W jw with i ≤ j. Recall that w = s(a1 , b1 ) · · · s(ak , bk ).
Lemma 8.12 Suppose that s(pi ) and s(p j ) both occur in a connected part of w which
satisfies condition (i) of Theorem 7.2. Then r ¹w r0 if and only if there exist r ¹ r 0 ¹ w in
W such that the class of r (resp. r 0 ) in Wiw (resp. W jw ) is r (resp. r0 ).
Proof: By Lemma 8.4, the partial ordering ¹w coincides with the one defined in Section 4.
It follows that the “if” part has already been proved using semi-standard Young tableaux.
Now suppose that r ¹w r0 , we define
¡
¢
¢ ¡
r := s ri−vi +u i , i − vi + u i s ri−vi +u i +1 , i − vi + u i + 1 · · · s(ri , i) ∈ W
where we let s(a − 1, a) to be the identity in W , and
¡
¢
¢ ¡
r 0 := s r 0j−v j +u j , j − v j + u j s r 0j−v j +u j +1 , j − v j + u j + 1 · · · s(r 0j , j) ∈ W if u i ≥ u j
and if u i < u j , we set
¡
¢
¡
¢ ¡
r 0 : = s ri−vi +u i , i − vi + u i s ri−vi +u i +1 , i − vi + u i + 1 · · · s r 0j−v j +u j ,
¢
j − v j + u j · · · s(r 0j , j)
Note that vi = v j . In the first case, it is easy to see that the class of r 0 in W j is r0 . For the
second, we use the fact that ri−v j +u j −1 < j − v j + u j − 1 (see the proof of Lemma 8.4).
0
for all m and
Moreover, these are clearly reduced expressions. Since r ¹w r0 , ri−m ≤ ri−m
0
2
we have r ¹ r ¹ w in W as required.
Lemma 8.13 Suppose that s(pi ) and s(p j ) both occur in a connected part of w which
satisfies condition (ii) of Theorem 7.2. Then r0 ¹w r if and only if there exist r 0 ¹ r ¹ w
in W such that the class of r (resp. r 0 ) in Wiw (resp. W jw ) is r (resp. r0 ).
Proof: Suppose that r0 ¹w r. This implies that l−1 ≤ rl0 ≤ rl ≤ l for l = i −vi +u i , . . . , i
(see the proof of Lemma 8.5). Note that here, r0 = st 0 st 0 +1 · · · s j where t 0 ≤ j is minimal
such that rt00 = t 0 and r = st st+1 · · · si where t ≤ i is minimal such that rt = t. In
particular t ≤ t 0 . We shall simply define r 0 := r0 = st 0 st 0 +1 · · · s j , r := st st+1 · · · si si+1 · · · s j
and r 0 ¹ r ¹ w as required.
On the other hand suppose that r 0 ¹ r ¹ w in W . As above, we have r0 = st 0 st 0 +1 · · · s j
where t 0 ≤ j is minimal such that rt00 = t 0 . Recall that we are working in a connected part
satisfying condition (ii) of Theorem 7.2.
0
for l ≤ i, and so rl0 ≤ rl . We then have r0 ¹w r.
If i < t 0 , then l − 1 = rl−1
LATTICE POLYTOPES FOR DEMAZURE MODULES
169
If t 0 ≤ i, then since r0 ¹ r 0 ¹ r , we have st 0 st 0 +1 · · · s j , and consequently st 0 st 0 +1 · · · si
can be written as subexpressions of a reduced expression for r . It follows that r0 ¹ r. 2
Lemma 8.14 Suppose that s(pi ) and s(p j ) occur in different connected parts. Then
r ¹w r0 if and only if there exist r ¹ r 0 ¹ w in W such that the class of r (resp. r 0 ) in Wiw
(resp. W jw ) is r (resp. r0 ).
Proof: By Remark 8.7, the partial ordering ¹w coincides with with the one defined in
Section 4. It follows that the “if” part has already been proved using semi-standard Young
tableaux.
Since we are working in different connected parts, and distinct connect parts commute,
r and r0 commute also. Therefore if we define r : = r and r 0 = rr0 in W . By definition, we
2
always have r ¹ r0 ¹ w.
Corollary 8.15 Let r1 , . . . , rm be elements of W w such that r1 ¹w · · · ¹w rm . Then there
exist liftings r1 , . . . , rm in W w such that the class of r j is r j (in the appropriate Wlw ) and
r1 ¹ · · · ¹ rm .
Corollary 8.16 Let λ be a dominant weight. There exists a bijection between the set of
lattice points of 1w
λ and the standard monomial basis of the Demazure module E w (λ) (see
[14, 17] ).
Proof: We can prove an analogue of Proposition
4.4. Using this and Theorem 8.10, we
Pm
w
can write a lattice point in 1w
λ as the sum
i=1 ri such that ri ∈ W and r1 ¹w r2 ¹w
· · · ¹w rm . Corollary 8.15 then implies that the set of lattice points of 1w
λ is in bijection
2
with the standard monomial basis of E w (λ).
Example 8.17 In the case of example 2 of 8.8, the reader can verify that the three chains
can be lifted as follows:
1 ¹ s2 ¹ s1 s2 ¹ s1 s2 , 1 ¹ s2 ¹ s2 , s2 ¹ s1 s2
and there are no liftings r of (0) and r 0 of (1, 2) such that r 0 ¹ r .
9.
An Example
Let us consider a slightly more complicated case. Let w = s1 s3 s2 s1 = s(1, 1)s(3, 1) be an
element of the Weyl group of sl4 . It satisfies condition (i) of Theorem 7.2. In this case, we
have p1 = (3), p2 = (1, 3) and p3 = (−1, −1, 3). Thus,
W1w = {1, s1 , s2 s1 , s3 s2 s1 }
W2w = {1, s2 , s3 s2 , s1 s2 , s1 s3 s2 }
W3w = {1, s3 }
170
DEHY AND YU
As before, we compute the following table:
s1
s3
s2
s1
e11
e33
e32
e31
s1
0
0
0
1
(1)
s 2 s1
0
0
1
1
(2)
s3 s2 s1
0
1
1
1
(3)
s2
0
0
1
0
(0, 2)
s 3 s2
0
1
1
0
(0, 3)
s 1 s2
1
0
1
0
(1, 2)
s1 s3 s2
1
1
1
0
(1, 3)
s3
0
1
0
0
(0, 1, 3)
The reader can verify for example for the adjoint representation as in Section 4 that the
w
number of lattice points in 1w
1 +13 is the dimension of the Demazure module E w (ω1 +ω3 )
which is 7.
10.
Some applications
We shall apply our results in this section to obtain a combinatorial description of the weight
multiplicities of a Demazure module, and we also present a polytope which is closely
connected toPthe weight.
n
Let λ = i=1
ai ωi be a dominant weight, w = s(a1 , b1 ) · · · s(ak , bk ) be an element of
W satisfying the conditions of Theorem 7.2 and denote by m w
λ (µ) the multiplicity of the
weight µ in E w (λ).
Let e pq be the standard basis as in Section 7 and αi , i = 1, . . . , n be the set of simple
roots as in the introduction.
Definition 10.1
We define a linear map
Aw : R`(w) −→ P ⊗Z R =: PR
by sending e pq to αq .
w
`(w)
We shall denote by Aw
to PR .
λ the affine map λ − A from R
Theorem 10.2 The character of the Demazure module E w (λ) is given by
X
w
e−A (x)
char E w (λ) = eλ
x
where the sum runs through the lattice points x of 1w
λ.
171
LATTICE POLYTOPES FOR DEMAZURE MODULES
Let µ ∈ P. To prove the theorem, it suffices to prove that
¡ `(w)
¡ w ¢−1 ¢
∩ 1w
(µ)
mw
λ (µ) = Card Z
λ ∩ Aλ
Proof:
Recall that the weight of a standard monomial of type λ is the weight of the corresponding
weight vector in E w (λ) (see [17, 14]). Therefore it suffices to show that for a lattice point
w
x of 1w
λ , Aλ (x) is the weight of the standard monomial T (x) corresponding to x as in the
proof of Theorem 7.2.
Let x be a lattice point of 1w
λ . According to Proposition 6.2, we can find {σi j }, i =
satisfying
the conclusions of the proposition
1, . . .P
, n and
j
=
1,
.
.
.
,
a
i
P and such that
n Pai
w
ϕ
(σ
).
It
follows
that
the
weight
of
T
(x)
is
the
sum
x = i=1
i, j σi j (ωi ). Since
Pnj=1Piai i j w w
w
Aλ (x) = i=1 j=1 Aωi (ϕi (σi j )), we are reduced to the case where λ is a fundamental
weight.
According to Lemma 2.3 and Theorem 3.3, there is a bijection between the weights of
Vωi and the vertices of 1i , which in turn are indexed by the elements of Wi . Explicitly, the
vertex (r1 , . . . , ri ) corresponds to the weight
sr1 · · · s1 sr2 · · · s2 · · · sri · · · si (ωi ) = ωi −
rp
i X
X
αq = ωi −
a p = pij q= j
αq
a p = pij q= j
p=1 q= p
On the other hand, (r1 , . . . , ri ) corresponds to the point
Therefore


rj
rj
X X
X X


e
−
αq
=
ω
Aw
pq
i
ωi
rj
X X
P
a p = pij
Pr j
q= j
e pq (see Section 7).
a p = pij q= j
and we are done since the multiplicity of any weight of Vωi is 1.
2
Aw
Corollary 10.3 The image of 1w
λ viaP
λ is the convex hull of {σ (λ) | σ ∈ W with σ ¹ w}.
w
In particular it is the Minkowski sum i ai Aw
ωi (1i ).
w −1
w
Corollary 10.4 Let µ be a weight of E w (λ), then 1w
λ (µ) := (Aλ ) (µ) ∩ 1λ is a convex
polytope with rational vertices containing all the points which correspond to the weight µ.
w
Moreover k1w
λ (µ) = 1kλ (kµ).
Acknowledgments
The authors would like to thank P. Littelmann for his continuous help, T. Delzant and F. du
Cloux for fruitful discussions, O. Mathieu for many useful suggestions and A. Zelevinsky for
his remarks on earlier versions of this paper. The authors wish also to thank the anonymous
referees for their remarks and suggestions. This work is mainly done during R.W.T. Yu’s
one-year detachment at the I.R.M.A. in Strasbourg and he would like to thank the C.N.R.S.
for giving him this opportunity.
172
DEHY AND YU
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