Table Of ContentA Littlewood-Richardson rule for evaluation
4
0
representations of U (sl )
0 q n
2
n
b
a Bernard LECLERC
J
8
]
T
R
Abstract
.
h We give a combinatorialdescription of the composition factors of the induction productof
t
a two evaluationmodules of the affine Iwahori-Heckealgebra of type GLm. Using quantum
m affineSchur-Weylduality,thisyieldsa combinatorialdescriptionofthecompositionfactors
[ ofthetensorproductoftwoevaluationmodulesofthequantumaffinealgebraUq(sln).
1 b
v
7 1 Introduction
7
0
1 1.1 Let Hm denote the Iwahori-Hecke algebra of type Am−1 over C(t). This is a semisimple
0 associative algebra isomorphic to the group algebra C(t)[S ] of the symmetric group. Hence
m
4
its simple modules S(λ) are parametrized by the partitions λ of m. Consider a decomposition
0
/ m = m1 + m2, and two partitions λ(1) and λ(2) of m1 and m2, respectively. Then we have a
h
H -moduleS(λ(1))andaH -moduleS(λ(2)),andwecanformtheinducedmodule
t m1 m2
a
m S(λ(1))⊙S(λ(2)):= IndHm S(λ(1))⊗S(λ(2)) .
: Hm1⊗Hm2 (cid:16) (cid:17)
v
Here, H ⊗ H is identified to a subalgebra of H in the standard way. Using again the
i m1 m2 m
X isomorphism H ∼= C(t)[S ], we see that the multiplicity of a simple H -module S(µ) in
m m m
r S(λ(1)) ⊙ S(λ(2)) is equal to the classical Littlewood-Richardson coefficient cµ (see e.g.
a λ(1)λ(2)
[Mcd]).
1.2 Let now H be the affine Iwahori-Hecke algebra over C(t) (see 2.1 below). For each
m
invertiblez ∈ C(t)wehaveasurjectiveevaluationhomomorphismτ :H → H . Pullingback
z m m
b
the simple H -module S(λ) via τ weobtain a simple H -module S(λ;z) called an evaluation
m z m
b
module. In analogy with 1.1, given two invertible elements z and z of C(t), we can then form
b 1 2
theinduced H -module
m
Sb(λ(1);z )⊙S(λ(2);z ):= IndHm S(λ(1);z )⊗S(λ(2);z ) .
1 2 Hbm1⊗Hm2 (cid:16) 1 2 (cid:17)
b b
1
It turns out that if we fix λ(1),λ(2) and vary the spectral parameters z ,z , this module is gener-
1 2
ically irreducible, that is, it is simple except for a finite number of values of the ratio z /z . In
1 2
[LNT,Theorem36]acombinatorial description ofthesespecial valueswasgiven.
In this note we shall make this result more precise by describing all the composition factors
of S(λ(1);z ) ⊙ S(λ(2);z ) at these critical values z /z . We shall also prove that, in contrast
1 2 1 2
withtheclassicalLittlewood-Richardson rule,allthecompositionfactorsappearwithmultiplicity
one. The composition factors occuring in a product will be described using the combinatorics of
Lusztig’s symbols, that is, of certain two-row arrays introduced by Lusztig for parametrizing the
irreduciblecomplexrepresentations oftheclassicalreductivegroupsoverfinitefields[Lu1,Lu2].
1.3 Wewillderiveourcombinatorialformulafromsomeexplicitcalculationsofcanonicalbases
in level 2representations ofthe quantum algebra U (sl )performed in[LM]. More precisely,
v n+1
bydualizing[LM,Theorem3],wegetaformulafortheexpansionoftheproductoftwoquantum
flagminorsonthedualcanonical basisofU (sl )(Theorem 4). UsingthenAriki’stheorem as
v n+1
in[LNT],weobtain immediately the above-mentioned Littlewood-Richardson rule forinduction
products oftwoevaluation modulesoveraffineHeckealgebras(Theorem2).
Finally, by means of the quantum affine analogue of the Schur-Weyl duality developed by
Cherednik, Chari-Pressley and Ginzburg-Reshetikhin-Vasserot, we can deduce from this rule a
similar one for the tensor product of two evaluation modules over the quantum affine algebra
U (sl ).
q N
b
2 Composition factors of induced H -modules
m
2.1 Let H be the affine Hecke algebra of typebGL over C(t). It has invertible generators
m m
T1,...,Tmb−1,y1,...,ym subjecttotherelations
T T T = T T T , (1 6 i6 m−2),
i i+1 i i+1 i i+1
T T = T T , (|i−j| > 1),
i j j i
(T −t)(T +1) = 0, (1 6i 6 m−1),
i i
y y = y y , (1 6 i,j 6 m),
i j j i
y T = T y , (j 6= i,i+1),
j i i j
T y T =ty , (1 6 i 6 m−1).
i i i i+1
Thesubalgebra Hm generated bytheTi’sistheIwahori-Hecke algebra oftypeAm−1.
Forany invertible z ∈ C(t) we have a unique algebra homomorphism τ : H → H such
z m m
that
b
τ (T ) = T , τ (y ) = z, (i = 1,...,m−1).
z i i z 1
Thisiscalledtheevaluation atz.
2
Wealsohaveanalgebraautomorphism σ : H → H suchthat
z m m
σ (T )= T , σ (y )= zyb, (bi = 1,...,m−1).
z i i z i i
Thisiscalledtheshiftbyz.
2.2 As mentioned in the introduction, given two partitions λ(1) and λ(2), the structure of the
induced H -module
m
S(λ(1);z )⊙S(λ(2);z )
b 1 2
depends essentially on the ratio z /z . Indeed, by twisting this module with the shift automor-
1 2
phismσ weobtaintheinduced module
z
S(λ(1);zz )⊙S(λ(2);zz ).
1 2
Forexample,itisknownthatifz /z 6∈tZthenS(λ(1);z )⊙S(λ(2);z )isirreducible. Therefore,
1 2 1 2
wecanassumewithoutlossofgenerality that
z = tai, a ∈ Z, a > ℓ(λ(i)), (i = 1,2), (1)
i i i
where as usual ℓ(λ) denotes the length of the partition λ. Since S(λ(1);z ) ⊙ S(λ(2);z ) and
1 2
S(λ(2);z )⊙S(λ(1);z )have the samecomposition factors withthe samemultiplicities, wecan
2 1
alsoassumethata 6 a .
1 2
2.3 Itwillbeconvenient towritepartitions inweaklyincreasing order. Givenapartition λand
an integer a > ℓ(λ) we can make λ into a non-decreasing sequence (λ ,...,λ ) of length a by
1 a
setting λ = 0forj = 1,...,a−ℓ(λ). Wecanthenassociate to(λ,a)theincreasing sequence
j
β = (β ,...,β ), β = j +λ . (2)
1 a j j
Inthisway,given(λ(i),a )(i = 1,2)asin2.2,weobtainasymbol
i
β(2) β(2),...,β(2)
S = = 1 a2 . (3)
(cid:18)β(1)(cid:19) (cid:18)β(1),...,β(1)(cid:19)
1 a1
Forexample,thesymbolattached tothepairs((1,1,2),3) and((2,3),5) is
1 2 3 6 8
S = .
(cid:18)2 3 5 (cid:19)
Conversely, givenasymbolS,i.e. atwo-rowarrayasinEq.(3)with
1 6β(i) < ··· < β(i) (i = 1,2),
1 ai
thereisauniquepair(λ(i),a )(i = 1,2)whosesymbolisS.
i
3
2.4 ThesymbolS ofEq.(3)issaidtobestandard ifβ(2) 6 β(1) fork 6 a . In[LM,§2.5]we
k k 1
havedefinedthepairsofastandardsymbolS,andthesetC(S)ofallsymbolsΣobtainedfromS
by permuting someofitspairs. Asshown in[LM, Lemma9], these notions areequivalent tothe
notionofadmissible involution ofLusztig[Lu4].
Fortheconvenience ofthereader weshall recallthese definitions. LetS = β beastandard
γ
symbol. We define an injection ψ : γ −→ β such that ψ(j) 6 j for all j ∈ (cid:0)γ.(cid:1)To do so it is
enough todescribe thesubsets
γl = {j ∈ γ | ψ(j) = j −l}, (0 6 l 6 n).
Wesetγ0 =γ ∩β andforl > 1weput
γl = {j ∈ γ−(γ0∪···∪γl−1)| j−l ∈β −ψ(γ0∪···∪γl−1)}.
Observethatthestandardness ofS impliesthatψ iswell-defined.
Example1 Take
1 3 5 8 9
S = .
(cid:18)3 6 7 10 (cid:19)
Then
γ0 = {3}, γ1 = {6,10}, γ2 = ··· = γ5 = ∅, γ6 = {7}.
Hence
ψ(3) = 3, ψ(6) = 5, ψ(7) = 1, ψ(10) = 9.
The pairs (j,ψ(j)) with ψ(j) 6= j (that is, j 6∈ β ∩γ) will be called the pairs of S. Given
a standard symbol S with p pairs, we denote by C(S) the set of all symbols obtained from S by
permuting some pairs in S and reordering the rows. Weconsider S itself as an element of C(S),
henceC(S)hascardinality 2p.
2.5 Givenapartition λand aninteger awecall Youngdiagram of(λ,a)theYoung diagram of
λinwhicheachcell(i,j)isfilledwiththeintegeri−j+a. Forinstance,ifλ = (2,3)anda= 5
thentheYoungdiagramof(λ,a)is
4 5
5 6 7
TherowsoftheYoungdiagram of(λ,a)yieldamultisegment
m(λ,a) := [k,k+λ −1].
k
16Xk6a
4
This is a formal sum (or multiset) of intervals in Z, in which we discard the empty intervals
corresponding tothek’swithλ = 0. Thus,continuing withthesameexample,wehave
k
m((2,3),5) =[4,5]+[5,7].
Similarly,weattachtoapair(λ(i),a )(i = 1,2)ortoitssymbolS themultisegment
i
m(S) = m(λ(1),a )+m(λ(2),a ).
1 2
2.6 Toeachmultisegment
m:= [α ,β ]
k k
Xk
isattachedanirreducible Hm-moduleLm,wherem = k(βk +1−αk)(seee.g. [LNT,§2.1]).
P
b
2.7 Letusassumethatthepair(λ(i),a )(i = 1,2) satisfiestheconditions of2.2. LetΣdenote
i
thesymbolattachedtothispair. Wecannowstate:
Theorem2 Thecomposition factors ofS(λ(1);ta1)⊙S(λ(2);ta2)arethemodules Lm(S) where
S runs through the set of standard symbols such that Σ ∈ C(S). Each of them occurs with
multiplicity one.
Theorem2willbededuced fromTheorem4below.
Example3 Let(λ(1),a ) = ((1,4),2) and(λ(2),a ) = ((1,2,3),4). Thecorresponding symbol
1 2
is
1 3 5 7
Σ = .
(cid:18)2 6 (cid:19)
Thestandard symbolsS suchthatΣ ∈ C(S)are
1 2 5 6 1 2 5 7 1 3 5 6 1 3 5 7
, , , .
(cid:18)3 7 (cid:19) (cid:18)3 6 (cid:19) (cid:18)2 7 (cid:19) (cid:18)2 6 (cid:19)
It follows that the composition factors of S((1,4);t2)⊙S((1,2,3);t4) are the Lm where m is
onethefollowingmultisegments:
n = [1,2]+[2,6]+[3,4]+[4,5], n = [1,2]+[2,5]+[3,4]+[4,6],
1 2
n = [1,1]+[2,2]+[2,6]+[3,4]+[4,5], n = [1,1]+[2,2]+[2,5]+[3,4]+[4,6].
3 4
5
3 Canonical bases
3.1 Fixn > 2andletg = sl . WeconsiderthequantumenvelopingalgebraU (g)overQ(v)
n+1 v
with Chevalley generators e ,f ,t (1 6 j 6 n). The simple roots and the fundamental weights
j j j
aredenotedbyα andΛ (1 6k 6 n)respectively. Theirreducible representation ofU (g)with
k k v
highest weightΛisdenoted byV(Λ). WedenotebyU (n)thesubalgebra ofU (g)generated by
v v
e (1 6 j 6 n).
j
3.2 LetB(resp.B∗)denotethecanonicalbasis(resp.thedualcanonicalbasis)ofU (n)([Lu3],
v
[BZ];seealso[LNT,§3]). TheelementsofBandB∗ arenaturallylabelledbythemultisegments
msupported on[1,n]. Weshalldenotethembybm andb∗m respectively.
Thevectorsb∗m forwhichmisoftheform
m = m(λ,a)
forsomepartitionλandsomeintegeraarecalledquantumflagminors. Indeed,by[BZ],theycan
be expressed as quantum minors of a triangular matrix whose entries are iterated brackets of the
e ’s(see[LNT,§5.2]).
i
3.3 Let(λ(i),a )(i = 1,2)beasin2.2. Wealsoassumethatthemultisegments
i
m = m(λ(i),a ) (i = 1,2)
i i
are supported on [1,n]. Let Σ be the symbol attached to the pair (λ(i),a ) (i = 1,2). For a
i
standard symbolS suchthatΣ ∈ C(S)wedenote byn(S,Σ)thenumberofpairsofS whichare
permuted to get Σ. Finally, we denote by N (λ,a) the number of cells of the Young diagram of
j
(λ,a)containing theintegerj.
Theorem4 Wehave
b∗m1b∗m2 = v−Na1(λ(2),a2) vn(S,Σ)b∗m(S)
XS
wherethesumrunsthrough allstandardsymbolsS suchthatΣ ∈C(S).
Example5 Wetake(λ(1),a )and(λ(2),a )asinExample3. Hence
1 2
m = [1,1]+[2,5], m = [2,2]+[3,4]+[4,6].
1 2
ThenN (λ(2), a ) =N ((1,2,3),4) = 1,andweobtain, usingthenotationofExample3,
a1 2 2
b∗m b∗m = v−1(v2b∗n +vb∗n +vb∗n +b∗n ).
1 2 1 2 3 4
6
3.4 Proof of Theorem 4. Following [LNT, §7.2], we will replace calculations of products of
elements of B∗ by calculations of dual canonical bases of finite-dimensional representations of
U (g).
v
3.4.1 Let U (n−) denote the subalgebra of U (g) generated by the f ’s, and let x 7→ x♯ denote
v v i
the algebra isomorphism from U (n) to U (n−) defined by e♯ = f (i = 1,...,n). Let Λ be a
v v i i
dominant integral weight and let u be a highest weight vector of the irreducible module V(Λ).
Λ
Thenthemapπ : x7→ x♯u projectsthecanonicalbasisBofU (n)totheunionofthecanonical
Λ Λ v
basisB(Λ)ofV(Λ)withtheset{0}. Thedualmapπ∗ givesanembedding ofthedualcanonical
Λ
basisB∗(Λ)ofV(Λ) ≃ V(Λ)∗ intothedualcanonical basisB∗ ofU (n) ≃ U (n)∗.
v v
3.4.2 In particular the subset of B∗ obtained by embedding the bases B∗(Λ ) (1 6 a 6 n) of
a
the fundamental representations is precisely the subset of quantum flag minors. It is well known
thatV(Λ )isaminusculerepresentationwhosebasesB∗(Λ )andB(Λ )coincide. Moreoverthe
a a a
elementsofthesebasesarenaturallylabelledbythepairs(λ,a)whoseYoungdiagram(asdefined
in 2.5) contains only cells numbered by integers between 1 and n. Denoting them by b∗ we
(λ,a)
have
∗ ∗ ∗
π (b )= b
Λa (λ,a) m(λ,a)
Equivalently, wecan also label the elements of B∗(Λ )by one-row symbols β as in Eq.(2)with
a
β 6 n+1.
i
3.4.3 Similarly,thebasisB∗(Λ )⊗B∗(Λ )isnaturallylabelledbythesetofsymbolsS asin
a1 a2
Eq.(3)withβ(i) 6 n+1(i = 1,2). Usingthetheory ofcrystal bases[K1,K2]onecanseethat
ai
the basis B∗(Λ + Λ ) has a natural labelling by the subset of standard symbols [LM, §2.3].
a1 a2
Moreover, denoting by b∗ the element of B∗(Λ +Λ ) labelled by the standard symbol S we
S a1 a2
have,usingalsothenotation of2.5,
∗ ∗ ∗
π (b ) = b .
Λa1+Λa2 S m(S)
3.4.4 Letι: V(Λ +Λ ) → V(Λ )⊗V(Λ )betheU (g)-moduleembeddingwhichmaps
a1 a2 a1 a2 v
u to u ⊗ u , and let ι∗ : V(Λ ) ⊗ V(Λ ) → V(Λ + Λ ) be its dual. Let
flb∗iaΛga∈1m+BΛina∗o2(rΛs.ai)ItΛ(iias1=sho1w,Λ2n)a2iannd[LdNenTo,te§7b.y2.7b∗m] tiha=a1t tπhΛ∗eaiim(ba∗ig)ae2(iof=b∗1,⊗2)b∗tah1uencdoerrraet2shpeocnodminpgoqsiutaionntuomf
1 2
tmheapBs∗π-Λ∗exa1p+anΛsai2on◦oι∗fbc∗moinbc∗midesituispetnooaugphowtoercaolfcuvlawteitthhethmeaptrroixduocftthbe∗mm1ba∗mp2ι.∗Hweitnhceretsopeccatlctoultahtee
1 2
basesB∗(Λ )⊗B∗(Λ )andB∗(Λ +Λ ).
a1 a1 a1 a2
7
3.4.5 The matrix of ι with respect to the bases B(Λ +Λ ) and B(Λ )⊗B(Λ ) was cal-
a1 a2 a1 a1
culatedin[LM,Theorem3]intermsofLusztig’ssymbols. Transposing thismatrixweobtainthe
desired matrixofι∗. Using3.4.2and3.4.3,wethengettheformulaofTheorem4. 2
3.5 ProofofTheorem2. By[LNT,§3.7]themultisegmentsmindexingthecompositionfactors
Lm of S(λ(1);ta1) ⊙ S(λ(2);ta2) are those occuring in the right-hand side of the formula of
Theorem 4. Moreover the composition multiplicities are obtained by specializing v to 1 in the
coefficients ofthisformula. Hencetheyareallequalto1. 2
4 Tensor products of U (sl )-modules
q N
4.1 Let U (sl ) be the quantized abffine algebra of type A(1) with parameter q a square root
q N N−1
oft(seeforexample[CP]forthedefiningrelations ofU (sl )). Thequantum affineSchur-Weyl
b q N
duality between H and U (sl ) [CP, Ch, GRV] gives a functor F from the category of
m q N b m,N
finite-dimensional H -modules to the category of level 0 finite-dimensional representations of
b m b
U (sl ). If N > m, F maps the simple modules of H to simple modules of U (sl ).
q N m,N m q N
b
However, the image of a non-zero simple H -module may be the zero U (sl )-module. More
b m b q N b
precisely,thesimpleHm-moduleLmismapbpedtoanon-zerosimpleUq(slN)b-moduleifandonly
if allthe segments occuring in mhave length 6 N −1. Inthis case the Drinfeld polynomials of
b b
Fm,N(Lm)areeasilycalculated fromm(see[CP]).
ThefunctorF transformsinductionproductintotensorproduct,thatis,forM inC and
m,N 1 m1
M inC onehas
2 m2
F (M ⊙M )= F (M )⊗F (M ).
m1+m2,N 1 2 m1,N 1 m2,N 2
4.2 TheimageunderF ofanevaluationmoduleforH isanevaluationmoduleforU (sl ),
m,N m q N
andallevaluation modulesofU (sl )canbeobtained inthisway,byvaryingm ∈ N∗.
q N b b
b
4.3 By application of the Schur functor F to Theorem 2 we thus obtain a combinatorial
m,N
descriptionofallcompositionfactorsofthetensorproductoftwoevaluationmodulesofU (sl ).
q N
Example6 We continue Example 3 and Example 5. The image of the H5-module Lm1 ubnder
F istheevaluation moduleV(m )ofU (sl )withDrinfeldpolynomials
5,N 1 q N b
P1(u) = u−b q−2,
P (u) = P (u) = 1,
2 3
P (u) = u−q−7,
4
P (u) = 1, (5 6 k 6 N −1).
k
8
Thisisanon-zeromoduleifandonlyifN >5. Similarly,theimageoftheH6-moduleLm2 under
F istheevaluation moduleV(m )ofU (sl )withDrinfeldpolynomials
6,N 2 q N b
P (u) = u−b q−4,
1
P (u) = u−q−7,
2
P (u) = u−q−10,
3
P (u) = 1, (4 6 k 6 N −1).
k
Thisisanon-zeromoduleifandonlyifN > 4. TheimagesoftheH11-modulesLm1,Lm2,Lm3,
Lm4 underF11,N arethemodulesV(n1),V(n2),V(n3),V(n4)wibthrespectiveDrinfeldpolyno-
mials
P (u) = 1,
1
P (u) = (u−q−3)(u−q−7)(u−q−9),
2
P (u) = P (u) = 1,
3 4
P (u) = u−q−8,
5
P (u) = 1, (6 6 k 6 N −1);
k
P (u) = 1,
1
P (u) = (u−q−3)(u−q−7),
2
P (u) = u−q−10,
3
P (u) = u−q−7,
4
P (u) = 1, (5 6 k 6 N −1);
k
P (u) = (u−q−2)(u−q−4),
1
P (u) = (u−q−7)(u−q−9),
2
P (u) = P (u) = 1,
3 4
P (u) = u−q−8,
5
P (u) = 1, (6 6 k 6 N −1);
k
P (u) = (u−q−2)(u−q−4),
1
P (u) = u−q−7,
2
P (u) = u−q−10,
3
P (u) = u−q−7,
4
P (u) = 1, (5 6 k 6 N −1).
k
9
ThemodulesV(n )andV(n )arenon-zeroonlyifN > 6. HenceV(m )⊗V(m )hasonlytwo
1 3 1 2
composition factors V(n ) and V(n ) for N = 5, and four composition factors V(n ),V(n ),
2 4 1 2
V(n ),V(n )forN > 6.
3 4
4.4 Wenotethatourresultimpliesthefollowing
Theorem7 All composition factors of the tensor product of two evaluation modules of U (sl )
q N
occurwithmultiplicity one.
b
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B. LECLERC: LaboratoiredeMathe´matiquesNicolasOresme,
Universite´deCaen,CampusII,
BldMare´chalJuin,BP5186,14032Caencedex,France
email: [email protected]
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