Journal of Algebraic Combinatorics 11 (2000), 269–275
c 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.
°
On the Connection Between Macdonald Polynomials
and Demazure Characters
YASMINE B. SANDERSON
Department of Mathematics, Rutgers University, New Brunswick, N.J. 08901
sanderso@math.rutgers.edu
Received March 30, 1998; Revised November 17, 1998; Accepted December 21, 1998
Abstract. We show that the specialization of nonsymmetric Macdonald polynomials at t = 0 are, up to multid This connection furnishes Lie-theoretic
plication by a simple factor, characters of Demazure modules for sl(n).
proofs of the nonnegativity and monotonicity of Kostka polynomials.
Keywords: affine Lie algebras, Macdonald polynomials, Demazure character
1.
Introduction
Macdonald defined a special class of polynomials Pλ (z, q, t), called symmetric Macdonald
polynomials, which form a basis of the symmetric polynomials in C(q, t)[z 1 , . . . , z n ]. These
polynomials are indexed by partitions λ ∈ Nn , λ1 ≥ λ2 ≥ · · · λn ≥ 0. They interpolate
between several classes of classical polynomials: Pλ (z, 0, t) are the Hall-Littlewood polynomials, which, in turn are the Schur functions when t = 0. By setting q = t α and letting t
go to 1, one obtains Jack polynomials. In [11], Macdonald mentions that there is no similar
interpretation of Pλ (z, q, 0). By using the theory of nonsymmetric Macdonald polynomials,
we show that the Pλ (z, q, 0) are the characters (up to factor) of certain Demazure modules
d This interpretation allows us to obtain Lie-theoretic proofs of the nonnegativity
of sl(n).
and monotonicity of Kostka polynomials. In addition, it gives us a branching rule for the
d
decomposition of certain integrable highest weight sl(n)-modules
under the action of sl(n).
The connection between Demazure characters and symmetric functions has already been
explored in [8] using a path realization of the crystal basis. The results in this paper intersect
somewhat with those in [8]. The main advantage of our approach is its simplicity and its explanation of the connection with Macdonald polynomials. Nonnegativity and positivity of
Kostka polynomials have already been proven by Lascoux-Schützenberger [9], Butler [1],
Lusztig [10]. The connection between the branching rule and Kostka polynomials was explored in [5]. A different representation-theoretic interpretation of Pλ (z, q, 0) is given in [4].
2.
Nonsymmetric Macdonald polynomials
These nonsymmetric analogues of the symmetric Macdonald polynomials were first introduced in [12, 14]. Nonsymmetric Macdonald polynomials E λ (z, q, t) are indexed by
compositions λ ∈ Nn and form a basis of C(q, t)[z 1 , . . . , z n ]. (See [2, 5] for their precise
270
SANDERSON
definition). In [6], Knop gives a recursive description of the E λ (z, q, t). We describe this recursion for when t = 0. In this case, we have E λ (z, q, 0) ∈ Z[q, q −1 ][z 1 , . . . , z n ]. For ease
of notation, we will denote E λ (z, q, 0) simply by E λ from now on. For i ∈ [1, . . . , n −1] let
si be the simple reflection that interchanges z i and z i+1 . Consider the following operators
on Z[q, q −1 ][z 1 , . . . , z n ]:
(1 − si )
for i ∈ [1, . . . , n − 1]
(z i − z i+1 )
8f (z 1 , . . . , z n ) := z n f (q −1 z n , z 1 , . . . , z n−1 )
H̄i := si − z i+1
H̄0 := 8 H̄1 8−1 = 8−1 H̄n−1 8
Then the recursion relations are given by [6]
Theorem 1 The E λ are generated by application of the H̄i (0 ≤ i < n) and 8 to 1.
More precisely, set E (0n ) := 1. The action of 8 and the H̄i on the set of E λ for λ ∈ Nn is as
follows:
q λ1 8E (λ1 ,...,λn ) = E (λ2 ,...,λn ,λ1 +1)
½
E si λ if λi < λi+1
for 1 ≤ i ≤ n − 1
H̄i E λ =
Eλ
if not
where si λ is the composition λ with λi and λi+1 interchanged.
½
E (λn −1,λ2 ,...,λn−1 ,λ1 +1) if λ1 > λn − 1
q λ1 −λn +1 H̄0 E λ =
if not
Eλ
To ease notation, we define the operators H̃0 and 8̃ on the set of nonsymmetric Macdonald
polynomials:
H̃0 E (λ1 ,...,λn ) := q λ1 −λn +1 H̄0 E (λ1 ,...,λn ) = E (λn −1,λ2 ,...,λn−1 ,λ1 +1)
8̃E (λ1 ,...,λn ) := q λ1 8E (λ1 ,...,λn ) = E (λ2 ,...,λn ,λ1 +1)
Although this definition of nonsymmetric Macdonald polynomials is given for only
λ ∈ Nn , we can easily extend it to compositions λ ∈ Zn by defining
E λ := 8̃−mn E λ+(m n ) = q −(m|λ|+nm(m+1)/2) 8−mn E λ+(m n )
where m is chosen large enough so that λ + (m n ) = (λ1 + m, . . . , λn + m) is in Nn . The E λ
are well-defined (don’t depend on the choice of m). In fact, let m 1 and m 2 , with m 1 ≤ m 2 ,
be two such choices. Then,
8̃−m 2 n E λ+(m n2 ) = 8̃−m 1 n 8̃−(m 2 −m 1 )n E λ+(m n2 ) = 8̃−m 1 n E λ+(m n1 ) ,
the last equality following from the well-definedness of the E µ for µ ∈ Nn . Note that the
E λ are elements of Z[q, q −1 ][z 1 , z 1−1 , . . . , z n , z n−1 ].
MACDONALD POLYNOMIALS AND DEMAZURE CHARACTERS
271
We now check that, for λ ∈ Zn \Nn , the E λ satisfy the recursion relations. For i 6= 0,
E si ·λ = 8̃−mn E si ·λ+(m n ) = 8̃−mn H̄i 8̃mn E λ = H̄i E λ
Now, let λ∗ := (λn − 1, λ2 , . . . , λn−1 , λ1 + 1) and choose m such that λ∗ + (m n ) ∈ Nn .
Then
E λ∗ = 8̃−mn E λ∗ +(m n ) = 8̃−mn H̃0 8̃mn E λ = q λ1 −λn +1 H̄0 E λ ,
the last equality following from the commutativity of the H̄i with 8̃n . This proves that, for
all λ ∈ Zn , the E λ satisfy the relations of Theorem 1.
Let Bm denote the Z[q, q −1 ]-vector space generated by all E λ with |λ| = m. Then the
H̄i Bm ⊂ Bm for all i and 8Bm ⊂ Bm+1 . The action of the H̄i (i 6= 0) and H̃0 on the E λ is
related to the action of the affine Weyl group on compositions:
si · (λ1 , . . . , λn ) := (λ1 , . . . , λi−1 , λi+1 , λi , λi+2 , . . . , λn )
so · (λ1 , . . . , λn ) := (λn − 1, λ2 , . . . , λn−1 , λ1 + 1)
The connection between compositions of a given degree, say m, and elements of the affine
Weyl group is as follows. Let m = kn + i where k ≥ 0 and 0 ≤ i < n. Then the smallest
composition is ηm := (k, . . . k, k + 1, . . . , k + 1) (i factors of k + 1 and n − i factors of k).
Every composition λ of degree m equals w · ηm where w is an affine Weyl group element.
For w = si1 , . . . , si j a reduced decomposition, we define H̄w := H̄i1 , . . . , H̄i j and H̃w
the same expression but with H̄0 replacedP
by H̃0 .
For a composition λ ∈ Zn , let u(λ) := i λi (λ2i −1) .
Theorem 2 We can write E λ = H̃w 8̃|λ| · 1 = q u(λ) H̄w 8|λ| · 1 where w is determined by
λ = wη|λ| .
Proof: By the commuting relations of 8 and the H̄i [6],
(
i = 1, . . . , n − 2
8 H̄i+1 = H̄i 8
82 H̄1 = H̄n−1 82
we need only prove that the power of q is u(λ) by induction. For i ≥ 1, the actions of the H̄i
do not involve any powers of q. The operator H̄0 equals 8 H̄1 8−1 by definition. Therefore,
we need only check that this holds for 8. Let µ = (λ1 , . . . , λn−1 , λn + 1). We have
E µ = q λn 8E λ = q λn q u(λ) 8Hw 8|λ| · 1 = q u(µ) Hw0 8|µ| · 1
where Hw0 is determined by the above commutation relations.
2
Remark We note that 8̃nk+i · 1 = q u(ηnk+i ) (z 1 , . . . , z n )k z n−i+1 · · · z n . The H̄i (all i)
commute with multiplication by q and the symmetric function z 1 · · · z n . Therefore, E λ =
q u(ηnk+i ) (z 1 , . . . , z n )k H̃w z n−i+1 · · · z n . We will use this information in Section 4.
272
3.
SANDERSON
d
Demazure modules of sl(n)
Let 3 be a dominant integral weight. Let V = V (3) be the unique (up to isomorphism)
d
irreducible highest weight sl(n)-module
with highest weight 3. Let W be the Weyl group
d
of sl(n). For each w ∈ W , the weight space Vw(3) of weight w(3) is one-dimensional.
We consider E w (3), the b-module generated by Vw(3) , where b is the Borel subalgebra.
The E w (3), called Demazure modules, are finite-dimensional vector spaces which form a
filtration of V which is compatible with the Bruhat order on W : w ≤ w0 ⇔ E w (3) ⊆
E w0 (3).
To each Demazure module E w (3), we can associate its character χ(E w (3):
X
(dim E w (3)µ )eµ
χ(E w (3)) :=
µ weight
Since the E w (3) are finite dimensional, the χ(E w (3)) are polynomials in the n simple
roots αi and lie in the group ring for the weight lattice P.
We now define Demazure operators. For each αi , we define an operator 1i on P:
1i :=
1 − e−αi si
1 − e−αi
where si is the simple reflection with respect to αi . Let w = si1 si2 · · · si j be a reduced decomposition. Then, we can define 1w := 1i1 1i2 · · · 1i j and 1w does not depend on the choice
of reduced decomposition. The connection between characters and Demazure operators is
given by [3, 7, 13]:
Theorem 3 χ(E w (3)) = 1w (e3 ).
4.
Macdonald polynomials and Demazure module characters
d defined by (3i , α j ) = δi j . Let
Let 30 , . . . , 3n−1 denote the n fundamental weights of sl(n)
Pn−1
δ = i=0 αi . Let π be the ring homomorphism π : Z[q, q −1 ][z 1 , . . . , z n ] → P defined
by: π(z i ) = e3i −3i−1 for i < n, π(z n ) = e30 −3n−1 and π(q) = e−δ .
Theorem 4 The operator H̄i is equivalent to the Demazure operator 1i in the sense that
the following diagram commutes:
Z[q, q −1 ][z 1 , . . . , z n ]
H̄i
↓
Z[q, q −1 ][z 1 , . . . , z n ]
→P
π
↓1 (e
→P
π
i
30
· )
Proof: We have that H̄i , i 6= 0 (resp. H̄0 ) commutes with multiplication by z j for j 6= i or
i +1 (resp. z 1 or z n ). Therefore, one only needs to verify this equivalence on the monomials
b
(resp. z 1a z nb ). This is done by direct computation.
2
z ia z i+1
MACDONALD POLYNOMIALS AND DEMAZURE CHARACTERS
273
Let C be the following “change of basis” operator on P: C(e30 ) = e3n−1 and C(e3i ) =
for 1 ≤ i ≤ n − 1.
e
3i−1 −δ
Theorem 5 The operator 8 is equivalent to the operator C in the sense that the following
diagram commutes:
Z[q, q −1 ][z 1 , . . . , z n ]
↓
8
Z[q, q −1 ][z 1 , . . . , z n ]
Proof:
→P
π
↓C
→P
π
2
By direct computation.
Theorems 1, 4, 5 along with the preceding Remark give us our main result:
Theorem 6 Through the π homomorphism, we can identify q −u(λ)+u(η|λ| ) E λ with
χ (E w (3i )) where i = |λ| mod n and where w is an affine Weyl group element defined
by λ = wη|λ| .
Proof:
We have that
E λ = q u(λ) H̄w 8|λ| · 1 = q u(λ)−u(η|λ| ) (z 1 · · · z n )k H̄w z n−i+1 · · · z n .
We have π(z 1 z 2 · · · z n ) = 1 and π(z n−i+1 · · · z n ) = e3i . Therefore,
π(E λ ) = q u(λ)−u(η|λ| ) 1w e3i .
2
Remark
1. π(E λ ) having nonnegative coefficients implies that E λ has nonnegative coefficients.
2. By setting q = 1, one obtains the real character of a Demazure module (see [15]). For
λ a partition, we have the factorization ([11], p. 324)
Pλ (z, 1, 0) = eλ0 (z) =
n
Y
λ −λi+1
ei i
(z)
i=1
where ei (z) is the ith elementary symmetric function. This gives us a similar factorization
of
χ(E w (3)) = q −u(λ)+u(η|λ| )
n−1
Y
ei (π(z))λi −λi+1 .
i=1
Previous examples of this factorization are found in [8, 15].
274
5.
SANDERSON
Positivity and monotonicity of Kostka polynomials
Recall that Pλ (z, q, t) denotes the symmetric Macdonald polynomial associated to the
partition λ.
Theorem 7
For λ a partition, we have E λ (z, q, 0) = Pλ (z, q, 0).
P
Proof: Consider w∈W H̄w E λ (z, q, t). It is symmetric and satisfies the same defining
conditions as Pλ (z, q, t) (see [6]), therefore is a scalar multiple of it. When t = 0, we have
H̄w E λ (z, q, 0) = E λ (z, q, 0). By comparing coefficients of the leading coefficient z λ in
2
both E λ (z, q, 0) and Pλ (z, q, 0), we see that we have equality.
Recall that one has the following order relation on partitions: two partitions γ and µ
such that |γ | = |µ| satisfy γ < µ if γ1 + · · · + γi ≤ µ1 + · · · + µi for all i with strict
inequality for some i.
P
It is known [[11], VI (8.11)] that Pλ (z, q, 0) = µ≤λ K µλ (q, 0)sµ (z) where K is the
Kostka function and the sµ are the Schur functions. In addition, it is known [[11], p. 355]
0
0
that K µλ (q, 0) = KP
µ0 λ0 (q) where µ (resp. λ ) is the dual partition of µ (resp. λ). It follows
that Pλ (z, q, 0) = µ≤λ K µ0 λ0 (q)sµ (z).
Theorem 8 The K µλ (q) have positive coefficients.
Proof: We have that Pλ (z, q, 0) is invariant under the H̄i (for i 6= 0). This is equivalent
to saying that the Demazure module E w (30 ) decomposes as a direct sum of simple sl(n)modules. In fact, we have the following decomposition:
E w (30 ) = ⊕ j∈Z (E w (30 )) jδ
where (E w (30 )) jδ is just the direct sum of weight spaces whose weights are of the form
υ = κ + jδ where κ is some weight for sl(n). (In other words, these are all weights that
satisfy hυ, di = j where d is the scaling element.) Since δ is orthogonal to the Cartan
subalgebra of sl(n), each (E w (30 )) jδ is a direct sum of irreducible sl(n)-modules. Let
λ = wν|λ| . The Pλ (z, q, 0) merely represents the character χ(E w (30 )) as seen in this light;
since the sµ (z) is a character of an irreducible sl(n)-module, the coefficient of q j in K µ0 λ0 (q)
is the multiplicity of the sl(n)-module of highest weight µ − jδ in E w (30 ). Therefore, the
2
K µ0 λ0 (q) have positive coefficients.
Remark A consequence of this theorem is that the Kostka numbers K µλ (1) are the multiplicities of the (finite-dimensional) sl(n)-modules in the Demazure modules E w (3).
d
of highest weight
Recall that V = V (3i ) is the irreducible highest weight sl(n)-module
3i . We have that χ (V ) = lim`(w)→∞ χ(E w (3i )). We can now describe the branching rule
for V in terms of Kostka polynomials (see [5]). Let {λ j } be an “increasing” sequence of
partitions in the sense that λ j := w j ν|λ j | where lim j `(w j ) = ∞ and where |λ j | = i(mod n).
We must choose ν|λ j | such that the resulting λ j are still partitions.
MACDONALD POLYNOMIALS AND DEMAZURE CHARACTERS
275
Corollary 1 The multiplicity of the sl(n)-module of weight µ in V is given by
lim q −u(λ
j
)+u(ν|λ j | )
j→∞
K µ0 λ j0 (q)
We also have a monotonicity result. Let K̃ λµ (q) := q −u(µ) K λµ (q). Recall that if λ = wνm
and γ = w0 νm , λ 6= γ are partitions, then λ < γ if and only if w < w0 in the Bruhat order,
where w and w 0 are chosen to have smallest length.
Theorem 9
K̃ λµ (q) − K̃ λν (q) has nonnegative coefficients when ν ≥ µ.
Proof: Let υ < γ be two partitions such that υ = w0 η|λ| and γ = wη|λ| . Then w0 < w
and E w0 (3) ⊂ E w (3). The coefficient of q j in K̃ ν 0 γ 0 (q) − K̃ ν 0 υ 0 (q) is the multiplicity of
the sl(n)-module of weight ν − jδ ( j ∈ Z) in E w (3)/E w0 (3). Therefore, it has positive
coefficients.
2
Acknowledgments
The connection between Demazure characters and Macdonald polynomials was pointed out
by I. Cherednik. The author thanks the referees for helpful comments.
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