Schur elements for the Ariki–Koike algebra and applications

DOI 10.1007/s10801-011-0314-4

Schur elements for the Ariki–Koike algebra and applications

Maria Chlouveraki·Nicolas Jacon

Received: 30 May 2011 / Accepted: 6 September 2011 / Published online: 27 September 2011

© Springer Science+Business Media, LLC 2011

Abstract We study the Schur elements associated to the simple modules of the Ariki–Koike algebra. We first give a cancellation-free formula for them so that their factors can be easily read and programmed. We then study direct applications of this result. We also complete the determination of the canonical basic sets for cyclotomic Hecke algebras of typeG(l, p, n)in characteristic 0.

Keywords Hecke algebras·Complex reflection groups·Schur elements·Blocks· Basic sets

1 Introduction

Schur elements play a powerful role in the representation theory of symmetric alge- bras. In the case of the Ariki–Koike algebra, that is, the Hecke algebra of the complex reflection groupG(l,1, n), they are Laurent polynomials whose factors determine when Specht modules are projective irreducible and whether the algebra is semisim- ple.

Formulas for the Schur elements of the Ariki–Koike algebra have been obtained independently, first by Geck, Iancu and Malle [13], and later by Mathas [18]. The first aim of this paper is to give a cancellation-free formula for these polynomials (Theorem3.2), so that their factors can be easily read and programmed. We then present a number of direct applications. These include a new formula for Lusztig’s a-function, as well as a simple classification of the projective irreducible modules for Ariki–Koike algebras (that is, the blocks of defect 0).

M. Chlouveraki

School of Mathematics, JCMB Room 5610, King’s Building, Edinburgh, EH9 3JZ, UK e-mail:maria.chlouveraki@ed.ac.uk

N. Jacon (

UFR Sciences et Techniques, 16 route de Gray, 25030 Besançon, France e-mail:njacon@univ-fcomte.fr

The second part of the paper is devoted to another aspect of the representation the- ory of these algebras in connection with these Schur elements: the theory of canonical basic sets. The main aim here is to obtain a classification of the simple modules for specializations of cyclotomic Hecke algebras in characteristic 0. In [6], we studied mainly the case of finite Weyl groups. In this paper, we focus on cyclotomic Hecke algebras of typeG(l, p, n). Using Lusztig’sa-function, defined from the Schur el- ements, the theory of canonical basic sets provides a natural and efficient way to parametrize the simple modules of these algebras.

The existence and explicit determination of the canonical basic sets is already known in the case of Hecke algebras of finite Weyl groups (see [11] and [6]). The case of cyclotomic Hecke algebras of typeG(l, p, n)has been partially studied in [11, 14], and recently in [7] using the theory of Cherednik algebras. Answering a question raised in [7], the goal of the last part of this paper is to complete the determination of the canonical basic sets in this case.

2 Preliminaries

In this section, we introduce the necessary definitions and notation.

2.1 A partitionλ=(λ₁, λ2, λ3, . . .)is a decreasing sequence of non-negative inte- gers. We define the length ofλto be the smallest integer(λ)such thatλi =0 for alli > (λ). We write|λ| :=

i≥1λi and we say thatλis a partition ofm, for some m∈Z>0, ifm= |λ|. We setn(λ):=

i≥1(i−1)λi. We define the set of nodes[λ]ofλto be the set

[λ] :=

(i, j )|i≥1, 1≤j≤λi

.

A nodex =(i, j ) is called removable if[λ] \ {(i, j )}is still the set of nodes of a partition. Note that if(i, j )is removable, thenj=λi.

The conjugate partition ofλis the partitionλdefined by λ_k:=#{i|i≥1 such thatλi≥k}. Obviously,λ₁=(λ). The set of nodes ofλsatisfies

(i, j )∈ [λ] ⇔(j, i)∈ [λ]. Note that if(i, λ_i)is a removable node ofλ, thenλ_λ

i=i.Moreover, we have n(λ)=

i≥1

(i−1)λi=1 2

i≥1

(λ_i−1)λ_i=

i≥1

λ_i 2

.

Ifx=(i, j )∈ [λ]andμis another partition, we define the generalized hook length ofxwith respect to (λ,μ) to be the integer:

h^λ,μ_i,j :=λ_i−i+μ_j−j+1.

Forμ=λ, the above formula becomes the classical hook length formula (giving the length of the hook ofλthat x belongs to). Moreover, we define the content ofx to be the differencej−i. The following lemma, whose proof is an easy combinatorial exercise (with the use of Young diagrams), relates the contents of the nodes of (the

“right rim” of)λwith the contents of the nodes of (the “lower rim” of)λ.

Lemma 2.2 Letλ=(λ₁, λ₂, . . .) be a partition and let k be an integer such that 1≤k≤λ₁. Letqandybe two indeterminates. Then we have

1 (q^λ1y−1)·

1≤i≤λ_k

q^λi−i+1y−1 q^λi−iy−1

= 1

(q^−λk+k−1y−1)·

k≤j≤λ1

q^−λj+j−1y−1 q^−λj+jy−1

.

2.3 Let l and n be positive integers. An l-partition (or multipartition) of n is an orderedl-tupleλ=(λ⁰, λ¹, . . . , λ^l−1)of partitions such that

0≤s≤l−1|λ^s| =n. We denote byΠ_n^l the set ofl-partitions ofn. The empty multipartition, denoted by∅, is anl-tuple of empty partitions. Ifλ=(λ⁰, λ¹, . . . , λ^l−1)∈Π_n^l, we denote byλ the l-partition(λ⁰, λ¹, . . . , λ^l−1).

2.4 Let R be a commutative domain with 1. Fix elementsq, Q₀, . . . , Q_l−1of R, and assume thatq is invertible inR. Set q:=(Q0, . . . , Q_l−1;q). The Ariki–Koike algebraH^qnis the unital associativeR-algebra with generatorsT0, T1, . . . , T_n−1and relations:

(T0−Q0)(T0−Q1)· · ·(T0−Ql−1)=0,

(T_i−q)(T_i+1)=0 for 1≤i≤n−1, T₀T₁T₀T₁=T₁T₀T₁T₀,

T_iT_i+1T_i=T_i+1T_iT_i+1 for 1≤i≤n−2,

T_iT_j=T_jT_i for 0≤i < j≤n−1 withj−i >1.

The last three relations are the braid relations satisfied byT0, T1, . . . , T_n−1.

The Ariki–Koike algebraH^qnis a deformation of the group algebra of the complex reflection groupG(l,1, n)=(Z/ lZ)Sn. Ariki and Koike [3] have proved thatH^qnis a freeR-module of ranklⁿn! = |G(l,1, n)|(see [2, Proposition 13.11]). In addition, whenRis a field, they have constructed a simpleH^qn-moduleV^λ, with characterχ^λ, for eachl-partitionλofn(see [2, Theorem 13.6]). These modules form a complete set of non-isomorphic simple modules in the case whereH^qnis split semisimple (see [2, Corollary 13.9]).

2.5 There is a useful semisimplicity criterion for Ariki–Koike algebras which has been given by Ariki in [1]. This criterion will be recovered from our results later (see Theorem4.2), so let us simply assume from now on thatH^qn is split semisim- ple. This happens, for example, when q, Q₀, . . . , Q_l−1 are indeterminates and R=Q(q, Q₀, . . . , Q_l−1).

Now, there exists a linear form τ :H^qn→R which was introduced by Bremke and Malle in [4], and was proved to be symmetrizing by Malle and Mathas in [16]

whenever allQi’s are invertible inR. An explicit description of this form can be found in any of these two articles. Following Geck’s results on symmetrizing forms (see [12, Theorem 7.2.6]), we obtain the following definition for the Schur elements associated to the irreducible representations ofH^qn.

Definition 2.6 Suppose thatRis a field and thatHn^qis split semisimple. The Schur elements ofH^qnare the elementss_λ(q)ofRsuch that

τ =

λ∈Π_n^l

1 sλ(q)χ^λ.

2.7 The Schur elements of the Ariki–Koike algebraH^qn have been independently calculated by Geck, Iancu and Malle [13], and by Mathas [18]. From now on, for all m∈Z>0, let[m]q:=(q^m−1)/(q−1)=q^m−1+q^m−2+ · · · +q+1. The formula given by Mathas does not demand extra notation and is the following:

Theorem 2.8 Letλ=(λ⁰, λ¹, . . . , λ^l−1)be anl-partition ofn. Then s_λ(q)=(−1)^n(l−1)(Q₀Q₁· · ·Q_l−1)⁻ⁿq^−α(λ)

·

0≤s≤l−1

(i,j )∈[λ^s]

Qs h^λ_i,j^s,λs

q·

0≤s<t≤l−1

X^λ_st,

where

α(λ)=

0≤s≤l−1

n λ^s

and

X_st^λ =

(i,j )∈[λ^t]

q^j−iQt−Qs

·

(i,j )∈[λ^s]

q^j−iQs−q^λt1Qt 1≤k≤λ^t₁

q^j−iQ_s−q^k−1−λtkQ_t q^j−iQ_s−q^k−λtkQ_t

.

The formula by Geck, Iancu and Malle is more symmetric, and describes the Schur elements in terms of beta numbers. Ifλ=(λ⁰, λ¹, . . . , λ^l−1)is an l-partition ofn, then the length of λ is (λ)=max{(λ^s)|0≤s≤l−1}. Fix an integer L such thatL≥(λ). TheL-beta numbers forλ^s are the integersβ_i^s=λ^s_i +L−ifori= 1, . . . , L. SetB^s= {β₁^s, . . . , β_L^s}fors=0, . . . , l−1. The matrix B=(B^s)0≤s≤l−1is called theL-symbol ofλ.

Theorem 2.9 Let λ=(λ⁰, . . . , λ^l−1) be an l-partition of n with L-symbol B= (B^s)_{0≤s≤l−1}, where L≥ (λ). Let a_L :=n(l−1)+_l

2

_L

2

and b_L:=lL(L− 1)(2lL−l−3)/12. Then

s_λ(q)=(−1)^aLq^bL(q−1)⁻ⁿ(Q₀Q₁. . . Q_l−1)⁻ⁿν_λ/δ_λ, where

νλ=

0≤s<t≤l−1

(Q_s−Q_t)^L

0≤s, t≤l−1

bs∈B^s

1≤k≤bs

q^kQ_s−Q_t

and

δλ=

0≤s<t≤l−1

(bs,bt)∈B^s×B^t

q^bsQs−q^btQt

0≤s≤l−1

1≤i<j≤L

q^βisQs−q^βsjQs

.

As the reader may see, in both formulas above, the factors ofs_λ(q)are not obvious.

Hence, it is not obvious for which values of q the Schur elementsλ(q)becomes zero.

3 A cancellation-free formula for the Schur elements

In this section, we will give a cancellation-free formula for the Schur elements ofH^qn. This formula is also symmetric.

3.1 LetXandY be multisets of rational numbers ordered so that their elements form decreasing sequences. We will writeX Y for the (ordered) multiset consisting of all the elements ofX andY together and such that the elements of X Y form a decreasing sequence. We have|X Y| = |X| + |Y|.

Theorem 3.2 Letλ=(λ⁰, λ¹, . . . , λ^l−1)be anl-partition ofn. Setλ¯:=

0≤s≤l−1λ^s. Then

sλ(q)=(−1)^n(l−1)q^−n(¯λ)(q−1)⁻ⁿ

0≤s≤l−1

(i,j )∈[λ^s]

0≤t≤l−1

q^h

λs ,λt

i,j Q_sQ⁻_t ¹−1 .

(1) Since the total number of nodes inλis equal ton, the above formula can be rewritten as follows:

sλ(q)=(−1)^n(l−1)q^−n(λ)¯

·

0≤s≤l−1

(i,j )∈[λ^s]

h^λ_i,j^s,λs

q

0≤t≤l−1, t=s

q^h

λs ,λt

i,j Q_sQ⁻_t ¹−1 . (2)

3.3 We will now proceed to the proof of the above result using the formula of Theo- rem2.8. The following lemma relates the termsq^−n(λ)¯ andq^−α(λ).

Lemma 3.4 Letλbe anl-partition ofn. We have

α(λ)+

0≤s<t≤l−1

i≥1

λ^s_iλ^t_i=n(λ).¯

Proof Following the definition of the conjugate partition, we haveλ¯_i=

0≤s≤l−1λ^s_i, for alli≥1. Therefore,

n(λ)¯ =

i≥1

λ¯_i 2

=

i≥1

0≤s≤l−1λ^s_i 2

=

i≥1 0≤s≤l−1

λ^s_i 2

+

0≤s<t≤l−1

λ^s_iλ^t_i

=α(λ)+

0≤s<t≤l−1

i≥1

λ^s_iλ^t_i.

Hence, to prove Equality (2), it is enough to show that, for all 0≤s < t≤l−1, X_st^λ =q^{−i≥1λsiλti}Q^|_s^λt|Q^|_t^λs|

(i,j )∈[λ^s]

q^h

λs ,λt

i,j Q_sQ⁻_t ¹−1

·

(i,j )∈[λ^t]

q^h

λt ,λs

i,j Q_tQ⁻_s¹−1

. (3)

3.5 We will proceed by induction on the number of nodes ofλ^s. We do not need to do the same forλ^t, because the symmetric formula for the Schur elements given by Theorem2.9implies the following: ifμis the multipartition obtained fromλby exchangingλ^s andλ^t, then

X^λ_st(Q_s, Q_t)=X^μ_st(Q_t, Q_s).

Ifλ^s= ∅, then

X_st^λ =

(i,j )∈[λ^t]

q^j−iQ_t−Q_s

=Q^|λ_s^t|

(i,j )∈[λ^t]

q^j−iQ_tQ⁻_s¹−1

=Q^|_s^λt|

1≤i≤λ^t₁

1≤j≤λ^t_i

q^j−iQtQ⁻_s¹−1

=Q^|_s^λt|

1≤i≤λ^t₁

1≤j≤λ^t_i

q^{λti−j+1−i}Q_tQ⁻_s¹−1

=Q^|_s^λt|

(i,j )∈[λ^t]

q^h

λt ,λs

i,j QtQ⁻_s¹−1 ,

as required.

3.6 Now assume that our assertion holds when #[λ^s] ∈ {0,1,2, . . . , N−1}. We want to show that it also holds when #[λ^s] =N≥1. Ifλ^s= ∅, then there existsisuch that (i, λ^s_i)is a removable node ofλ^s. Letνbe the multipartition defined by

ν_i^s:=λ^s_i −1, ν_j^s:=λ^s_j for allj=i, ν^t:=λ^t for allt=s.

Then[λ^s] = [ν^s] ∪ {(i, λ^s_i)}. Since (3) holds forX_st^ν and

X^λ_st=X^ν_st·

q^λsi−iQs−q^λt1Qt 1≤k≤λ^t₁

q^λsi−iQ_s−q^k−1−λtkQ_t q^λsi−iQ_s−q^k−λtkQ_t

,

it is enough to show that (to simplify notation, from now on setλ:=λ^sandμ:=λ^t):

q^λi−iQ_s−q^μ1Q_t

1≤k≤μ₁

q^λi−iQ_s−q^k−1−μkQ_t q^λi−iQ_s−q^k−μkQ_t

=q^−μλiQt

q^{λi−i+μλi−λi+1}QsQ⁻_t ¹−1

·A·B, (4) where

A:=

1≤k≤λi−1

q^{λi−i+μk−k+1}Q_sQ⁻_t ¹−1 q^{λi−i+μk−k}Q_sQ⁻_t ¹−1 and

B:=

1≤k≤μ

λi

q^{μk−k+λλi−λi+1}Q_tQ⁻_s¹−1 q^{μk−k+λλi−λi}Q_tQ⁻_s¹−1

.

Note that, since(i, λ_i)is a removable node ofλ, we haveλ_λ

i=i. We have

A=q^λi−1

1≤k≤λi−1

q^λi−iQ_s−q^k−1−μkQ_t q^λi−iQ_s−q^k−μkQ_t .

Moreover, by Lemma2.2, fory=q^i−λiQ_tQ⁻_s¹,we obtain B= (q^μ1+i−λiQtQ⁻_s¹−1)

(q^{−μλi+λi−1+i−λi}Q_tQ⁻_s¹−1)

·

λ_i≤k≤μ₁

q^{−μk+k−1+i−λi}QtQ⁻_s¹−1 q^{−μk+k+i−λi}QtQ⁻_s¹−1

,

i.e.,

B=Q⁻_t ¹q^{μλi−λi+1} (q^λi−iQs−q^μ1Qt) (q^{μλi−λi+1+λi−i}Q_sQ⁻_t ¹−1)

·

λi≤k≤μ1

q^λi−iQ_s−q^k−1−μkQ_t q^λi−iQ_s−q^k−μkQ_t

.

Hence, Equality (4) holds.

4 First consequences

We give here several direct applications of Formula (2) obtained in Theorem3.2.

4.1 A first application of Formula (2) is that we can easily recover a well-known semisimplicity criterion for the Ariki–Koike algebra due to Ariki [1]. To do this, let us assume thatq, Q₀, . . . , Q_l−1are indeterminates andR=Q(q, Q₀, . . . , Q_l−1).

Then the resulting “generic” Ariki–Koike algebraH^qn is split semisimple. Now as- sume thatθ:Z[q^±1, Q^±₀¹, . . . , Q^±_l−¹₁] →Kis a specialization and letKH^qn be the specialized algebra, whereKis any field. Note that for allλ∈Π_n^l, we havesλ(q)∈ Z[q^±1, Q^±₀¹, . . . , Q^±_l−¹₁]. Then by [12, Theorem 7.2.6],KH^qnis (split) semisimple if and only if, for allλ∈Π_n^l, we have θ (s_λ(q))=0. From this, we can deduce the following:

Theorem 4.2 (Ariki) Assume thatKis a field. The algebraKH^qnis (split) semisimple if and only ifθ (P (q))=0, where

P (q)=

1≤i≤n

1+q+ · · · +qⁱ⁻¹

0≤s<t≤l−1

−n<k<n

q^kQ_s−Q_t .

Proof Assume first thatθ (P (q))=0. We distinguish three cases:

(a) If there exists 2≤i≤n such that θ (1+q+ · · · +qⁱ⁻¹)=0, then we have θ ([h^λ_1,n−i+^0,λ0 ₁]q)=0 for λ=((n),∅, . . . ,∅)∈Π_n^l. Thus, for thisl-partition, we haveθ (s_λ(q))=0, which implies thatKHn^qis not semisimple.

(b) If there exist 0≤s < t≤l−1 and 0≤k < nsuch thatθ (q^kQ_s−Q_t)=0, then we haveθ (q^h

λs ,λt

1,n−kQsQ⁻_t ¹−1)=0 forλ∈Π_n^l such thatλ^s=(n),λ^t = ∅. We haveθ (sλ(q))=0 andKH^qnis not semisimple.

(c) If there exist 0≤s < t≤l−1 and−n < k <0 such thatθ (q^kQs−Qt)=0, then we haveθ (q^h

λt ,λs

1,n+kQtQ⁻_s¹−1)=0 forλ∈Π_n^l such thatλ^s= ∅,λ^t=(n). Again, we haveθ (sλ(q))=0 andKH^qnis not semisimple.

Conversely, ifKH^qnis not semisimple, then there existsλ∈Π_n^l such thatθ (s_λ(q))= 0. As for all 0≤s, t≤l−1 and(i, j )∈ [λ^s], we have−n < h^λ_i,j^s,λt< n, we conclude

thatθ (P (q))=0.

4.3 We now consider a remarkable specialization of the generic Ariki–Koike algebra.

Letube an indeterminate. Letr∈Z>0and letr₀, . . . , r_l−1be any integers. Set r:=

(r₀, . . . , r_l−1)andη_l:=exp(2√

−1π/ l). For alli=0, . . . , l−1, we setm_i:=r_i/r and we define m:=(m₀, . . . , m_l−1)∈Q^l. Assume thatR=Z[q^±1, Q^±₀¹, . . . , Q^±_l−¹₁] and consider the morphism

θ:R→Z[η_l] u^±1

such thatθ (q)=u^r andθ (Q_j)=η^j_lu^rj forj =0,1, . . . , l−1. We will denote by H^m,rn the specialization of the Ariki–Koike algebraH^qn viaθ. The algebraHn^m,r is

called a cyclotomic Ariki–Koike algebra. It is defined overZ[η_l][u^±1]and has a pre- sentation as follows:

• generators:T₀, T₁, . . . , T_n−1,

• relations:

T₀−u^r0

T₀−η_lu^r1

· · ·

T₀−η_l^l−1u^rl−1

=0, T_j−u^r

(T_j+1)=0 forj=1, . . . , n−1 and the braid relations symbolized by the diagram

⁴ · · ·

T₀ T₁ T₂ T_n−1

.

We setK:=Q(ηl). The algebra K(u)H^m,rn , which is obtained by extension of scalars toK(u), is a split semisimple algebra. As a consequence, one can apply Tits’s Deformation Theorem (see, for example, [12, §7.4]), and see that the set of simple K(u)H^m,rn -modules Irr(K(u)H^m,rn )is given by

Irr

K(u)H^m,r_n

=

V^λ|λ∈Π_n^l .

Using the Schur elements, one can attach to every simpleK(u)H^m,rn -moduleV^λ a rational number a^(m,r)(λ), by setting a^(m,r)(λ)to be the negative of the valuation of the Schur element ofV^λinu, that is, the negative of the valuation ofθ (sλ(q)). We call this number the a-value ofλ. By [11, §5.5], this value may be easily computed combinatorially: Letλ∈Π_n^l and lets∈Z>0such thats > (λ). LetBbe the shifted m-symbol of λ of sizes∈Z>0. This is thel-tuple(B⁰, . . . ,B^l−1)where, for all j=0, . . . , l−1 and for alli=1, . . . , s+ [m_j](where[m_j]denotes the integer part ofm_j), we have

B^j_i =λ^j_i −i+s+m_j and B^j= B^j_s+[m

j], . . . ,B^j₁ .

Write

κ1(λ)≥κ2(λ)≥ · · · ≥κh(λ)

for the elements ofBwritten in decreasing order (allowing repetitions), whereh= ls+

0≤j≤l−1[m_j]. Letκ_m(λ)=(κ₁(λ), . . . , κ_h(λ))∈Q^h_≥0and define n_m(λ):=

1≤i≤h

(i−1)κi(λ).

Then, by [11, Proposition 5.5.11], the a-value ofλis a^(m,r)(λ)=r

nm(λ)−nm(∅) , where∅denotes the empty multipartition.

Generalizing the dominance order for partitions, we will writeκm(λ)κm(μ)if κ_m(λ)=κ_m(μ)and

1≤i≤tκ_i(λ)≥

1≤i≤tκ_i(μ)for allt≥1. The following result [11, Proposition 5.5.16] will be useful in then next sections:

Proposition 4.4 Assume thatλandμare twol-partitions with the same rank such thatκ_m(λ)κ_m(μ). Then a^(m,r)(μ) >a^(m,r)(λ).

Now, Formula (2) allows us to give an alternative description of the a-value ofλ:

Proposition 4.5 Letλ∈Π_n^l. The a-value ofλis a^(m,r)(λ)=r

n(λ)−

0≤s≤l−1

(i,j )∈[λ^s]

0≤t≤l−1,t=s

min

h^λ_i,j^s,λt +ms−mt,0 .

4.6 We now consider another type of specialization. Letv₀, . . . , v_l−1be any integers.

Letkbe a subfield ofCand letηbe a primitive root of unity of ordere >1. Assume thatR=Z[q^±1, Q^±₀¹, . . . , Q^±_l−¹₁]and consider the morphism

θ:R→k(η)

such thatθ (q)=η andθ (Q_j)=η^vj for j =0,1, . . . , l−1. By Theorem 4.2, the specialized algebrak(η)H^qn is not generally semisimple, and a result by Dipper and Mathas which will be specified later (see Sect.5.2) implies that the study of this algebra is enough for studying the non-semisimple representation theory of Ariki–

Koike algebras in characteristic 0. Let D= V^λ:M

λ∈Πn^l, M∈Irr(k(η)H^qn)

be the associated decomposition matrix (see [12, §7.4]), which relates the irreducible representations of the split semisimple Ariki–Koike algebraH^qn and the specialized Ariki–Koike algebrak(η)H^qn. We are interested in the classification of the blocks of defect 0. That is, we want to classify the l-partitions λ∈Π_n^l which are alone in their blocks in the decomposition matrix. These correspond to the modulesV^λ which remain projective and irreducible after the specializationθ. By [17, Lemme 2.6] (see also [12, Theorem 7.2.6]), these elements are characterized by the property thatθ (s_λ(q))=0. In our setting, using Formula (2), we obtain the following:

Proposition 4.7 Under the above hypotheses,λ∈Π_n^l is in a block of defect 0 if and only if, for all 0≤s, t≤l−1 and(i, j )∈ [λ^s],edoes not divideh^λ_i,j^s,λt +v_s−v_t. Remark 4.8 As pointed out by M. Fayers and A. Mathas, the above proposition can also be obtained using [10].

5 Canonical basic sets for Ariki–Koike algebras

In this part, we generalize some known results on basic sets for Ariki–Koike algebras, using a fundamental result by Dipper and Mathas. This will help us determine the canonical basic sets for cyclotomic Ariki–Koike algebras in full generality.

5.1 We consider the cyclotomic Ariki–Koike algebraH^m,rn defined in Sect.4.3, re- placing from now on the indeterminateuby the indeterminateq(following the usual notation). Letθ:Z[η_l][q^±1] →K(η)be a specialization such that θ (q)=η∈C^∗. We obtain a specialized Ariki–Koike algebraK(η)H^m,rn . The relations between the generators are the usual braid relations together with the following ones:

T0−η^r0

T0−η_lη^r1

· · ·

T0−η^l_l⁻¹η^rl−1

=0, T_j−η^r

(T_j+1)=0 forj =1, . . . , n−1.

Let

D= V^λ:M

λ∈Πn^l, M∈Irr(K(η)H^m,rn )

be the associated decomposition matrix (see [12, §7.4]). The matrixD relates the irreducible representations of the split semisimple Ariki–Koike algebraK(q)H^m,rn

and the specialized Ariki–Koike algebraK(η)H^m,rn . The goal of this section is to study the form of this matrix in full generality.

First assume thatηis not a root of unity. Then, for all 0≤i=j ≤l−1, we have ηⁱ_l^−jη^ri−rj=η^rd

for alld∈Z>0. By the criterion of semisimplicity due to Ariki (Theorem4.2), this implies that the algebraK(η)H^m,rn is split semisimple, and thus D is the identity matrix. Hence, from now, one may assume thatηis a primitive root of unity of order e >1. Then there existsk∈Z>0such that gcd(k, e)=1 andη=exp(2√

−1π k/e).

5.2 We will now use a reduction theorem by Dipper and Mathas, which will help us understand the form ofD. Set I:= {0,1, . . . , l−1}. There is a partition

I=I1 I2 · · · Ip

such that

• for all 1≤α < β≤p,(i, j )∈Iα×Iβ andd∈Z>0, we have η^rd−ηⁱ_l^−jη^ri−rj=0

• for all 1≤α≤pand(i, j )∈Iα×Iα, there existsd∈Z>0such that η^rd−ηⁱ_l^−jη^ri−rj=0.

For allj=1, . . . , p, we setl[j] := |Ij|and we consider Ijas an ordered set Ij=(i_j,1, i_j,2, . . . , i_j,l[j]) withi_j,1< i_j,2<· · ·< i_j,l[j]. We define

πj: Q^l → Q^li

(x0, x1, . . . , xl−1)→(xi_j,1, xi_j,2, . . . , xi_j,l[j])