Journal of Algebra 233, 594᎐613 Ž2000.
doi:10.1006rjabr.2000.8441, available online at http:rrwww.idealibrary.com on
Hecke Algebra Actions on the Coinvariant Algebra
Ron M. Adin1
Department of Mathematics and Computer Science, Bar-Ilan Uni¨ ersity,
Ramat Gan 52900, Israel
E-mail: radin@math.biu.ac.il
Alexander Postnikov
Department of Applied Mathematics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
E-mail: apost@math.mit.edu
and
Yuval Roichman1
Department of Mathematics and Computer Science, Bar-Ilan Uni¨ ersity,
Ramat Gan 52900, Israel
E-mail: yuvalr@math.biu.ac.il
Communicated by G. D. James
Received August 12, 1999
Two actions of the Hecke algebra of type A on the corresponding polynomial
ring are studied. Both are deformations of the natural action of the symmetric
group on polynomials, and keep symmetric functions invariant. We give an explicit
description of these actions, and deduce a combinatorial formula for the resulting
graded characters on the coinvariant algebra.
䊚 2000 Academic Press
1. INTRODUCTION
1.1. The symmetric group Sn acts on the polynomial ring Pn s
F w x 1 , . . . , x n x Žwhere F is a field of characteristic zero. by permuting
variables. Let In be the ideal of Pn generated by the symmetric i.e.,
1
Research supported in part by the Israel Science Foundation and by internal research
grants from Bar-Ilan University.
594
0021-8693r00 $35.00
Copyright 䊚 2000 by Academic Press
All rights of reproduction in any form reserved.
595
HECKE ALGEBRA ACTIONS
Ž Sn-invariant. polynomials without a constant term. The coin¨ ariant algebra
of type A is the quotient PnrIn . Schubert polynomials, constructed in the
seminal papers wBGG, Dex, form a distinguished basis for the coinvariant
algebra. These polynomials correspond to Schubert cells in the corresponding flag variety.
1.2. In this paper we present two deformations of this action. For these
deformations we can take F s CŽ q ., the field of rational functions in an
indeterminate q. Most of the results actually hold when F is replaced by
the ring Zw q x of polynomials in q with integer coefficients.
Let T1 , . . . , Tny1 be the standard generators of the Hecke algebra HnŽ q .
of type A; for definitions see Section 2.1 below.
The first action ␳ 1: HnŽ q . ª Hom F Ž Pn , Pn . is defined using q-commutators
␳ 1 Ž Ti . [ ⭸ i X i y qX i ⭸ i
Ž 1 F i - n. ,
Ž 1.1.
where
⭸i [
1
x i y x iq1
Ž 1 y si .
is the divided difference operator Žsee Section 2.2., and X i denotes
multiplication by x i . This action belongs to a family introduced in wLSx Žsee
Section 7.1 below.. For a geometric interpretation see wDKLLSTx. In
wDKLLST, Sect. 1x such families of operators are attributed to Hirzebruch
wHrx.
The second action is naturally defined on monomials by the formula
¡qx
␳ 2 Ž Ti . Ž
␤
x i␣ x iq1
m
␤ ␣
i x iq1 m,
. [~ Ž 1 y q . x
¢x
␣ ␤
i x iq1 m,
␣ ␤
i x iq1 m
if ␣ ) ␤ ,
q
␣
x i␤ x iq1
m,
if ␣ - ␤ ,
if ␣ s ␤ .
Ž 1.2.
Here m is a monomial involving neither x i nor x iq1.
For a closely related action Ždefined in the context of quantum groups.
see wJix.
Claim. The ideal In is invariant under both actions. The resulting
graded characters on the coinvariant algebra have a common combinatorial formula.
This shows, in particular, that ␳ 1 and ␳ 2 lead to equivalent representations of HnŽ q . on the coinvariant algebra. For q s 1 they both reduce to
the natural Sn action.
596
ADIN, POSTNIKOV, AND ROICHMAN
1.3. Since the ideal In is invariant under both ␳ 1 and ␳ 2 , the coinvariant algebra PnrIn carries appropriate actions ␳˜1 and ␳˜2 . Let ␹ 1k and ␹ 2k
be the characters of these representations on the kth homogeneous
component of PnrIn . We shall give an explicit formula for these characters, using the following combinatorial function.
For any permutation w g Sn , define
¡Ž yq .
~
m Ž w. [
¢0,
q
m
,
if there exists a unique 0 F m - n so that
w Ž 1 . )⭈⭈⭈) w Ž m q 1 . - w Ž m q 2 . -⭈⭈⭈-w Ž n . ,
otherwise.
Ž 1.3.
Let ␮ [ Ž ␮ 1 , . . . , ␮ t . be a partition of n, and let S␮ [ S␮ 1 = ⭈⭈⭈ = S␮ t be
the corresponding Young subgroup of Sn . For any permutation w g Sn
write w s r ⭈ Ž w 1 = ⭈⭈⭈ = wt ., where wi g S␮ i Ž1 F i F t . and r is a representative of minimal length for the left coset wS␮ in S n . Define
weight q␮ Ž w . [
THEOREM.
t
Ł m q Ž wi . .
Ž 1.4.
is1
For all k G 0 and ␮ & n,
␹ 1k Ž T␮ . s ␹ 2k Ž T␮ . s
Ý
weight q␮ Ž w . ,
wgS n : l Ž w .sk 4
where T␮ [ T1T2 ⭈⭈⭈ T␮ 1y1T␮ 1q1 ⭈⭈⭈ ⭈⭈⭈ T␮ 1q ⭈ ⭈ ⭈ q ␮ ty1 is the subproduct of
T1T2 ⭈⭈⭈ Tny1 omitting T␮ 1q ⭈ ⭈ ⭈ q ␮ i for all 1 F i - t.
The proof relies on an explicit description of the action with respect to
the Schubert basis of the coinvariant algebra. See Theorems 4.1 and 6.5
below.
Remark. This character formula is a natural q-analogue of a weight
formula for Sn presented in wRo2x. A formally similar result appears also in
Kazhdan᎐Lusztig theory. Kazhdan᎐Lusztig characters may be represented
as sums of exactly the same weights, but over different summation sets
wRo1, Corollary 4; Ra2x.
1.4. The rest of this paper is organized as follows. Preliminaries and
necessary background are given in Section 2. In Section 3 we introduce
q-commutators and study their representation matrices. The character
formula for q-commutators is proved in Section 4. Natural randomized
operators are introduced in Section 5. In Section 6 we show that the
representations induced by the two different actions are equivalent. Sec-
HECKE ALGEBRA ACTIONS
597
tion 7 concludes the paper with remarks regarding related families of
operators, connections with Kazhdan᎐Lusztig theory, and open problems.
2. PRELIMINARIES
2.1. The Hecke Algebra of Type A
The symmetric group Sn is generated by n y 1 involutions s1 , s2 , . . . ,
sny 1 satisfying the Moore᎐Coxeter relations
si siq1 si s siq1 si siq1
Ž 1 F i - n y 1.
Ž 2.1.
and
si s j s s j si
if < i y j < ) 1.
Ž 2.2.
These involutions are known as the Coxeter generators of S n .
All reduced expressions of a permutation w g Sn with respect to these
generators have the same length, denoted by l Ž w ..
The Hecke algebra HnŽ q . of type A is the algebra over F [ CŽ q .
generated by n y 1 generators T1 , . . . , Tny1 , satisfying the Moore᎐Coxeter
relations Ž2.1. and Ž2.2. as well as the following ‘‘deformed involution’’
relation:
Ti 2 s Ž 1 y q . Ti q q
Ž 1 F i - n. .
Ž 2.3.
It should be noted that the last relation is slightly non-standard; this is
done in order to get more elegant q-analogues. In order to shift to the
standard version, one should replace each Ti by yTi .
Let w be a permutation in Sn and let si1 ⭈⭈⭈ si lŽ w . be a reduced expression for w. It follows from the above relations that Tw [ Ti1 ⭈⭈⭈ Ti lŽ w . is
independent of the choice of reduced expression; the set Tw ¬ w g Sn4
forms a linear basis for HnŽ q ..
Let ␮ s Ž ␮ 1 , . . . , ␮ t . be a partition of n. Define T␮ g HnŽ q . to be the
product
T␮ [ T1T2 ⭈⭈⭈ T␮ 1y1T␮ 1q1T␮ 1q2 ⭈⭈⭈ T␮ 1q ␮ 2y1T␮ 1q ␮ 2q1 ⭈⭈⭈ ⭈⭈⭈ T␮ 1q ⭈ ⭈ ⭈ ␮ ty1 .
This is the subproduct of the product T1T2 ⭈⭈⭈ Tny1 of all generators Žin
the usual order., obtained by omitting T␮1q ⭈ ⭈ ⭈ q ␮ i for all 1 F i - t. These
elements play an important role in the character theory of HnŽ q .. For
q s 1, the elements T␮ are representatives of all conjugacy classes in Sn . It
follows that, for q s 1, a character is determined by its values at these
elements. This is also the case for arbitrary q, as the following theorem
shows.
598
ADIN, POSTNIKOV, AND ROICHMAN
THEOREM 2.1 wRa1, Theorem 5.1x.
linear combination
Cw s
For each w g Sn there exists a
Ý aw , ␮ T␮ g Hn Ž q . ,
␮
with a w, ␮ g Z w q x, such that
␹ Ž Tw . s ␹ Ž Cw .
for all characters ␹ of the Hecke algebra HnŽ q ..
Let ␮ s Ž ␮ 1 , . . . , ␮ t . be a partition of n. Each permutation w g Sn has
an associated weight, weight q␮ Ž w ., as defined in Ž1.3., Ž1.4.. The irreducible
characters of HnŽ q . are indexed by the partitions of n. These characters
may be represented as weighted sums over Knuth equivalence classes.
THEOREM 2.2 wRo1, Corollary 4x.
shape ␭. Then
␹ ␭ Ž T␮ . s
Let C be a Knuth equi¨ alence class of
Ý
weight q␮ Ž w . ,
wg C
where ␹ ␭ is the irreducible character of HnŽ q . corresponding to the shape ␭.
2.2. Schubert Polynomials and the Coin¨ ariant Algebra
2.2.1. Basic Actions on the Polynomial Ring
Let x 1 , x 2 , . . . , x n be independent variables, and let Pn be the polynomial ring F w x 1 , x 2 , . . . , x n x. The symmetric group Sn acts on Pn by permuting the variables x i . Let ⌳ n s ⌳w x 1 , x 2 , . . . , x n x be the subring of symmetric functions Ži.e., polynomials which are invariant under the action of Sn ..
Denote by ⌳ nŽ i . the ring of all polynomials which are invariant under the
action of si for a fixed i, 1 F i - n. Clearly, f g ⌳ nŽ i . if and only if f is
symmetric in the variables x i and x iq1. We call the polynomials in ⌳ nŽ i .
i-symmetric polynomials.
For 1 F i - n define a divided difference operator ⭸ i : Pn ª Pn by
⭸ i [ Ž x i y x iq1 .
y1
Ž 1 y si . .
If f g Pn is a homogeneous polynomial of degree d, which is not i-symmetric, then ⭸ i Ž f . is homogeneous of degree d y 1. For i-symmetric
polynomials ⭸ i Ž f . s 0.
HECKE ALGEBRA ACTIONS
599
The operators ⭸ i satisfy the nil-Coxeter relations wMa, Ž2.1.x
⭸ i2 s 0
Ž 1 F i - n. ,
⭸ i ⭸ iq1 ⭸ i s ⭸ iq1 ⭸ i ⭸ iq1
⭸i ⭸j s ⭸j ⭸i
Ž 1 F i - n y 1. ,
if < i y j < ) 1.
Ž 2.4.
Ž 2.5.
Ž 2.6.
Let X i be the operator on Pn corresponding to multiplication by x i .
Clearly, X i increases degree by 1.
The algebra generated by the operators ⭸ i , 1 F i - n, and X i , 1 F i F n
was studied in wDe, BGGx. The generators satisfy the following commutation relations:
⭸i Xj s Xj ⭸i
if < i y j < ) 1,
Ž 2.7.
⭸ i X i s 1 q X iq1 ⭸ i
Ž 1 F i - n. ,
Ž 2.8.
X i ⭸ i s 1 q ⭸ i X iq1
Ž 1 F i - n. ,
Ž 2.9.
2.2.2. Schubert Polynomials
For any sequence a s Ž a1 , . . . , a k . of positive integers less than n,
define ⭸a [ ⭸a1 ⭈⭈⭈ ⭸a k. It follows from the relations Ž2.5., Ž2.6. that if a, b
are two reduced expressions for the same permutation w g S n then
⭸a s ⭸ b . We can therefore use the notation ⭸w for w g Sn , and in
particular ⭸s i [ ⭸ i for 1 F i - n.
The relation ⭸ i2 s 0 implies that for any w g S n and any 1 F i - n
⭸ i ⭸w s
½
⭸s i w ,
if l Ž si w . ) l Ž w . ,
0,
if l Ž si w . - l Ž w . .
Ž 2.10.
For each w g Sn we define the Schubert polynomial ᑭ w by
ᑭ w [ ⭸w y1 w 0Ž x 1ny 1 x 2ny 2 ⭈⭈⭈ x ny1 . ,
where w 0 is the longest element in Sn .
By definition, ᑭ w is a homogeneous polynomial of degree l Ž w ..
It follows from Ž2.10. that
⭸i Ž ᑭ w . s
½
ᑭ w si ,
if l Ž wsi . - l Ž w . ,
0,
if l Ž wsi . ) l Ž w . .
Ž 2.11.
600
ADIN, POSTNIKOV, AND ROICHMAN
Denote ᑭ s i by ᑭ i . For any 1 F i - n,
ᑭ i s x 1 q ⭈⭈⭈ qx i .
Ž 2.12.
See wMa, Ž4.4.x. The following is an important variant of Monk’s formula.
MONK’S FORMULA wMa, Ž4.11.x.
Let 1 F i - n and w g Sn . Then
ᑭiᑭw s
Ý ᑭ wt ,
t
where the sum extends o¨ er all transitions t s t jk interchanging j and k, with
1 F j F i - k F n and l Ž wt . s l Ž w . q 1.
The description of the action of the operator X i on Schubert polynomials follows from Monk’s formula and Ž2.12.,
X i Ž ᑭ w . s Ž ᑭ i y ᑭ iy1 . ᑭ w s
Ý
jFi-k
s
Ý
ᑭ w t jk y
jsi-k
Ý
ᑭ w t jk ,
ᑭ w t jk y
Ý
ᑭ w t jk
j-iFk
Ž 2.13.
j-isk
where all summations are over the transpositions t s t jk satisfying l Ž wt . s
l Ž w . q 1, with j and k in the indicated ranges.
2.2.3. The Coin¨ ariant Algebra
Recall that ⌳ n s ⌳w x 1 , . . . , x n x is the subring of Pn consisting of symmetric functions, and let In be the ideal of Pn generated by symmetric
functions without a constant term. The quotient PnrIn is called the
coin¨ ariant algebra of S n . S n acts naturally on this algebra. The resulting
representation is isomorphic to the regular representation of the symmetric group. See, e.g., wHu, Sect. 3.6; Hi, Sect. II.3x.
Let R k Ž0 F k F Ž n2 .. be the kth homogeneous component of the
Žn2 .
coinvariant algebra: PnrIn s [ks0
R k . Each R k is an F w Sn x-module; let
k
␹ be the corresponding character. The set ᑭ w ¬ w g Sn4 of Schubert
polynomials forms a basis for PnrIn , and the set ᑭ w ¬ l Ž w . s k 4 forms a
basis for R k .
The action of the simple reflections on Schubert polynomials is described by the following proposition, which is a reformulation of wBGG,
Theorem 3.14Žiii.x.
601
HECKE ALGEBRA ACTIONS
PROPOSITION 2.3.
¡ᑭ
si Ž ᑭ w . s
For any simple reflection si and any w g Sn ,
w,
~yᑭ
qÝ
¢
w
q Ýk - i
k ) iq1
if l Ž wsi . ) l Ž w . ,
ᑭ wŽ k , iq1, i. y Ý k - i ᑭ wŽ k , i , iq1.
ᑭ wŽ k , i , iq1. y Ý k ) iq1 ᑭ wŽ k , iq1, i. ,
if l Ž wsi . - l Ž w . ,
where Ž k, i, i q 1., Ž k, i q 1, i . are cycles of length 3, and the sums extend
o¨ er those ¨ alues of k Ž in the prescribed ranges. for which w Ž k, i, i q 1.
Ž respecti¨ ely, w Ž k, i q 1, i .. has the same length as w.
Note that the signs in this proposition may depend on notational
conventions.
Let ␮ be a partition of n, and let ␹ k be the Sn-character on R k as
above. The following character formula is analogous to Theorem 2.2.
THEOREM 2.4 wRo2, Theorem 2x. With the notations of Theorem 2.2,
␹ k Ž w␮ . s
Ý
weight 1␮ Ž w . ,
l Ž w .sk
where weight 1␮ Ž w . is the weight Ž1.4. with q s 1, and w␮ is any permutation
of cycle-type ␮.
The goal of this paper is to define a Hecke algebra action on the
polynomial ring Pn which produces a q-analogue of Theorem 2.4.
3. q-COMMUTATORS
For 1 F i - n define the q-commutator w ⭸ i , X i x q as follows:
w ⭸ i , X i x q [ ⭸ i X i y qX i ⭸ i .
It should be noted that for q s 1, w ⭸ i , X i x1 s si . Let A i [ w ⭸ i , X i x q .
Claim 3.1. The operators A i , 1 F i - n, satisfy the Hecke algebra
relations Ž2.1. ᎐ Ž2.3..
Proof. Combine the nil-Coxeter relations Ž2.4. ᎐ Ž2.6. for the operators
⭸ i with the commutation relations Ž2.7. ᎐ Ž2.9. for the operators ⭸ i and X j .
It follows that the mapping Ti ¬ A i Ž1 F i - n. may be extended to a
representation ␳ 1 of HnŽ q . on Pn :
␳ 1 Ž Ti . [ A i s w ⭸ i , X i x q .
602
ADIN, POSTNIKOV, AND ROICHMAN
Remark. The polynomial action of the Coxeter generators of S n is
multiplicati¨ e; i.e., for any generator si and any two polynomials f, g g Pn ,
si Ž fg . s si Ž f . si Ž g . .
Ž 3.1.
Thus each si acts on Pn as an algebra automorphism. It follows that if f is
i-symmetric Žsee Section 2.2.1. then si Ž fg . s fsi Ž g .. In contrast to that,
the operators A i are not multiplicative. Actually, Ž2.3. implies that the
eigenvalues of any linear action of a Hecke algebra generator Ti are 1 and
yq, and taking f to be a Žyq .-eigenvector of A i , one would get Žif A i
were multiplicative. that f 2 is a q 2-eigenvector, which is impossible for
generic q.
Claim 3.2. For any 1 F i - n, any i-symmetric polynomial f g ⌳ nŽ i .,
and any polynomial g g Pn ,
Ai Ž f . s f
Ž 3.2.
A i Ž fg . s fA i Ž g . .
Ž 3.3.
and
Proof. Ž3.2. is the special case g s 1 of Ž3.3.. The latter follows from
the fact that for arbitrary polynomials f, g g Pn
⭸ i Ž fg . s ⭸ i Ž f . g q si Ž f . ⭸ i Ž g . .
Therefore, if f g ⌳ nŽ i . then ⭸ i Ž fg . s f⭸ i Ž g ., so that
A i Ž fg . s ⭸ i Ž x i fg . y qx i ⭸ i Ž fg . s f ⭸ i Ž x i g . y qx i ⭸ i Ž g . s fA i Ž g . .
It follows that the ideal In of Pn is invariant under all the operators A i ,
giving rise to a representation ␳˜1 of HnŽ q . on the quotient PnrIn , namely,
on the coinvariant algebra. Let ␹ 1k be the character of this representation
on the kth homogeneous component R k of PnrIn Ž0 F k F Ž 2n ...
Recall from Section 2.2.3 that the set of Schubert polynomials ᑭ w ¬ l Ž w .
s k 4 forms a basis for R k .
The representation ␳˜1 yields a q-analogue of Proposition 2.3.
603
HECKE ALGEBRA ACTIONS
For any 1 F i - n and w g Sn ,
THEOREM 3.3.
¡ᑭ
␳˜1 Ž Ti . Ž ᑭ w . s
w,
~yqᑭ
qÝ
¢
w
q qÝ k - i
k ) iq1
if l Ž wsi . ) l Ž w . ,
ᑭ wŽ k , iq1, i. y Ý k - i ᑭ wŽ k , i , iq1.
ᑭ wŽ k , i , iq1. y qÝ k ) iq1 ᑭ wŽ k , iq1, i. ,
if l Ž wsi . - l Ž w . ,
where Ž k, i, i q 1., Ž k, i q 1, i . are cycles of length 3, and the sums extend
o¨ er those ¨ alues of k Ž in the prescribed ranges. for which w Ž k, i, i q 1.
Ž respecti¨ ely, w Ž k, i q 1, i .. has the same length as w.
Proof. By the commutation relation Ž2.8.,
A i s 1 q Ž X iq1 y qX i . ⭸ i .
Applying Ž2.11. and Ž2.13. completes the proof.
4. CHARACTERS OF q-COMMUTATORS
In this section we prove the following q-analogue of Theorem 2.4.
THEOREM 4.1.
For any partition ␮ & n and k G 0,
␹ 1k Ž T␮ . s
weight q␮ Ž w .
where
in Section 2.1.
Ý
weight q␮ Ž w . ,
l Ž w .sk
is defined as in Ž1.4., and the subproduct T␮ is defined as
First recall that, by Theorem 3.3, for any 1 F i - n and w g S n
à i Ž ᑭ w . s
½
ᑭw ,
if l Ž wsi . ) l Ž w . ,
yqᑭ w q Ý lŽ z s i .) lŽ z .slŽ w . a w , z Ž q . ᑭ z ,
if l Ž wsi . - l Ž w . ,
Ž 4.1.
where A˜i [ ␳˜1ŽTi ., a w, z Ž q . g Zw q x, and the summation is over all z g Sn
with l Ž zsi . ) l Ž z . s l Ž w ..
Denote by ² ⭈ , ⭈ : the inner product on PnrIn defined by ² ᑭ ¨ , ᑭ w : [
␦¨ w , where ␦¨ w is the Kronecker delta. In order to prove Theorem 4.1 we
need the following lemma.
LEMMA 4.2. Let w g Sn be a permutation satisfying l Ž wsi . - l Ž w .. Then,
for any ␲ g S n ,
¦A˜i A˜␲ Ž ᑭ w . , ᑭ w;s yq² A˜␲ Ž ᑭ w . , ᑭ w: .
604
ADIN, POSTNIKOV, AND ROICHMAN
Proof of Lemma 4.2. It follows from Ž4.1. that if l Ž wsi . - l Ž w . and
¨ g S n then
² Ã i Ž ᑭ ¨ . , ᑭ w: s
½
yq,
0,
if ¨ s w,
if ¨ / w.
Ž 4.2.
Substituting Ž4.2. into
¦A˜i A˜␲ Ž ᑭ w . , ᑭ w;s
¦˜žÝ
Ai
¨
/ ;
² A˜␲ Ž ᑭ w . , ᑭ ¨ : ᑭ ¨ , ᑭ w
ݲ A˜␲ Ž ᑭ w . , ᑭ ¨:² A˜i Ž ᑭ ¨ . , ᑭ w:
s
¨
we obtain the desired conclusion.
Proof of Theorem 4.1. In order to prove Theorem 4.1, it suffices to
prove that for any partition ␮ s Ž ␮ 1 , . . . , ␮ t . of n
¦A˜ Ž ᑭ
␮
w
. , ᑭ w;s weight q␮ Ž w . ,
where A˜␮ s ␳˜1ŽT␮ . is the subproduct of A˜1 A˜2 ⭈⭈⭈ A˜ny1 obtained by omitting A˜␮1q ⭈ ⭈ ⭈ q ␮ i for all 1 F i - t.
Assume now that there is an index i such that A˜i and A˜iq1 are factors
of A˜␮ , l Ž wsi . ) l Ž w ., and l Ž wsiq1 . - l Ž w .. Then, by Lemma 4.2,
¦A˜
iq1
A˜␮ Ž ᑭ w . , ᑭ w s yq A˜␮ Ž ᑭ w . , ᑭ w .
;
¦
;
Ž 4.3.
On the other hand, by the Hecke algebra relations, A˜iq1 A˜␮ s A˜␮ A˜i .
Hence
¦A˜
iq1
A˜␮ Ž ᑭ w . , ᑭ w s A˜␮ A˜i Ž ᑭ w . , ᑭ w s A˜␮ Ž ᑭ w . , ᑭ w . Ž 4.4.
; ¦
; ¦
;
The last equality follows from Ž4.1..
Comparing Ž4.3. and Ž4.4. we obtain
yq A˜␮ Ž ᑭ w . , ᑭ w s A˜␮ Ž ᑭ w . , ᑭ w .
¦
; ¦
;
We conclude that, if there is an index i so that A˜i and A˜iq1 are factors of
A˜␮ , l Ž wsi . ) l Ž w ., and l Ž wsiq1 . - l Ž w ., then Žsince q is indeterminate.
¦Ã Ž ᑭ
␮
w
. , ᑭ w;s 0.
Note that in this case i, i q 1, and i q 2 belong to the same ‘‘block’’ in the
partition ␮ , and w Ž i . - w Ž i q 1. ) w Ž i q 2.. Thus indeed
weight q␮ Ž w . s 0.
605
HECKE ALGEBRA ACTIONS
It remains to check the case in which there is no index i so that both A˜i
and A˜iq1 appear as factors in the product A˜␮ , with l Ž wsi . ) l Ž w . and
l Ž wsiq1 . - l Ž w ..
In this case, the relation A˜i A˜j s A˜j A˜i for < i y j < ) 1 gives
A˜␮ s A˜i1 ⭈⭈⭈ A˜i m A˜i mq 1 ⭈⭈⭈ A˜i␮
1
q ⭈ ⭈ ⭈ q ␮ tyt ,
where l Ž wsi j . - l Ž w . for j F m, and l Ž wsi j . ) l Ž w . for j ) m. Applying
Ž4.1. and Lemma 4.2 iteratively implies
¦Ã Ž ᑭ
␮
m
w
. , ᑭ w;s Ž yq . s weight q␮ Ž w . ,
where m s 噛 i ¬ l Ž wsi . - l Ž w . and A˜i is a factor of A˜␮ 4 .
5. RANDOMIZED OPERATORS
In this section we define a natural ‘‘randomized’’ action of the Coxeter
generators on the polynomial ring Pn , and show that this action satisfies
the Hecke algebra relations. This action will be defined initially on
monomials, and then extended by linearity to all polynomials in Pn .
␤
Let e␣ , ␤ , m [ x i␣ x iq1
m, where m g Pn is a monomial involving neither
x i nor x iq1 , and ␣ , ␤ are nonnegative integers. Note that the linear
subspace V␣ , ␤ , m [ span e␣ , ␤ , m , e␤ , ␣ , m 4 is invariant under the action of si .
In this space si acts as a transposition of the two basis elements Žif
␣ / ␤ ..
A natural randomization of e␣ , ␤ , m is Ž1 y q . e␣ , ␤ , m q qe␤ , ␣ , m , where
the parameter q may be interpreted as transition probability 0 F q F 1.
Motivated by well-known asymmetric physical processes Žsimulated annealing etc.., we define
RUi Ž e␣ , ␤ , m . [
½
e␤ , ␣ , m ,
if ␣ G ␤ ,
Ž 1 y q . e␣ , ␤ , m q qe␤ , ␣ , m ,
if ␣ - ␤ ,
Ž 5.1.
and extend this randomized action to the whole polynomial ring Pn by
linearity. See also wJix.
Claim 5.1. The operators RUi , 1 F i - n, satisfy the Hecke algebra
relations Ž2.1. ᎐ Ž2.3..
Proof. This is easily verified by an explicit calculation of the action on
the monomials e␣ , ␤ , m .
606
ADIN, POSTNIKOV, AND ROICHMAN
The operators RUi lead, therefore, to a representation of HnŽ q . on Pn .
Unfortunately, the symmetric functions are not invariant under this action.
Consider, therefore, the operators whose representation matrices with
respect to the basis of monomials are the transposes of those representing
RUi ; i.e., define
¡qe ,
. [~ Ž 1 y q . e
¢e ,
if ␣ ) ␤ ,
␤, ␣ , m
R i Ž e␣ , ␤ , m
␣, ␤, m
q e␤ , ␣ , m ,
␣, ␤, m
if ␣ - ␤ ,
Ž 5.2.
if ␣ s ␤ .
Of course, the operators R i , 1 F i - n, also satisfy the Hecke relations
Ž2.1. ᎐ Ž2.3.. It follows that the mapping Ti ¬ R i Ž1 F i - n. may be extended to a representation ␳ 2 of HnŽ q . on Pn :
␳ 2 Ž Ti . [ R i .
The following claim is analogous to Claim 3.2.
Claim 5.2. For any 1 F i - n, any i-symmetric polynomial f g ⌳ nŽ i .,
and any polynomial g g Pn ,
Ri Ž f . s f
Ž 5.3.
R i Ž fg . s fR i Ž g . .
Ž 5.4.
and
Proof. This may be shown by direct calculation.
It follows from the first part of the claim that symmetric functions are
pointwise invariant under ␳ 2 Ž HnŽ q ... By the second part, the ideal In is
also invariant under ␳ 2 Ž HnŽ q ... Thus, ␳ 2 gives rise to a representation ␳˜2
of HnŽ q . on the coinvariant algebra PnrIn .
The action of R i on monomials is transparent. Section 6 is devoted to a
better understanding of the action on the coinvariant algebra.
6. PROPERTIES OF THE RANDOMIZED ACTION
The following sequence of assertions concerns the connections between
the operators A i and R i .
Claim 6.1. The operators A i and R i have the same invariant vectors,
ker Ž A i y 1 . s ker Ž R i y 1 . s ⌳ n Ž i . ,
607
HECKE ALGEBRA ACTIONS
where ⌳ nŽ i . is the set Žactually, subalgebra . of all polynomials invariant
under si .
Proof. By the definition of A i and the commutation relations Ž2.8.,
ker Ž A i y 1 . s ker Ž X iq1 y qX i . ⭸ i s ker ⭸ i s ⌳ n Ž i . .
As for R i y 1, let V␣ , ␤ , m [ span e␣ , ␤ , m , e␤ , ␣ , m 4 as in the beginning of
Section 5. Note that
Pn s
[
Ž ␣ , ␤ , m .N ␣ G ␤ 4
V␣ , ␤ , m
is a decomposition of Pn into a direct sum of R i-invariant subspaces.
By Ž5.2., in V␣ , ␤ , m ,
¡ye
. s~yqe
¢0,
␣, ␤, m
Ž R i y 1 . Ž e␣ , ␤ , m
q qe␤ , ␣ , m ,
if ␣ ) ␤ ,
q e␤ , ␣ , m ,
if ␣ - ␤ ,
␣, ␤, m
if ␣ s ␤ .
Thus
ker Ž R i y 1 . l V␣ , ␤ , m s
½
span e␣ , ␤ , m q e␤ , ␣ , m 4 ,
if ␣ / ␤ ,
span e␣ , ␤ , m 4 ,
if ␣ s ␤ ,
implying
ker Ž R i y 1 . s
[
Ž ␣ , ␤ , m .N ␣ G ␤ 4
span e␣ , ␤ , m q e␤ , ␣ , m 4 s ⌳ n Ž i . .
Claim 6.2. Ža. For any positive integers i - n, j F n and m,
Ž A i y R i . Ž x jm . s Ž 1 y q . ⭸ i Ž x jmq 1 . .
Ž 6.1.
Žb. For any polynomial f g Z w x 1 , . . . , x n x, the polynomial Ž A i y R i . f
is i-symmetric and divisible by 1 y q:
Ž Ai y R i . f g ⌳ n Ž i . l Ž 1 y q . ⭈ Zw x1 , . . . , x n x .
Proof. Ža. If j f i, i q 14 then both sides of Ž6.1. equal zero. If j s i
and m G 1 then
Ž A i y R i . Ž x im . s Ž 1 q Ž x iq1 y qx i . ⭸ i y R i . Ž x im .
608
ADIN, POSTNIKOV, AND ROICHMAN
m
s x im q Ž x iq1 y qx i .
ty1
m
y qx iq1
Ý x imyt x iq1
ts1
m
s Ž1 y q.
t
s Ž 1 y q . ⭸ i Ž x imq1 . .
Ý x imy t x iq1
ts0
If j s i q 1 and m G 1 then, by Claim 6.1.
m
m
y x im . s Ž A i y R i . Ž yx im .
Ž A i y R i . Ž x iq1
. s Ž A i y R i . Ž x im q x iq1
mq 1
s y Ž 1 y q . ⭸ i Ž x imq 1 . s Ž 1 y q . ⭸ i Ž x iq1
..
Žb. It suffices to prove this claim for monomials x 1k 1 x 2k 2 ⭈⭈⭈ x nk n . Any
such monomial has the form gx jm , where g g ⌳ nŽ i ., j g i, i q 14 , and m
is a nonnegative integer. If m s 0 then Ž A i y R i .Ž g . s 0. Otherwise, it
follows from Ž3.3., Ž5.4., and Ž6.1. that
Ž A i y R i . Ž gx jm . s g Ž A i y R i . Ž x jm . s Ž 1 y q . g ⭸ i Ž x jmq 1 . , Ž 6.2.
as claimed.
LEMMA 6.3. ⌳ nŽ i . is spanned, as a ⌳ n-module, by the Schubert polynomials ᑭ w with l Ž wsi . ) l Ž w .. The same holds when the ground field F is
replaced by Z.
Proof. First of all, if l Ž wsi . ) l Ž w . then, by Proposition 2.3, si Ž ᑭ w . s
ᑭ w and therefore ᑭ w g ⌳ nŽ i ..
By the same proposition, for any w g Sn
Ž 1 q si . Ž ᑭ w . g span ᑭ z N l Ž zsi . ) l Ž z . s l Ž w . 4
Ž in PnrIn . ,
and therefore, for any f g PnrIn ,
Ž 1 q si . Ž f . g span ᑭ z N l Ž zsi . ) l Ž z . 4
Ž in PnrIn . .
If f g ⌳ nŽ i .rIn then Ž1 q si .Ž f . s 2 f, so that ⌳ nŽ i .rIn is spanned, as a
vector space, by the above Schubert polynomials. Since In is the ideal of Pn
generated by the homogeneous elements in ⌳ n of positive degree, a
standard argument yields the claimed result for ⌳ nŽ i . as a ⌳ n-module.
The result for Z instead of F follows from the fact that Schubert
polynomials also form a ‘‘basis’’ for polynomials with integer coefficients.
609
HECKE ALGEBRA ACTIONS
The following proposition provides a description of the action of
␳˜2 Ž HnŽ q .. on Schubert polynomials.
PROPOSITION 6.4.
␳˜2 Ž Ti . Ž ᑭ w
For each 1 F i - n and w g Sn ,
¡ᑭ
¢
if l Ž wsi . ) l Ž w . ,
w,
. s~yqᑭ
w
q Ý lŽ z s i .) lŽ z . Ž 1 y q . bw , z q c w , z ᑭ z ,
if l Ž wsi . - l Ž w . ,
where bw, z g Z, c w, z g y1, 0, 14 , and the sum extends o¨ er all permutations
z g Sn with l Ž zsi . ) l Ž z . s l Ž w ..
Proof. Since ᑭ w g ⌳ nŽ i . for w g Sn with l Ž wsi . ) l Ž w ., Claim 6.1
implies that ␳ 2 ŽTi .Ž ᑭ w . s ᑭ w for these w.
Homogeneous components of Pn are invariant under the action of each
R i , 1 F i - n. It follows that the homogeneous components of the coinvariant algebra are invariant under ␳˜2 Ž HnŽ q .., so that each ␳˜2 ŽTi .Ž ᑭ w . is
spanned by Schubert polynomials of degree l Ž w .. Combining this fact with
Claim 6.2Žb. and Lemma 6.3 shows that for any 1 F i - n and w g Sn
Ž ␳˜2 Ž Ti . y ␳˜1Ž Ti . . Ž ᑭ w . s Ž 1 y q .
Ý
dw, z ᑭ z ,
Ž 6.3.
l Ž zs i .)l Ž z .sl Ž w .
where d w, z g Z, and the sum extends over all permutations z g Sn with
l Ž zsi . ) l Ž z . s l Ž w ..
Combining Ž6.3. with Ž4.1. gives, for any w g Sn with l Ž wsi . - l Ž w .,
␳˜2 Ž Ti . Ž ᑭ w . s ␳˜1 Ž Ti . Ž ᑭ w . q Ž ␳˜2 Ž Ti . y ␳˜1 Ž Ti . . Ž ᑭ w .
s yqᑭ w q
Ý
aw , z ᑭ z
Ý
dw, z ᑭ z
Ý
Ž 1 y q . d w , z q aw , z ᑭ z ,
l Ž zs i .)l Ž z .sl Ž w .
qŽ 1 y q .
l Ž zs i .)l Ž z .sl Ž w .
s yqᑭ w q
l Ž zs i .)l Ž z .sl Ž w .
where a w, z g 0, " 1, " q4 , d w, z g Z, and the sum extends over all z g Sn
with l Ž zsi . ) l Ž z . s l Ž w .,
Substituting
bw , z s
½
dw, z ,
d w , z y a w , zrq,
if a w , z / "q
if a w , z s "q
610
ADIN, POSTNIKOV, AND ROICHMAN
and
cw , z s
½
if a w , z / "q
if a w , z s "q
aw , z ,
a w , zrq,
completes the proof.
Imitating the proof of Theorem 4.1 we obtain
THEOREM 6.5.
␹ 2k Ž T␮ . s
Ý
weight q␮ Ž w . ,
l Ž w .sk
where weight q␮ Ž w . are the same weights as in Theorem 4.1.
Combining Theorems 6.5 and 4.1 together with Ram’s result ŽTheorem
2.1. shows that
THEOREM 6.6. The representation of HnŽ q . induced by the q-commutators
A i on the homogeneous components of the coin¨ ariant algebra and the
representation induced by the transposed randomized operators R i on these
components are equi¨ alent.
Problem 6.7. Calculate the coefficients bw, z in Proposition 6.4.
We conjecture that bw, z g y1, 0, 14 .
7. FINAL REMARKS
7.1. Related Families of Operators
Consider the following family of q-commutators:
Bi [ y w ⭸ i , X iq1 x q
Ž 1 F i - n. .
This family is closely related to the q-commutators A i .
Fact 7.1. The operators Bi satisfy the Hecke algebra relations
Ž2.1. ᎐ Ž2.3..
PROPOSITION 7.2. Let Di be operators on the polynomial ring Pn of the
form c i q Pi Ž X i , X iq1 . ⭸ i , 1 F i - n, where Pi are polynomials of two ¨ ariables, and c i are constants. If
Ž1.
Ž2.
Di satisfy the Hecke algebra relations Ž2.1. ᎐ Ž2.3. Ž with q / 0, y1.,
Di are degree preser¨ ing Ž as operators on Pn .,
611
HECKE ALGEBRA ACTIONS
Ž3. ⌳ n , the subring of symmetric functions, is pointwise in¨ ariant under
all Di , 1 F i - n,
then, for n ) 2, either Di s A i Ž᭙ i . or Di s Bi Ž᭙ i ..
Proposition 7.2 is related to a general theorem of Lascoux and
Schutzenberger.
¨
LS THEOREM wLS, Theorem 1x. Let x 1 , x 2 , x 3 be ¨ ariables, and let si
i s 1, 2 be the simple transpositions as abo¨ e. Let Di , i s 1, 2 be linear
operators on the ring of rational functions CŽ x 1 , x 2 , x 3 . Ž considered as a
¨ ector space o¨ er C. defined by
Di s Pi q Q i si ,
where Pi , Q i g CŽ x i , x iq1 . are rational functions of the corresponding pair of
¨ ariables.
Assume that:
Ž1.
Ž2.
D1 D 2 D1 s D 2 D1 D 2 ;
D 1 is in¨ ertible and P1 / 0.
Then D 1 and D 2 preser¨ e the ring of polynomials C w x 1 , x 2 , x 3 x if and only if
there exist ␣ , ␤ , ␥ , ␦ , ␩ g C, so that ⌬ [ ␣␦ y ␤␥ / 0, ␩ / 0, ␩ / ⌬, and
Pi Ž x i , x iq1 . s Ž x i y x iq1 .
y1
Ž ␣ x i q ␤ . Ž ␥ x iq1 q ␦ .
and
Q i s ␩ y Pi .
Also, in that case, both D 1 and D 2 satisfy Di2 s ⌬ Di q ␩ Ž␩ y ⌬ ..
Obviously, the initial conditions in this theorem are quite different from
those of Proposition 7.2; but the two families A i and Bi are common
solutions of both problems Žfor the LS theorem in the special case
⌬ s 1 y q, ␩ s 1.. Note that for ⌬ s 1 y q, ␩ s yq one gets two other
families of the q-commutator type, for which ⌳ n is not pointwise invariant.
It should be mentioned that the family R i of Sections 5 and 6 is not
obtainable from the LS theorem Žor from Proposition 7.2..
7.2. Connections with Kazhdan᎐Lusztig Theory
Theorem 3.3 has a remarkable analogue in Kazhdan᎐Lusztig theory.
In their seminal paper wKLx Kazhdan and Lusztig constructed a canonical
basis to Hecke algebra representations. A basic theorem in this theory
describes the action of the generators Ts on the canonical basis
elements Cw .
612
ADIN, POSTNIKOV, AND ROICHMAN
THEOREM 7.3 wKL, Ž2.3a. ᎐ Ž2.3c.x. Let W be a Coxeter group, s a Coxeter
generator of W, w g W, and Cw the corresponding Kazhdan᎐Lusztig basis
element. Then
Ts Ž Cw . s
½
yCw ,
qCw q q
if l Ž sw . - l Ž w . ,
1r2
Ý lŽ s z .- lŽ z .slŽ w . a w , z C z ,
if l Ž sw . ) l Ž w . ,
where the coefficients a w, z g Z are independent of q.
This analogy leads to similar character formulas in the two theories; see
Theorems 2.2 and 4.1. This surprising phenomenon seems to warrant
further study.
7.3. Probabilistic Aspects
The parameter q in the definition of the Hecke algebra may be
interpreted as a transition probability. This gives a natural interpretation to
the appearance of the coefficients q and 1 y q in the basic Hecke relation
Ž2.3.. This observation was fundamental to the definition of the randomized action in Section 5. The operators defined there interpolate between
two well-studied extreme cases: sorting Ž q s 0. and mixing Ž q s 1. by
means of adjacent transpositions. They also form an interesting link
between algebra and physics-motivated optimization.
7.4. Other Weyl Groups
Extension of all the above to other Weyl and Coxeter groups is highly
desirable. Preliminary computations indicate that this may not be straightforward.
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