Modular representations of cyclotomic Hecke algebras of type G(r,p,n)

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Modular representations of cyclotomic Hecke algebras of type G(r,p,n)

Gwenaelle Genet, Nicolas Jacon

To cite this version:

Gwenaelle Genet, Nicolas Jacon. Modular representations of cyclotomic Hecke algebras of type G(r,p,n). International Mathematics Research Notices, Oxford University Press (OUP), 2006, pp.Art.

ID 93049. �hal-00002867�

ccsd-00002867, version 1 - 17 Sep 2004

Modular representations of cyclotomic Hecke algebras of type G ( r, p, n )

Gwena¨ elle Genet

and Nicolas Jacon

Abstract

We give a classification of the simple modules for the cyclotomic Hecke algebras overCin the modular case. We use the unitriangular shape of the decomposition matrices of Ariki-Koike algebras and Clifford theory.

1 Introduction

Letr, pand nbe integers such thatpdividesr. The complex reflection group of typeG(r, p, n) is defined to be the groups ofn×npermutation matrices such that the entries are either 0 orr^th roots of unity and the product of the nonzero entries is a ^r_p-root of unity.

In [2] and [5], Ariki and Brou´e-Malle have defined a Hecke algebra associated to each complex reflection groupG(r, p, n). It can be seen as a deformation of the algebra of the group G(r, p, n). According to a conjecture of Brou´e and Malle, such algebras, called cyclotomic Hecke algebras, should occur as endo- morphism algebras of the Lusztig induced character. The representation theory of cyclotomic Hecke algebras of typeG(r,1, n) (also known as Ariki-Koike alge- bras) is beginning to be well understood. They are cellular algebras, the simple modules have been classified in both semi-simple and modular cases and the decomposition matrices are known in characteristic 0 (see [21] for a survey of these results).

In [2], Ariki has shown that a Hecke algebra of typeG(r, p, n) can be consid- ered as the 0-component of a graded system for a Hecke algebra of typeG(r,1, n) with a special choice of parameters. As a consequence, in the semi-simple case, he has given a complete set of non isomorphic simple modules by using Clifford theory. In the modular case, partial results have been obtained by Hu in [17]

(see also [16] and [18] where the case r =p = 2 which corresponds to Hecke algebras of type Dn is studied). The main problem is that the restrictions of the simple modules for non semi-simple Ariki-Koike algebras are much more complicated to describe than for semi-simple ones.

The purpose of this paper is precisely to give a parametrization of the simple modules for the non semi-simple cyclotomic Hecke algebras of type G(r, p, n)

∗Address: UFR Math´ematiques - Case 7012 - Universit´e Denis Diderot - Paris 7 - 75251 PARIS Cedex 05, France. E-mail address: genet@math.jussieu.fr

†Address: Universit´e de Caen, UFR des Sciences D´epartement de Math´ematiques Laboratoire LMNO, Campus II 14032 Caen cedex, France. E-mail address: ja- con@math.unicaen.fr

over C. It comes from the “canonical basic set” introduced by Geck and Rouquier in [13] and extended by the second author in [20]. This set induces a parametrization of the simple modules of non semi-simple Ariki-Koike alge- bras by some FLOTW multipartitions (this kind of multipartitions has been defined by Foda, Leclerc, Okado, Thibon and Welsh in [9]). We proceed in an analogous way than in [14, Theorem 2.1], using the unitriangular shape of the decomposition matrices of Ariki-Koike algebras and Clifford theory.

2 Cyclotomic Hecke algebras, Clifford theory

2.A Cyclotomic Hecke algebras

Letr, pandnbe integers such that pdividesr. We denote d:= r

p. LetRbe an integral ring,qa unit ofRand a sequencex= (x1, . . . , xd) of elements inR.

Denote H^q,x_r,p,n(R), the Hecke R-algebra associated to the complex reflection group G(r, p, n), for the parameters q, x, defined by the following presenta- tion:

generators: a0, . . . , an,

relations: (a0−x1). . .(a0−xd) = 0 (ai−q)(ai+ 1) = 0, 1≤i≤n, a1a3a1=a3a1a3,

aiai+1ai=ai+1aiai+1, 2≤i≤n−1, (a1a2a3)²= (a3a1a2)²,

a1ai=aia1, 4≤i≤n,

aiaj=ajai, 2≤i < j ≤n, j≥i+ 2, a0a1a2= (q⁻¹a1a2)^2−ra2a0a1+ (q−1)

r−2

X

i=1

(q⁻¹a1a2)^1−ka0a1,

=a1a2a0, a0ai=aia0, 3≤i≤n.

We have the following special cases:

• ifr=p= 1,H^q,x_r,p,n(R) is the Hecke algebra of typeAn−1,

• ifr= 2 and p= 1,H^q,x_r,p,n(R) is the Hecke algebra of typeBn,

• ifr=p= 2,H^q,x_r,p,n(R) is the Hecke algebra of typeDn,

• ifp= 1,H^q,x_r,p,n(R) is the Ariki-Koike algebra defined in [4].

Suppose the ringR contains ap^th root of unity ηp and ap^th root yi ofxi, for each i ∈ [1, d]. We define a sequence Q = (Q1, . . . , Qr) of elements in R from the sequence x: for j ∈ [1, r], j = sp+t with s ∈ [0, d−1] and t ∈ [1, p], let Qj := ys+1η_p^t−1. Then, H^q,Q_r,1,n(R) is the R-algebra defined by

generators: T1, . . . , Tn,

relations: (T₁^p−x1). . .(T₁^p−xd) = 0, (Ti−q)(Ti+ 1) = 0, 2≤i≤n, T1T2T1T2=T2T1T2T1,

TiTi+1Ti=Ti+1TiTi+1, 2≤i≤n−1, TiTj=TjTi, 1≤i < j≤n, j≥i+ 2.

We use [2, Proposition 1.6] and identifya0 with T₁^p, a1 withT₁⁻¹T2T1 and ai with Ti, for i ∈ [2, n], such that H^q,x_r,p,n(R) is considered as a subalgebra of H^q,Q_r,1,n(R). Then by [15, §4.1], the Ariki-Koike algebraH^q,Q_r,1,n(R) is graded over the cyclotomic Hecke algebraH^q,x_r,p,n(R) by a cyclic group of orderp:

H^q,Q_r,1,n(R) =

p−1

M

i=0

T₁ⁱH^q,x_r,p,n(R).

Put H:=H^q,Q_r,1,n(R) andH^′ :=H^q,x_r,p,n(R). We can define an automorphism g_H^H′ ofH^′ by

g^H_H′(h) :=T₁⁻¹hT1, for allh∈H^′.

We also have an automorphismf_H^H′ ofHwhich is defined on the grading by fH^H′(T1jh) =η^j_pT₁^jh, forj∈[0, p−1], h∈H^′.

The next theorem of Dipper and Mathas will be very useful in the following of this paper.

Theorem 2.1 ([8]) With the previous notations, assume that we have a parti- tion of Q:

Q=Q¹a Q²a

. . .a Q^s such that

fΓ(q, Q) = Y

1≤α<β≤s

Y

Qi∈Qα Qj∈Qβ

Y

−n<a<n

(q^aQi−Qj)

is a unit of R. Then, H^q,Q_r,1,n(R)is Morita-equivalent to the following algebra:

M

n1,...,ns>0 n1+...+ns=n

H^q,Q_r ¹

1,1,n1(R)⊗. . .⊗H^q,Q_r ^s

s,1,ns(R) where for i∈[1, s],ri=|Qⁱ|.

2.B Clifford theory

Lety1,y2,...,yd andv be indeterminates overC. Fori∈[1, d], put xi:=y^p_i

and fori∈[1, r], i=sp+t withs∈[0, d−1],t∈[1, p], ηp:= exp(^2iπ_p ), Qi:=η^t−1_p ys+1.

Form the sequencesx:= (x1, . . . ,xd) andQ:= (Q1, . . . ,Qr).

Let A := C[x1,x⁻¹₁ , . . . ,xd,x⁻¹_d ,v,v⁻¹], K := C(x1, . . . ,xd,v) its field of fractions.

Letθ:A→Cbe a ring homomorphism such thatCis the field of fractions ofθ(A).

We put fori∈[1, d],

xi:=θ(xi), fori∈[1, r],

Qi:=θ(Qi) and

v:=θ(v).

Note that for i ∈ [1, r], i = sp+t with s ∈ [0, d−1], t ∈ [1, p], then Qi:=η_p^t−1ys+1, for a complex number ys+1 such thaty_s+1^p =xs+1.

We also form the sequencesx:= (x1, . . . , xd) andQ:= (Q1, . . . , Qr).

We will note for shortH, resp. H, for the Ariki-Koike algebrasH^v,Q_r,1,n(K), resp. H^v,Q_r,1,n(C) andH^′, resp. H^′, for the cyclotomic Hecke algebrasH^v,x_r,p,n(K), resp. H^v,x_r,p,n(C).

Since H, resp. H, is graded overH^′, resp. H^′, we will give some results which comes from Clifford theory.

Let H be the Ariki-Koike algebra H, resp. H, and H^′ be the cyclotomic Hecke algebra H^′, resp. H^′. In both cases, we will denote by Res the functor of restriction from the category of the H-modules to the category of the H^′- modules and by Ind the functor H⊗H^′−from the category of theH^′-modules to the category of theH-modules.

LetV be aH-simple module andU be a simple submodule of ResV. Put of,V := min{k∈N_>0|^fkV ≃V}

where f := f_H^H′ and for i ∈ [0, of,V −1], ^fiV is the H-module with the same underlying space as V and the structure of module is given by composing the original action ofHwithfⁱ.

In the same way, let

og,U:= min{k∈N_>0| ^gkU ≃U}

where g :=g^H_H′ and for i ∈ [0, og,U −1], ^giU is the H^′-module with the same underlying space asU with the action ofH^′ twisted bygⁱ.

Denote [X :Y] the multiplicity of a simple moduleY in the semi-simple mod- uleX.

Lemma 2.2 Denote for shortg:=gH^H′ andf :=fH^H′. Every simpleH^′-moduleU appears as a direct summand of the restriction ResV of a simple H-module V with multiplicity-free. With such modules U and V, a conjugate ^giU of U, i ∈ [0, og,U −1], occurs in the restriction of the conjugate ^fjV of V, for all

j ∈[0, of,V −1]and the conjugates of V are the only simple H-modules whose restrictions contain such a ^giU. Besides

ResV ≃

of,V−1

M

i=0

fⁱV, (1)

IndU ≃

og,U−1

M

i=0

gⁱU, (2)

p = og,Uof,V. (3)

Proof: It is easy to see that

ResV ≃[ResV :U]

og,U−1

M

i=0

gⁱU, (4)

(see [6, Theorem 11.1]). Using [14, Proposition 2.2], we get

IndU ≃[IndU :V]

of,V−1

M

i=0 fⁱV

as well as p

of,Vog,U

= [ResV :U][IndU :V]. (5) As ResV is g-stable, [6, Proposition 11.14] implies that EndH(Ind ResV) is graded over EndH^′(ResV) by a cyclic group of orderp. By (4) and (5), the di- mension of EndH(Ind ResV) overCis _o^p_f,V² and by the grading over EndH^′(ResV), this dimension is equal top[ResV :U]²og,U. By comparison with (5), it follows that

[ResV :U] = [IndU :V]. (6)

Now, by definition of of,V, for shorto, there exists anH-automorphism t:V →^foV.

So, t^po is anH-isomorphism. By Schur’s lemma,t^po is a scalar. There exists a complex numberc such that (_c^t)^po = IdV. Then

ResV ≃

p o−1

M

i=0

Ker(t

c−η^o_pⁱIdV).

It is easy to see that for each i ∈[0,^p_o−1], Ker(_c^t−η_p^oiIdV) is an H^′-module isomorphic to ^giKer(^t_c −IdV). So the semi-simple module ResV decomposes into more than ^p_o simple modules and by (5) by less than ^p_o simple modules.

Therefore ResV decomposes into exactly ^p_o simple modules and [IndU :V] = 1 so [ResV :U] = 1 by (6).

As the module induced from a simpleH^′-module toHis semi-simple, the asser- tions of the Lemma are proved.

2.C The semi-simple case

We keep the notations of the previous subsection.

We study the representation theory of the algebras H := H^v,Q_r,1,n(K) and H^′:=H^v,x_r,p,n(K). By [1, Main Theorem],His a split semi-simple algebra. This implies thatH^′ is also split semi-simple.

A complete set of non isomorphic modules forHhas been given in [4] and [7].

Let Π^r_n be the set of r-partitions of n. We say that λ = λ(i)

i∈[1,r] is a r- partition of sizenif for everyi,λ(i) is a partition and Σ^r_i=1|λ(i)|=n.

For each r-partition λ ∈ Π^r_n, we can associate an H^v,Q_r,1,n(A)-module S^v,Q(λ) called a “Specht module” that is free over A. Here, we use the definition of

“classical” Specht module contrary to [7] where the dual Specht modules are defined. We have the following theorem.

Theorem 2.3 ([4],[7]) The following set is a complete set of non isomorphic absolutely irreducible H^v,Q_r,1,n(K)-modules:

nS_K^v,Q(λ) :=K⊗AS^v,Q(λ)| λ∈Π^r_no .

For eachλ∈ Π^r_n, the set of standard tableaux of shape λis a basis of the underlyingC-vector space ofS_K^v,Q(λ) (see [4]). Using [4, Propositions 3.16, 3.17]

and some combinatorial properties, it is easy to get the

Proposition 2.4 Letλ∈Π^r_n. The map

h^v,Q_λ :^fHH′S_K^v,Q(λ)→S_K^v,Q ̟(λ)

(7) that sends a standard λ-tableau T = (T_i)i∈[1,r] to the standard ̟(λ)-tableau

̟(T) := (T_̟−1(i))i∈[1,r]is anH-isomorphism where̟is the permutation ofS_r defined byQ̟(i):=ηpQi,for alli∈[1, r]and̟(λ) = (λ(̟⁻¹(1)), . . . , λ(̟⁻¹(r))).

Define

oλ:= min{k∈N_>0|̟^k(λ) =λ}.

It is clear that oλ=o_f,S^v,Q

K (λ)and ResS_K^v,Q(λ)≃

p oλ−1

M

i=0

Ker(h^o_λ^λ −η^o_p^λiId_S^v,Q

K (λ)), whereh^o_λ^λ:^foλS_K^v,Q(λ)→S_K^v,Q λ

sends a standardλ-tableauTto the standard λ-tableau̟^oλ(T). Fori∈[0,_o^p

λ −1], put

S_K^v,Q(λ, i) := Ker(h^o_λ^λ−η^o_p^λiId_S^v,Q

K (λ)). (8)

LetL be a set of representatives ofr-partitions for the action of the cyclic group generated by̟ over Π^r_n. It follows from Lemma 2.2, the

Proposition 2.5 The set

{S_K^v,Q(λ, i) ; i∈[0, p oλ

−1], λ∈ L}

is a complete set of non isomorphic simple H^′-modules.

IfTis a standardλ-tableau, then forj∈[0,_o^p

λ −1], the element (T, j) :=

p oλ−1

X

i=0

η_p^−oλij̟^oλi(T) belongs to the eigenspace Ker(h^o_λ^λ−η^o_p^λjId_S^v,Q

K (λ)). It comes from the fact that T = oλ

p

p oλ−1

X

j=0

(T, j), that for every j ∈ [0,_o^p

λ −1], the eigenspace Ker(h^o_λ^λ − η^o_p^λjId_S^v,Q

K (λ)) is generated by the (T, j), Ta standard λ-tableau. So we have just described the simpleH^′-modules, see also [2].

The following part is now concerned with the non semi-simple case: simple H^q,Q_r,1,n(C)-modules andH^q,x_r,p,n(C)-modules.

2.D The non semi-simple case

We keep the notations of§2.B.

2.D.1 Decomposition maps

The aim of this part is to define the decomposition maps for H := H^q,Q_r,1,n(C) and H^′ := H^q,x_r,p,n(C). Recall that A := C[x1,x⁻¹₁ , . . . ,xd,x⁻¹_d ,v,v⁻¹], K :=

C(x1, . . . ,xd,v) and thatθ:A→Cis a ring homomorphism.

Using [10], there exists a discrete valuation ringOwith maximal idealI(O) such that A ⊂ O and I(O)∩A = ker(θ). Let k0 := O/I(O) be the residue field of O. Denote by : O →k0 the canonical map. We obtain the algebras H^q,Q_r,1,n(k0) andH^q,x_r,p,n(k0). The fieldk0 can be considered as an extension of C that is the quotient field of θ(A). By [12, Lemma 7.3.4], there is an isomor- phism between the Grothendieck groups of finitely generated H-modules and H^q,Q_r,1,n(k0)-modules (resp. H^′-modules andH^q,x_r,p,n(k0)-modules):

R0(H)≃R0(H^q,Q_r,1,n(k0)) and R0(H^′)≃R0(H^q,x_r,p,n(k0)).

As a consequence, we obtain well-defined decomposition maps by choosingO- forms for the simpleH^q,Q_r,1,n(K)-modules (resp. H^q,x_r,p,n(K)-modules) and reducing them moduloI(O):

d^r,1,n:R0(H)→R0(H),

d^r,p,n:R0(H^′)→R0(H^′).

For a simpleH-moduleV andW a simpleH^′-module, there exist non neg- ative integers d⁽¹⁾_V,M with M a simple H-module and d^(p)_W,N with N a simple H^′-module such that:

d^r,1,n([V]) = X

M∈Irr(H)

d⁽¹⁾_V,M[M]

and d^r,p,n([W]) = X

N∈Irr(H^′)

d^(p)_W,N[N].

By using the same argument as in [11, Lemma 5.2] (see also [12, Theorem 7.4.3.c]), we obtain the following result.

Proposition 2.6 The following diagram commutes:

R0(H) −−−−→^Res R0(H^′)

d^r,1,n



 y



 y^d

r,p,n

R0(H) −−−−→^Res R0(H^′)

Moreover, for any simple H-moduleV and any simpleH-moduleM, we have d⁽¹⁾_V,M =d⁽¹⁾_fH

H′V,^{f HH′}M

,

and for any simpleH^′-moduleW and any simpleH^′-moduleN, we have:

d^(p)_W,N =d^(p)_gH H′W,^gHH′N

.

In the following, we will see that under some additionnal hypotheses, the above property leads to some interesting results about the decomposition map of cyclotomic Hecke algebras of typeG(r, p, n) and about the simpleH^′-modules.

2.D.2 Simple modules of non semi-simple Ariki-Koike algebras We will work under the following hypothesis. We assume thatv is a primitive root of unity,

v:=ηe= exp(2iπ e ), for a positive integere >1. Let

f := gcd(e, p), e=f e^′, p=f p^′, thenηf=η^p_p^′ =η^e_e^′.

By the definition of Q and Theorem 2.1, without lost of generality, we can suppose that there exist integers v1 ≤. . . ≤ vδ that belong to [0, e^′−1] such that Qconsists of the complex numbersη_e^viη^j_p, whereiranges over [1, δ] andj ranges over [0, p−1]. Thenr=δf p^′ and we can splitQintop^′ sets:

Q=Q¹a . . .a

Q^p′,

withQ^j formed by theη^v_eⁱη_p^lp′+j,i∈[1, δ] andl∈[0, f−1]. We considerQ^j as an ordered sequence

Q^j = (η^j−1_p η_e^v1, . . . , η_p^j−1η_e^vδ, η^p_p^′+j−1η^v_e¹, . . . , η_p^p′+j−1η_e^vδ, . . . η^(f−1)p_p ^′+j−1η_e^v1, . . . , η^(f−1)p_p ^′+j−1η_e^vδ). (9)

Remark: First note that, forj∈[1, p^′],Q^j=η^j−1_p Q¹and

Q¹= (η^v_e¹, . . . , η^v_e^δ, η^v_e^1+e′, . . . , η_e^vδ+e′, . . . , η_e^{v1+(f−1)e′}, . . . , η_e^{vδ+(f−1)e′}), withv1≤. . .≤vδ ≤v1+e^′ ≤. . .≤vδ+ (f−1)e^′.

Note also that the quotient of two elements that belong toQ^j, forj∈[1, p^′], is a power ofv=ηewhile the quotient of an element ofQ^j by an element ofQ^l, l∈[1, p^′],l6=j is not such a power, by definition off.

If we order the elements of Q by the ordered elements of Q¹, then the ordered elements ofQ², etc, Theorem 2.1 implies that the Ariki-Koike algebra H is Morita equivalent to the algebra

M

n1,...,np′ ≥0 n1+...+np′=n

H^q,Q_{f δ,1,n}¹

1(C)⊗. . .⊗H^q,Qp

′

f δ,1,n_p′(C).

So it is clear thatH is also Morita equivalent to the algebra M

n1,...,np′ ≥0 n1+...+np′=n

H^q,Q_{f δ,1,n}¹

1(C)⊗. . .⊗H^q,Q_{f δ,1,n}¹

p′(C).

We orderQand we split it intop^′ sets following what we have done forQ:

Q=Q¹a . . .a

Q^p′.

2.D.3 Parametrization of simple H-modules

We can now give a parametrization for the simple H-modules. By [3, Theo- rem 2.5], the simple modules of the Ariki-Koike algebraH^q,Q_{f δ,1,n}¹

i(C),i∈[1, p^′], are the quotient modules D^v,QC ¹(λ) :=SC^v,Q1(λ)/J(SC^v,Q1(λ)) of the specialisa- tions of the Specht modules S_C^v,Q1(λ) := C⊗S^v,Q1(λ) by their radical of Ja- cobson, with λa Kleshchev f δ-partition (for the parametersQ¹) of sizeni. So the simple modules ofHare labelled by thep^′-tuples of Kleshchevf δ-partitions (associated toQ¹) (λ¹, . . . , λ^p′) with

p^′

X

i=1

|λⁱ|=n. We will denote this set by Λ⁰ and forλ∈Λ⁰,D^v,QC (λ) the corresponding simpleH-module.

In [20], another parametrization for the simpleH^q,Q_{f δ,1,n}¹

i(C)-modules has been found by using Lusztig’s a-function (see Theorem 2.7 below). Following this paper, we will associate to each r-partition an a-value. To do this, we first describe the a-value of af δ-partitionµ= (µ(1), µ(2), ..., µ(f δ)).

Fork= 1, ..., δ,s= 1, ..., f andj= (s−1)δ+kwe put wj :=vk+ (s−1)e^′ so thatwj is the power of thej^th-component of Q¹.

Then, forj:= 1, ..., f δ, we define:

m^(j):=wj−je f δ +e and fors= 1, ..., n, we put:

B_s^′(j):=µ(j)s−s+n+m^(j)

where we use the convention that µ(j)p := 0 ifpis greater than the height of µ(j). Forj = 1, ..., f δ, letB^′(j)= (B^′(j)₁ , ..., B^′(j)n ). Then, we define:

a⁽¹⁾_{f δ}(µ) := X

0≤i≤j<f δ (a,b)∈B′(i)×B′(j)

a>bif i=j

min{a, b} − X

1≤i,j≤f δ a∈B′(i)

1≤k≤a

min{k, m^(j)}+g(n).

whereg(n) is a rational number which only depends on them^(j) and onn(the expression ofg is given in [20]).

Let λ ∈ Π^r_n, write λ = (λ(1), . . . , λ(r)) as λ = (λ¹, . . . , λ^p′) where, for i∈[1, p^′],λⁱ = λ(f δ(i−1) + 1), . . . , λ(f δ(i−1) +f δ)

. Finally, we define

a⁽¹⁾_r (λ) :=

p^′

X

i=1

a⁽¹⁾_{f δ}(λⁱ).

To each Specht module S_K^v,Q1(λ) of H^q,Q_{f δ,1,n}¹ _′(K), n^′ ∈ [0, n] andλ an f δ- partition, we also associate ana-value

a⁽¹⁾_{f δ} S_K^v,Q1(λ)

:=a⁽¹⁾_{f δ}(λ).

To each simpleH^q,Q_{f δ,1,n}¹ _′(C)-module D_C^v,Q1(λ), n^′ ∈[0, n] andλa Kleshchev f δ-partition of sizen^′, we again associate ana-value

a⁽¹⁾_{f δ}(D^v,QC ¹(λ)) := min{a⁽¹⁾_{f δ}(µ)| µ∈Π^{f δ}_n′, d⁽¹⁾

S_K^v,Q1(µ),D^v,Q_C ¹(λ)6= 0}.

Theorem 2.7 ([19, Theorem 2.3.8]) Let λ be a Kleshchev f δ-partition of n^′ for a non-negative integer n^′. There exists a unique µ := κ(λ) such that a⁽¹⁾_{f δ}(DC^v,Q1(λ)) =a⁽¹⁾_{f δ}(µ) andd⁽¹⁾

S_K^v,Q1(µ),D^v,Q_C ¹(λ)6= 0. The function κis a bijec- tion between the set of the Kleshchev f δ-partitions of size n^′ and the set of the FLOTW f δ-partitions of sizen^′.

Recall what is a FLOTWf δ-partitionλ= λ(1), . . . , λ(f δ)

of sizen^′associ- ated to the parametersv andQ¹, see [9]. Denote each partitionλ(i), i∈[1, f δ], as (λ(i)1, λ(i)2, . . .). The multipartitionλsatisfies

1. fori∈[1, f],j ∈[1, δ−1],kpositive integer, λ (i−1)δ+j

k ≥λ (i−1)m+j+ 1

k+vj+1−vj, 2. fori∈[1, f−1],kpositive integer,

λ(iδ)k≥λ(iδ+ 1)k+v1+e^′−vδ, 3. for kpositive integer,

λ(f δ)k ≥λ(1)k+v1+e^′−vδ, as well as

4. to each node of the diagram ofλwhich is located in the a^th row and the b^th column of thec^th partition ofλ(for two poditive integersaandband 1 ≤ c ≤ f δ), we associate its residue η^b−a+v_e ^c. Then, for any positive integerk, the cardinality of the set of the residues associated to the nodes in both thek^th columns and the right rims of the Young diagrams ofλis note.

In the same way, to each Specht module S_K^v,Q(λ) of H, λ ∈ Π^r_n, λ = (λ¹, . . . , λ^p′), we can definea⁽¹⁾r (S_K^v,Q(λ)) :=a⁽¹⁾r (λ) that is the sum

p^′

X

i=1

a⁽¹⁾_{f δ}(λⁱ).

To each simpleH-moduleD^v,QC (λ),λ∈Λ⁰, we associate thea-value a⁽¹⁾_r (D^v,Q_C (λ)) := min{a⁽¹⁾_r (µ)|µ∈Π^r_n, d⁽¹⁾

S^v,Q_K (µ),D^v,Q_C (λ)6= 0}. (10) With Theorem 2.1, it is clear that there exists a unique µ:= κ^′(λ) such that a⁽¹⁾r (DC^v,Q(λ)) =a⁽¹⁾r (µ) andd⁽¹⁾

S^v,Q_K (µ),D^v,QC (λ)6= 0. In fact, κ^′(λ) = (κ(λ¹), . . . , κ(λ^p′)) ifλ= (λ¹, . . . , λ^p′).

The function κ^′ is a bijection between Λ⁰ and Λ¹ the set of the p^′-tuples of FLOTWf δ-partitions (λ¹, . . . , λ^p′) of size

p^′

X

i=1

|λⁱ|=n.

2.D.4 Preliminary results

In order to prove the main Theorem, we will need some preliminary results.

First, recall that we have ordered Q by the ordered elements of Q¹, then by the ordered elements ofQ² etc as in §2.D.2. Let ̟ be the permutation of S_r defined in Proposition 2.4 and let

λ:= (λ¹[1], ..., λ¹[f δ], λ²[1], ..., λ²[f δ], ..., λ^p′[1], ..., λ^p′[f δ])

be anr-partition. Hence, following the notations of the previous paragraph, we haveλ^j[i] =λ(f δ(j−1) +i). Then, it is easy to verify that

• fori∈[1,(p^′−1)f δ], then ̟(i) =i+f δ,

• fori∈[r−f δ+ 1, r−δ], then̟(i) =i−r+ (f+ 1)δ,

• fori∈[r−δ+ 1, r], then ̟(i) =i−r+δ.

Then, by Proposition 2.4, we have:

S_K^v,Q(λ)≃S_K^v,Q ̟(λ)

where ̟(λ) = (λ^p′[f δ−δ+ 1], . . . , λ^p′[f δ],λ^p′[1], . . . , λ^p′[f δ−δ], λ¹[1], . . . , λ¹[f δ], . . . , λ^p′−1[1], . . . , λ^p′−1[f δ]).

Proposition 2.8 Letλ∈Π^r_n, thena⁽¹⁾r (S_K^v,Q(λ)) =a⁽¹⁾r (^fHH′S_K^v,Q(λ)). Besides, if λ∈Λ⁰, thena⁽¹⁾r (D^v,QC (λ)) =a⁽¹⁾r (^fHH′D^v,QC (λ)).

Proof: By the definition of thea-value (see§2.D.3), we have:

a⁽¹⁾_r (S^v,Q_K (λ)) :=

p^′

X

i=1

a⁽¹⁾_{f δ}(λⁱ[1], ..., λⁱ[f δ])

Thus, by using the above properties, it is sufficient to show thata⁽¹⁾_{f δ} (λ^p′[1], . . . , λ^p′[f δ])=a⁽¹⁾_{f δ}(λ^p′[f δ−δ+ 1], . . . , λ^p′[f δ], λ^p′[1], . . . , λ^p′[f δ−δ]).

Using the notations of §2.D.3, for i = 1, ..., δ and j = 1, ..., f we write m⁽ⁱ⁾[j] :=m^{((j−1)δ+i)}. Then, all we have to do is to prove thatm⁽ⁱ⁾[j] doesn’t depend onj. We havem⁽ⁱ⁾[j] =vi−((j−1)δ+i)e

f δ + (j−1)e^′=vi−ie

f because e^′f =e. Hence, we obtain the desired result.

With the definition (10), it is trivial that the previous result about Specht modules implies that, for anyλ∈Λ⁰,a⁽¹⁾r (D^v,Q_C (λ)) =a⁽¹⁾r (^fHH′D^v,Q_C (λ)).

With Proposition 2.8 and Lemma 2.2, we can associate ana-value to each simpleH^′-moduleU. If U appears in the restriction ResS_K^v,Q(λ), for λ∈Π^r_n, put

a^(p)_r (U) :=a⁽¹⁾_r S_K^v,Q(λ) .

In the same way, we can associate ana-value to each simpleH^′-moduleW. IfW appears in the restriction ResD^v,QC (λ), forλ∈Λ⁰, then

a^(p)_r (W) :=a⁽¹⁾_r (D^v,QC (λ)).

Another useful proposition is the following one.

Proposition 2.9 Let λ ∈ Λ¹, then ̟(λ) ∈ Λ¹, where ̟ is defined in the Proposition 2.4.

Proof: Writeλ∈Λ¹ as λ= (λ¹, . . . , λ^p′) where λⁱ = (λⁱ[1], . . . , λⁱ[f δ]) is a FLOTWf δ-partition forQ¹. We will see that̟(λ)∈Λ¹so the result will be proved. The r-partition ̟(λ) is equals to (λ^p′[f δ−δ+ 1] , . . . , λ^p′[f δ],λ^p′[1]

, . . . , λ^p′[f δ−δ], λ¹[1], . . . , λ¹[f δ], . . . λ^p′−1[1], . . . , λ^p′−1[f δ]). It belongs to Λ¹ if and only if (λ^p′[f δ−δ+ 1] , . . . , λ^p′[f δ], λ^p′[1] , . . . , λ^p′[f δ−δ]) is a FLOTW f δ-partition. It is easy to verify that the three first conditions hold.

Now, the set of the residues associated to the nodes in both thek^thcolumns and the right rims of the Young diagrams of this last multipartition consists of the residues of λ^p′ but multiplied by η_e^e′. So the fourth condition about FLOTW multipartitions hold as well as the Proposition.

3 A parametrization of the simple H

-modules

We keep the above notations. The aim of this section is to describe a parametriza- tion of the simpleH^′-modules, withH^′=H^v,Q_r,p,n(C). The proof is highly inspired by [14, Theorem 2.1].

Theorem 3.1 For all simple H^′-module N, there exists a simple H^′-module WN such that

d^r,p,n([WN]) = [N] + X

a^(p)r (L)<a^(p)r (N)

dWN,L[L],

anda^(p)r (N) =a^(p)r (WN).

Let Λ^1′ := {WN | N ∈ Irr(H^′)}. Then Λ^1′ consists in the S_K^v,Q(λ, i), with λ∈Λ¹∩ L,i∈[0,_o^p

λ−1]with the notations of the Proposition 2.5 and the map N ∈Irr(H^′)7→WN ∈Λ^1′ is bijective. SoΛ^1′ is a parametrization of the simple H^′-modules.

Proof : Fix a simpleH^′-moduleN. By Lemma 2.2, there existsλ∈Λ⁰such that N appears in the restriction ResD_C^v,Q(λ). We have

d^r,1,n([SC^v,Q κ(λ)

]) = [DC^v,Q(λ)] + X

a(p) r (Dv,Q

C (µ))<

a(p) r (Dv,Q

C (λ))

d⁽¹⁾

S^v,Q_K (κ(λ)),DC^v,Q(µ)[DC^v,Q(µ)],

(11) anda⁽¹⁾r SC^v,Q(κ(λ))

=a⁽¹⁾r (D^v,QC (λ)). With the notations of (8) and Proposi- tion 2.6, this equality implies that

p oκ(λ)−1

X

i=0

d^r,p,n([SC^v,Q(κ(λ), i)]) =

p of H

H′,Dv,Q C (λ)

−1

X

i=0

[^(gHH′)iN] + [N^′], (12)

whereN^′is a sum ofH^′-modules,D, witha-valuea^(p)r (D)< a^(p)r (D^v,Q_C (λ)) and, for alli,j,a^(p)r (SC^v,Q(κ(λ), i)) =a^(p)r (^(gHH′)iN).

Denote by τ(λ)∈Λ⁰, the multipartition defined byDC^v,Q(τ(λ)) :=^fHH′D^v,QC (λ).

If we twist the action ofHbyf_H^H′, the equality (11) gives thatκ(τ(λ)) =̟(κ(λ)) by Proposition 2.9 and by definition ofκ. As this map is bijective, it is clear that

o_fH

H′,S^v,Q_K (κ(λ)) =o_fH

H′,DC^v,Q(λ)

and therefore for alli∈[0,_o^p

κ(λ) −1], o_gH

H′,S^v,Q_K (κ(λ),i)=o_g^H

H′,N

with the last equality of Lemma 2.2.

Recall that for i∈[0,_oκ(λ)^p −1],S_K^v,Q(κ(λ), i) is conjugate to S_K^v,Q(κ(λ),0) by (g_H^H′)ⁱ. The equality (12) implies that there exists an unique i∈ [0,_oκ(λ)^p −1]

such that

d^r,p,n([S_C^v,Q(κ(λ), i)]) = [N] + [N^′′], (13) where N^′′ is a sum of H^′-modules, D, with a-value a^(p)r (D) < a^(p)r (N). This construction makes clear all the statements of the Theorem.

Remark: The above theorem is only concerns with the case wherevis a primitive e^th-root of unity. Ifv isn’t a root of unity, it is readily checked that an analogue of this theorem holds by replacing Λ¹by the set of Kleshchev multipartitions Λ⁰ at e=∞. The proof of this result can be easily obtained following the outline of [14].

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