JMMO Fock space and Geck-Rouquier classification of simple modules for Hecke algebras

0.29 MB·English

by  Nicolas Jacon

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0
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2 JMMO Fock space and Geck-Rouquier classification of

n simple modules for Hecke algebras
a
J
Nicolas Jacon
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1

Abstract. Using Lusztig a-function, M.Geck and R.Rouquier have recently
]
T proved the existence of a canonical set B in natural bijection with the set of
simple modules for Hecke algebras. In this paper, we recall the definition of
R
thissetandwereportrecentresultswhichshowthatthedefinitionofcanonical
h. basic set can be extended to the case of Ariki-Koike algebras. Moreover, we
t giveanexplicitdescriptionofthissetforallHeckealgebrasandforallAriki-
a Koikealgebras.
m
[

1
v
1. Introduction
7
6 Let W be a finite Weyl group with set of simple reflections S ⊂ W, let v be
2 an indeterminate, u = v2 and let A = C[v,v−1]. The Hecke algebra H of W over
1
A is the associative A-algebra with basis {T | w ∈ W} and the multiplication
0 w
5 between two elements of the basis is determined by the following rule. Let s ∈ S
0 and w ∈W, then:
/
h
T if l(sw)>l(w),
at TsTw = sw
m (uTsw+(u−1)Tw if l(sw)<l(w),
: where l is the usual length function of W. Such algebras play an important role,
v
i for example in the representationtheory offinite groupsof Lie type (see [DG] and
X
[Gb]) or in the theory of knots and links (see [GP, Chapter 4]).
r Let K be the field of fractions of A and let θ : A → C be a homomorphism
a
into the field of complex numbers. Let HK := K ⊗A H and let HC := C⊗A H.
On the one hand, the representationtheory of H is relatively well-understood: it
K
is known that this is a split semisimple algebra which is isomorphic to the group
algebraC[W]anditssimplemodulesareinnaturalbijectionwiththesimpleC[W]-
modules. Ontheotherhand,thesimplemodulesofHC aremuchmorecomplicated
to describe because HC is notsemisimple in general. In fact, this problemis linked
to the problem of determining a map d between the Grothendieck group R (H )
θ 0 K
of finitely generated HK-modules and the Grothendieck group R0(HC) of finitely
generatedHC-modules. This mapis called the “decompositionmap” andit relates
the simple HK-modules with the simple HC-modules via a process of modular
reduction:
dθ :R0(HK)→R0(HC).
1
2 NICOLASJACON
The decomposition maps associated to Hecke algebras of exceptional types are
almostallexplicitelyknown,see[Mu],[Ge1],[Ge2]and[GL](infact,forW =E ,
8
we only have an “approximation” of this map, see [Mu]). In this paper, we are
mainly interesting about the classical types that is type A , type B and type
n−1 n
D .
n
Forthesetypes,worksofAriki[Ab],Dipper-James[DJ],Dipper-James-Murphy
[DJM],andHu[H]provideadescriptionofthesimpleHC-modules. Infact,Hecke
algebrasoftypeA andB areparticularcasesofAriki-Koikealgebras(orcyclo-
n−1 n
tomic Hecke algebras of type G(d,1,n), see [AK]). In [A2], Ariki has shown that
thecomputationofthedecompositionmapforthesealgebrascanbeeasilydeduced
fromthe computationofthe Kashiwara-Lusztigcanonicalbasisofirreduciblehigh-
est weight U (sl )-modules. In particular, these results lead to a parametrization
q e
of the simple modules for Ariki-Koike algebras (and so, for Hecke algebras of type
A and B )busing a class of multipartitions which appears in the crystal graph
n−1 n
theoryofU (sl )-modules,namelytheKleshchevmultipartitions. FortypeD ,Hu
q e n
hasobtainedaclassificationofthesimplemodulesbyusingthe factthattheHecke
algebraoftypbeD canbeseenasasubalgebraofaHeckealgebraoftype B (with
n n
unequal parameters). One of the main problem is that, for type B and D , these
n n
results lead to a recursive parametrization of the simple modules.
In [Gk] and in [GR], Geck and Rouquier have given another approch for the
descriptionofthesimpleHC-modules. Theyhaveshowntheexistenceofacanonical
setB ⊂Irr(H )by usingLusztiga-fonctionandKazhdan-Lusztigtheory. Thisset
K
iscalledthe“canonicalbasicset”anditisinnaturalbijectionwithIrr(HC). Hence,
it gives a way to classify the simple HC-modules. Moreover, the existence of the
canonical basic set implies that the matrix associated to d has a lower triangular
θ
shape with 1 along the diagonal.
The description of the canonical basic set is now complete for all finite Weyl
groupsW andforallspecializationsθ. Inthispaper,wereporttheserecentresults
which show that this set can be indexed by another class of multipartitions which
also appears in the crystal graph theory of the U (sl )-modules. The proof also
q e
requiresAriki’stheorembutwhatweobtainhereisanonrecursiveparametrization
of the simple HC-modules. Moreover, we will see bthat all these results can be
extendedtothecaseofAriki-Koikealgebrasevenifwedon’thaveKazhdan-Lusztig
type basis for Ariki-Koike algebras.
2. Representations of semisimple Hecke algebras
LetH beanIwahori-HeckealgebraofafiniteWeylgroupW overA:=C[v,v−1]
asitisdefinedintheintroduction. LetK =C(v)andletH bethecorresponding
K
Hecke algebra. Then A is integrally closed in K and H is a split semisimple
K
algebra. ByTitsdeformationtheorem(see[GP,Theorem8.1.7]),H isisomorphic
K
tothe groupalgebraC[W]. Hence, the simpleH -modules areinnaturalbijection
K
with the simple C[W]-modules. In fact, for type A and type B , the simple
n−1 n
H -modulescanbeexplicitlydescribedbyusingthetheoryofcellularalgebras(see
K
[GrL]) while for type D , the simple H -modules are obtained by using Clifford
n K
theory. We obtain the following parametrizations for the classical types of Weyl
groups:
GECK-ROUQUIER CLASSIFICATION OF SIMPLE MODULES FOR HECKE ALGEBRAS 3
2.1. Type A . Assume that W is a Weyl group of type A .
n−1 n−1
s s s
1 2 n−1
t t ` ` ` t
Let λ be a partition of rankn, then, we can construct a H-module Sλ, free over A
which is called a Specht module (see the construction of “dual Specht modules” in
[Ab, Chapter 13] in a more general setting). Moreover,we have:
Irr(H )={Sλ :=K⊗ Sλ | λ∈Π1},
K K A n
where we denote by Π1 the set of partitions of rank n.
n
2.2. Type B . Assume that W is a Weyl group of type B .
n n
s s s s
1 2 3 n
t t t ` ` ` t
Let (λ(0),λ(1)) be a bi-partition of rank n, then, we can construct a H-module
S(λ(0),λ(1)), free over A which is called a Specht module. Moreover,we have:
Irr(H )={S(λ(0),λ(1)) :=K⊗ S(λ(0),λ(1)) | (λ(0),λ(1))∈Π2},
K K A n
where we denote by Π2 the set of bi-partitions of rank n.
n
2.3. Type D . Assume that W is a Weyl group of type D .
n n
s
Ht1
H
HH(cid:8)st3 ` ` ` stn
(cid:8)
(cid:8)
(cid:8)t
s
2
Then,H canbeseenasasubalgebraofaHeckealgebraH oftypeB withunequal
1 n
parameters (see [Gex] for more details).
Similarytotheequalparametercase,forall(λ(0),λ(1))∈Π2,wecanconstruct
n
a H -module S(λ(0),λ(1)), free over A which is called a Specht module. We have:
1
Irr(H )={S(λ(0),λ(1)) :=K⊗ S(λ(0),λ(1)) | (λ(0),λ(1))∈Π2}.
1,K K A n
We have an operation of restriction Res between the set of H -modules and the
1,K
set of H -modules, for (λ(0),λ(1))∈Π2:
K n
• if λ(0) 6= λ(1), we have Res(S(λ(0),λ(1))) ≃ Res(S(λ(1),λ(0))) and the H -
K K K
module V[λ(0),λ(1)] :=Res(S(λ(0),λ(1))) is a simple H -module.
K K
• if λ(0) = λ(1), we have Res(S(λ(0),λ(1))) = V[λ(0),+] ⊕ V[λ(0),−] where
K
V[λ(0),+] and V[λ(0),−] are non isomorphic simple H -modules.
K
Moreover,we have:
Irr(H )= V[λ,µ] | λ6=µ, (λ,µ)∈Π2 V[λ,±] | λ∈Π1 .
K n n
2
n o[n o
4 NICOLASJACON
Now,weturntotheproblemofdeterminingaclassificationofthesetofsimple
modules for Hecke algebras in the case where v is no longer an indeterminate but
a complex number.
3. Modular representations and canonical basic sets for Hecke algebras
Let θ :A→C be a ring homomorphism. We put:
f
O := | f,g ∈C[v], g(θ(v))6=0 .
g
(cid:26) (cid:27)
O is a discrete valuation ring and we have A ⊂ O. By [GP, Theorem 7.4.3], we
obtain a well-defined decomposition map
dθ :R0(HK)→R0(HC).
whereR0(HK)(resp. R0(HC))istheGrothendieckgroupoffinitelygeneratedHK-
modules (resp. HC-modules). This is defined as follows: let V be a simple HK-
module. Then,by[GP, §7.4],thereexistsaH -moduleV suchthatK⊗ V =V.
O O
By reducing V modulo the maximal ideal of O, m := (v −θ(v))O, we obtain a
HC-module C⊗OV . Then, we put: b b
b
d ([V])=[C⊗ V].
b θ O
d is well-defined and for V ∈ Irr(H ), there exist numbers (d ) such
θ K b V,M M∈Irr(HC)
that:
d ([V])= d [M].
θ V,M
M∈XIrr(HC)
The matrix (d ) is called the decomposition matrix. For more details
V,M V∈Irr(HK)
M∈Irr(HC)
about the construction of decomposition maps, even in a more general setting, see
[Gb].
Now, we will recall results of Geck and Rouquier which show that the decom-
position map has always a unitriangular shape with one along the diagonal.
Let {C } be the Kazhdan-Lusztig basis of H. For x,y ∈W, the multipli-
w w∈W
cation between two elements of this basis is given by:
C C = h C
x y x,y,z z
z∈W
X
where h ∈ A for all z ∈ W. For any z ∈ W, there is a well-defined integer
x,y,z
a(z)≥0 such that
va(z)h ∈Z[v] for all x,y ∈W,
x,y,z
va(z)−1h ∈/ Z[v] for some x,y ∈W.
x,y,z
We obtain a function which is called the Lusztig a-function:
a: W → N
z 7→ a(z)
GECK-ROUQUIER CLASSIFICATION OF SIMPLE MODULES FOR HECKE ALGEBRAS 5
Now, following [L1, Lemma 1.9], to any M ∈ Irr(HC), we can attach an a-value
a(M) by the requirement that:
C .M =0 for all w∈W with a(w)>a(M),
w
C .M 6=0 for some w ∈W with a(w)=a(M).
w
Wecanalsoattachana-valuea(V)toanyV ∈Irr(H ),inananalogousway. Note
K
that there is an equivalent definition of the a-value of a simple H -module using
K
the fact that Hecke algebras are symmetric algebras, this will be important in the
context of Ariki-Koike algebras where we don’t have Kazhdan-Lusztig theory. Let
τ : H → A be the symmetrizing trace of H (see [GP, §7.1]) which is defined by
K
τ(T ) = 0 if w 6= 1 and τ(T ) = 1. Then, for each V ∈ Irr(H ), there exists a
w 1 K
Laurent polynomial s ∈A such that:
V
1
τ = χ ,
V
s
V
V∈IXrr(HK)
where χ is the character of V ∈ Irr(H ). s is called the Schur element of the
V K V
simple H -module V. Then, Lusztig ([L2]) has shown that we have:
K
1
a(V)= min{l∈Z | vls ∈Z[v]}.
V
2
Wecannowgivethetheoremofexistenceofthecanonicalbasicset. Themaintool
of the proof is the Lusztig asymptotic algebra.
Theorem 3.1 (Geck [Gk], Geck-R.Rouquier [GR]). We define the following
subset of Irr(H ):
K
B :={V ∈Irr(HK) | dV,M 6=0 and a(V)=a(M) for some M ∈Irr(HC)}.
Then there exists a unique bijection
Irr(HC) → B
M 7→ V
M
such that the following two conditions holds:
(1) For all V ∈B, we have d =1 and a(V )=a(M).
M VM,M M
(2) If V ∈ Irr(HK) and M ∈ Irr(HC) are such that dV,M 6= 0, then we have
a(M)≤a(V), with equality only for V =V .
M
The set B is called the canonical basic set with respect to the specialization θ.
Note thata descriptionofthe setB wouldleadto anaturalparametrizationof
the set of simple HC-modules. If HC is semisimple, we know by Tits deformation
theorem that the decomposition matrix is just the identity. Hence, we obtain the
following result.
Proposition 3.2. Assumethat θ is such that HC is a split semisimple algebra.
Then, we have:
B =Irr(H ).
K
Now, We want to give an explicit description of B in the non semisimple case.
By [GP, Theorem 7.4.7],HC is semisimple unless θ(u) is a root of unity. Thus, we
can restrict ourselves to the case where θ(u) is a root of unity.
The idea is to use results of Ariki which give an interpretation of the decom-
position map using the theory of canonical basis of Fock spaces. This theory can
6 NICOLASJACON
not be applied to all Hecke algebras. In fact, this is concerned with the class of
Ariki-KoikealgebraswhichcontainsHeckealgebrasoftypeA andB asspecial
n−1 n
cases.
4. Ariki-Koike algebras
a)First,werecallthedefinitionofAriki-Koikealgebras(see[Ma]foracomplete
surveyoftherepresentationtheoryofthesealgebras). LetRbeacommutativering,
let d∈N , n∈N and let v, u , u ,..., u be d+1 parameters in R. The Ariki-
>0 0 1 d−1
Koike algebra H := H (v;u ,...,u ) (or cyclotomic Hecke algebra of type
R,n R,n 0 d−1
G(d,1,n)) over R is the unital associative R-algebra with presentation by:
• generators: T , T ,..., T ,
0 1 n−1
• braid relations symbolised by the following diagram:
T T T T
0 1 2 n−1
t t t ` ` ` t
and the following ones:
(T −u )(T −u )...(T −u )=0,
0 0 0 1 0 d−1
(T −v)(T +1)=0 (i≥1).
i i
These relations are obtained by deforming the relations of the wreath product
(Z/dZ)≀S . We have the following special cases:
n
• if d=1, H is the Hecke algebra of type A over R,
R,n n−1
• if d=2, H is the Hecke algebra of type B over R.
R,n n
It is known that the simple modules of (Z/dZ)≀S are indexed by the d-tuples of
n
partitions. The same is true for the semisimple Ariki-Koike algebras defined over
a field. We say that λ is a d-partition of rank n if:
• λ = (λ(0),..,λ(d−1)) where, for i = 0,...,d−1, λ(i) = (λ(i),...,λ(i)) is a
1 ri
partition of rank |λ(i)| such that λ(i) ≥...≥λ(i) >0,
1 ri
d−1
• |λ(k)|=n.
k=0
X
We denote by Πd the set of d-partitions of rank n.
n
For each d-partition λ of rank n, we can associate a H -module Sλ which is
R,n R
free over R. This is called a Specht module1. These modules generalize the Specht
modules previously defined for Hecke algebras of type A and B .
n−1 n
AssumethatRisafield. Then,foreachd-partitionofrankn,thereisanatural
bilinearformwhichisdefinedovereachSλ. Wedenotebyradtheradicalassociated
R
to this bilinear form. Then, the non zero Dλ := Sλ/rad(Sλ) form a complete set
R R R
of non-isomorphic simple H -modules (see for example [Ab, chapter 13]). In
R,n
particular, if H is semisimple, we have rad(Sλ) = 0 for all λ ∈ Πd and the
R,n R n
set of simple modules are given by the Sλ. We have the following criterion of
R
semisimplicity:
Theorem 4.1 (Ariki [A1]). H is split semisimple if and only if we have:
R,n
1Here,weusethedefinitionoftheclassicalSpechtmodules. ThepassagefromclassicalSpecht
modulestotheirdualsisprovidedbythemap(λ(0),λ(1),...,λ(d−1))7→(λ(d−1)′,λ(d−2)′,...,λ(0)′)
where,fori=0,...,d−1,λ(i)′ denotes theconjugate partition.
GECK-ROUQUIER CLASSIFICATION OF SIMPLE MODULES FOR HECKE ALGEBRAS 7
• for all i6=j and for all l ∈Z such that |l|<n, we have:
vlu 6=u ,
i j
n
• (1+v+...+vi−1)6=0.
i=1
Y
Assume that R is a field of characteristic 0, using results of Dipper-Mathas
[DM],Ariki-Mathas[AM]andMathas[Ms],thecasewhereH isnotsemisimple
R,n
can be reduced to the case where R = C and where all the u are powers of v. In
i
this paper, we will mostly concentrate upon the case where v is a primitive root of
unity of order e.
b) Let η := exp(2iπ) and let v , v ,..., v be integers such that 0 ≤ v ≤
e e 0 1 d−1 0
...≤vd−1 <e. WeconsidertheAriki-KoikealgebraHC,n overCwiththe following
choice of parameters:
u =ηvj for j =0,...,d−1,
j e
v =η .
e
Thisalgebraisnotsemisimpleingeneral. HencethesimpleHC,n-modulesaregiven
by the non zero Dλ. We denote:
C
Γn :={λ∈Πd | Dλ 6=0}.
0 n C
Now,we wish to describe the notionof decompositionmapin the contextof Ariki-
Koike algebras.
Let y be anindeterminate and letv:=yd. Let A=C[y,y−1]. We assume that
we have d invertible elements u ,...,u in A such that we have for all i 6= j and
0 d−1
l∈Z with |l|≤n:
vlu 6=u
i j
We considerthe Ariki-KoikealgebraH with the following choiceof parameters:
A,n
u =u for j =0,...,d−1,
j j
v =v.
Let K be the field of fractions of A and let H := K ⊗ H . By the
K,n A A,n
above criterion of semisimplicity, H is semisimple and its simple modules are
K,n
given by the Specht modules Sλ defined over K. Now, let θ : A → C such that
K
θ(u)=η :=exp(2iπ). Assume that we have θ(u )=ηvj for j =0,...,d−1. Then,
e e j e
we have HC,n =C⊗AHA,n and the decomposition map is defined as follows:
dθ : R0(HK,n) → R0(HC,n)
[SKλ] 7→ [SCλ]= dSKλ,DCµ[DCµ]
µ∈Γn
X0
We will now explain the connections between this decomposition map and the
theory of Fock spaces.
5. Quantum groups and Fock spaces
The aim of this part is to introduce Ariki’s theorem which provides a way to
compute the decomposition maps for Ariki-Koike algebras. For details, we refer to
[Ab]. We keep the notations of the previous section.
8 NICOLASJACON
5.1. Quantum group of type A(1) . Let h be a free Z-module with basis
e−1
{h ,d | 0≤i<e} and let {Λ ,δ | 0≤i<e} be the dual basis with respect to the
i i
pairing:
h , i:h∗×h→Z
such that hΛ ,h i = δ , hδ,di = 1 and hΛ ,di = hδ,h i = 0 for 0 ≤ i,j < e. For
i j ij i j
0≤i<e, we define the simple roots of h∗ by:
2Λ −Λ −Λ +δ if i=0,
α = 0 e−1 1
i 2Λ −Λ −Λ if i>0,
i i−1 i+1
(cid:26)
where Λ :=Λ . The Λ are called the fundamental weights.
e 0 i
Letq beanindeterminateandletU bethequantumgroupoftypeA(1) . This
q e−1
is a unitalassociativealgebraoverC(q) which is generatedby elements {e ,f |i∈
i i
{0,...,e−1}} and {k | h∈h} (see for example [Ab] for the relations).
h
For j ∈N and l∈N, we define:
qj −q−j
• [j] := ,
q q−q−1
• [j]! :=[1] [2] ...[j] ,
q q q q
l [l]!
q
• = .
j [j]![l−j]!
(cid:20) (cid:21)q q q
Let A = Z[q,q−1]. We consider the Kostant-Lusztig A-form of U which is
q
denoted by U : this is a A-subalgebra of U generated by the divided powers
A q
el fl
e(il) := [l]i! , fj(l) := [l]j! for 0 ≤ i,j < e and l ∈ N and by khi, kd, kh−i1, kd−1 for
q q
0 ≤ i < e. Now, if S is a ring and u an invertible element in S, we can form the
specialized algebra U :=S⊗ U by specializing the indeterminate q to u∈S.
S,u A A
For any n≥0, let F be the C(q) vector space with basis consisting of all the
n
d-partitions of rank n. The Fock space is the direct sum:
F :=⊕n∈NFn.
We will now see that this space can be endowed with two different structures of
U -module. To describe these actions, we need some combinatorial definitions.
q
Let λ = (λ(0),...,λ(d−1)) be a d-partition of rank n. The diagram of λ is the
following set:
[λ]= (a,b,c) | 0≤c≤d−1, 1≤b≤λ(c) .
a
n o
The elements of this diagram are called the nodes of λ. Let γ = (a,b,c) be a
node of λ. The residue of γ associated to the set {e;v ,...,v } is the element of
0 d−1
Z/eZ defined by:
res(γ)=(b−a+v )(mod e).
c
If γ is a node with residue i, we say that γ is an i-node. Let λ and µ be two
d-partitions of rank n and n+1 such that [λ] ⊂ [µ]. There exists a node γ such
that[µ]=[λ]∪{γ}. Then, we denote [µ]/[λ]=γ. Ifres(γ)=i, we saythat γ is an
addable i-node for λ and a removable i-node for µ.
GECK-ROUQUIER CLASSIFICATION OF SIMPLE MODULES FOR HECKE ALGEBRAS 9
5.2. Hayachi realization of Fock spaces. In this part, we consider the
following order on the set of removable and addable nodes of a d-partition: we say
that γ =(a,b,c) is below γ′ =(a′,b′,c′) if c<c′ or if c=c′ and a<a′.
This order will be called the AM-order and the notion of normal nodes and
good nodes below are linked with this order (in the next paragraph, we will give
another order on the set of nodes which is distinct from this one).
Let λ and µ be two d-partitions of rank n and n+1 such that there exists an
i-node γ such that [µ]=[λ]∪{γ}. We define:
Na(λ,µ)=♯{addable i−nodes of λ above γ}
i
−♯{removable i−nodes of µ above γ},
Nb(λ,µ)=♯{addable i−nodes of λ below γ}
i
−♯{removable i−nodes of µ below γ},
N (λ)=♯{addable i−nodes of λ}
i
−♯{removable i−nodes of λ},
Nd(λ)=♯{0−nodes of λ}.
Theorem 5.1 (Hayashi [Ha]). F becomes a U -module with respect to the
q
following action:
eiλ= q−Nia(µ,λ)µ, fiλ= qNib(λ,µ)µ,
res([λX]/[µ])=i res([µX]/[λ])=i
khiλ=qNi(λ)λ, kdλ=q−Nd(λ)λ,
where i=0,...,e−1.
Let M be the U -submodule of F generated by the empty d-partition. It is
q
isomorphictoanintegrablehighestweightmodule. In[K]and[L3],Kashiwaraand
Lusztighaveindependantly shownthe existence ofaremarkablebasisforthis class
of modules: the canonical basis. We will see the links between the canonical basis
of M and the decomposition map for HC,n. First, it is known that the elements of
this basis are labeled by the vertices of a certain graph called the crystal graph.
BasedonMisraandMiwa’sresult,ArikiandMathasobservedthatthevertices
ofthisgrapharegivenbythesetofKleshchevd-partitionswhichwewillnowdefine.
Let λ be a d-partition andlet γ be ani-node,we saythat γ is a normali-node
of λ if, whenever η is an i-node of λ below γ, there are more removable i-nodes
between η and γ than addable i-nodes between η and γ. If γ is the highest normal
i-node of λ, we say that γ is a good i-node.
We can now define the notion of Kleshchev d-partitions associated to the set
{e;v ,...,v }:
0 d−1
Definition 5.2. The Kleshchevd-partitionsaredefined recursivelyasfollows.
• The empty partition ∅:=(∅,∅,...,∅) is Kleshchev.
• If λ is Kleshchev, there exist i ∈ {0,...,e−1} and a good i-node γ such
that if we remove γ from λ, the resulting d-partition is Kleshchev.
We denote by Λ0,n the set of Kleshchev d-partitions associated to the
{e;v0,...,vl−1}
set {e;v ,...,v }. If there is no ambiguity concerning {e;v ,...,v }, we denote
0 d−1 0 d−1
it by Λ0,n. Now, the crystal graph of M is given by:
10 NICOLASJACON
• vertices: the Kleshchev d-partitions,
i
• edges: λ→µ if and only if [µ]/[λ] is a good i-node.
Thus, the canonical basis B of M is labeled by the Kleshchev d-partitions:
B={G(λ) | λ∈Λ0 , n∈N}.
{e;v0,...,vd−1}
This set is a basis of the U -module M generated by the empty d-partition
A A
and for any specialization of q into an invertible element u of a field R, we obtain
a basis of the specialized module M by specializing the set B.
R,u
By the characterization of the canonical basis, for each λ ∈ Λ0,n, there exist
polynomials d (q) ∈ Z[q] and a unique element G(λ) of the canonical basis such
µ,λ
that:
G(λ)= d (q)µ and G(λ)=λ (mod q).
µ,λ
µX∈Πdn
Now, we have the following theorem of Ariki which shows that the problem of
computing the decomposition numbers of H can be translated to that of com-
R,n
puting the canonical basis of M. This theorem was first conjectured by Lascoux,
Leclerc and Thibon ([LLT]) in the case of Hecke algebras of type A .
n−1
Theorem 5.3 (Ariki [A2]). There exist a bijection j : Λ0,n → Γn such that
0 0
for all µ∈Πd and λ∈Λ0,n, we have:
n
d (1)=d ,
µ,λ SKµ,DCj0(λ)
where we recall that:
Γn :={λ∈Πd | Dλ 6=0}.
0 n C
Moreover,we have:
Theorem 5.4 (Ariki [A3], Ariki-Mathas [AM]). We have Γn = Λ0,n and
0
j =Id.
0
Hence the above theorem gives a first classification of the simple modules by
thesetofKleshchevd-partitions. As notedintheintroduction,the problemofthis
parametrization of the simple HC,n-modules is that we only know a recursive de-
scriptionofthe Kleshchevd-partitions. We nowdealwith anotherparametrization
of this set found by Foda et al. which uses almost the same objects as Ariki and
Mathas.
5.3. JMMO realization of Fock space. This action has been defined in
[JMMO]andhasbeenusedandstudiedin[FLOTW]. Weneedtodefineanother
order on the set of nodes of a d-partitions.
Here, we say that γ =(a,b,c) is above γ′ =(a′,b′,c′) if:
b−a+v <b′−a′+v or if b−a+v =b′−a′+v and c>c′.
c c′ c c′
This order will be called the FLOTW order and it allows us to define functions
Na(λ,µ) and Nb(λ,µ) given by the same way as Na(λ,µ) et Nb(λ,µ) for the AM
i i i i
order.
Now, we have the following result: