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 type*G(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 group*G(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 type*G(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 type*G(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 partition***λ*=*(λ*_{1}*, λ*2*, λ*3*, . . .)*is a decreasing sequence of non-negative inte-
gers. We define the length of*λ*to be the smallest integer*(λ)*such that*λ**i* =0 for
all*i > (λ). We write*|*λ*| :=

*i*≥1*λ**i* and we say that*λ*is a partition of*m, for some*
*m*∈Z*>0*, if*m*= |*λ*|. We set*n(λ)*:=

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

[*λ*] :=

*(i, j )*|*i*≥1, 1≤*j*≤*λ**i*

*.*

A node*x* =*(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, then*j*=*λ**i*.

The conjugate partition of*λ*is the partition*λ*^{}defined by
*λ*^{}* _{k}*:=#{

*i*|

*i*≥1 such that

*λ*

*i*≥

*k*}

*.*Obviously,

*λ*

^{}

_{1}=

*(λ). The set of nodes ofλ*

^{}satisfies

*(i, j )*∈ [*λ*^{}] ⇔*(j, i)*∈ [*λ*]*.*
Note that if*(i, λ*_{i}*)*is a removable node of*λ, thenλ*^{}_{λ}

*i*=*i.*Moreover, we have
*n(λ)*=

*i*≥1

*(i*−1)λ*i*=1
2

*i*≥1

*(λ*^{}* _{i}*−1)λ

^{}

*=*

_{i}*i*≥1

*λ*^{}* _{i}*
2

*.*

If*x*=*(i, j )*∈ [*λ*]and*μis 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.

For*μ*=*λ, the above formula becomes the classical hook length formula (giving the*
length of the hook of*λ*that *x* *belongs to). Moreover, we define the content ofx* to
be the difference*j*−*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)*λ*with the contents of the nodes of (the “lower rim” of)*λ*^{}.

* Lemma 2.2 Letλ*=

*(λ*

_{1}

*, λ*

_{2}

*, . . .)*

*be a partition and let*

*k*

*be an integer such that*1≤

*k*≤

*λ*

_{1}

*. Letqandybe two indeterminates. Then we have*

1
*(q*^{λ}^{1}*y*−1)·

1≤*i*≤*λ*^{}_{k}

*q*^{λ}^{i}^{−}^{i}^{+}^{1}*y*−1
*q*^{λ}^{i}^{−}^{i}*y*−1

= 1

*(q*^{−}^{λ}^{}^{k}^{+}^{k}^{−}^{1}*y*−1)·

*k*≤*j*≤*λ*1

*q*^{−}^{λ}^{}^{j}^{+}^{j}^{−}^{1}*y*−1
*q*^{−}^{λ}^{}^{j}^{+}^{j}*y*−1

*.*

**2.3 Let** *l* and *n* be positive integers. An *l*-partition (or multipartition) of *n* is an
ordered*l-tuple λ*=

*(λ*

^{0}

*, λ*

^{1}

*, . . . , λ*

^{l}^{−}

^{1}

*)*of partitions such that

0≤*s*≤*l*−1|*λ** ^{s}*| =

*n. We*denote by

*Π*

_{n}*the set of*

^{l}*l-partitions ofn. The empty multipartition, denoted by*

**∅, is**an

*l-tuple of empty partitions. If*=

**λ***(λ*

^{0}

*, λ*

^{1}

*, . . . , λ*

^{l}^{−}

^{1}

*)*∈

*Π*

_{n}*, we denote by*

^{l}

**λ**^{}the

*l-partition(λ*

^{0}

*, λ*

^{1}

*, . . . , λ*

^{l}^{−}

^{1}

*).*

**2.4 Let** *R* be a commutative domain with 1. Fix elements*q, Q*_{0}*, . . . , Q*_{l}_{−}_{1}of *R,*
and assume that*q* is invertible in*R. Set q*:=*(Q*0*, . . . , Q*_{l}_{−}1;*q). The Ariki–Koike*
*algebraH*^{q}*n*is the unital associative*R-algebra with generatorsT*0*, T*1*, . . . , T** _{n−}*1and
relations:

*(T*0−*Q*0*)(T*0−*Q*1*)· · ·(T*0−*Q**l*−1*)*=0,

*(T** _{i}*−

*q)(T*

*+1)=0 for 1≤*

_{i}*i*≤

*n*−1,

*T*

_{0}

*T*

_{1}

*T*

_{0}

*T*

_{1}=

*T*

_{1}

*T*

_{0}

*T*

_{1}

*T*

_{0}

*,*

*T*_{i}*T*_{i}_{+}_{1}*T** _{i}*=

*T*

_{i}_{+}

_{1}

*T*

_{i}*T*

_{i}_{+}

_{1}for 1≤

*i*≤

*n*−2,

*T*_{i}*T** _{j}*=

*T*

_{j}*T*

*for 0≤*

_{i}*i < j*≤

*n*−1 with

*j*−

*i >*1.

*The last three relations are the braid relations satisfied byT*0*, T*1*, . . . , T** _{n−}*1.

The Ariki–Koike algebra*H*^{q}*n*is a deformation of the group algebra of the complex
reflection group*G(l,*1, n)=*(*Z*/ l*Z*)*S*n*. Ariki and Koike [3] have proved that*H*^{q}*n*is
a free*R-module of rankl*^{n}*n! = |G(l,*1, n)|(see [2, Proposition 13.11]). In addition,
when*R*is a field, they have constructed a simple*H*^{q}*n*-module*V**^{λ}*, with character

*χ*

*, for each*

^{λ}*l-partition*of

**λ***n*(see [2, Theorem 13.6]). These modules form a complete set of non-isomorphic simple modules in the case where

*H*

^{q}*n*is 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 that*H*^{q}*n* is split semisim-
ple. This happens, for example, when *q, Q*_{0}*, . . . , Q*_{l}_{−}_{1} are indeterminates and
*R*=Q*(q, Q*_{0}*, . . . , Q*_{l}_{−}_{1}*).*

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

whenever all*Q**i*’s are invertible in*R. 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 of*H*^{q}*n*.

**Definition 2.6 Suppose that**Ris a field and thatH*n*^{q}*is split semisimple. The Schur*
*elements ofH*^{q}*n**are the elementss*_{λ}*(q)ofRsuch that*

*τ* =

* λ*∈

*Π*

_{n}

^{l}1
*s***λ***(q)χ*^{λ}*.*

**2.7 The Schur elements of the Ariki–Koike algebra***H*^{q}*n* 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*=

*(λ*

^{0}

*, λ*

^{1}

*, . . . , λ*

^{l}^{−}

^{1}

*)be anl-partition ofn. Then*

*s*

_{λ}*(q)*=

*(*−1)

^{n(l}^{−}

^{1)}

*(Q*

_{0}

*Q*

_{1}· · ·

*Q*

_{l}_{−}

_{1}

*)*

^{−}

^{n}*q*

^{−}

^{α(λ)}·

0≤*s*≤*l*−1

*(i,j )*∈[*λ** ^{s}*]

*Q**s* *h*^{λ}_{i,j}^{s}^{,λ}^{s}

*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}^{−}^{i}*Q**t*−*Q**s*

·

*(i,j )*∈[*λ** ^{s}*]

*q*^{j}^{−}^{i}*Q**s*−*q*^{λ}^{t}^{1}*Q**t*
1≤*k*≤*λ*^{t}_{1}

*q*^{j}^{−}^{i}*Q** _{s}*−

*q*

^{k}^{−}

^{1}

^{−}

^{λ}

^{t}

^{k}*Q*

_{t}*q*

^{j}^{−}

^{i}*Q*

*−*

_{s}*q*

^{k−λ}

^{t}

^{k}*Q*

_{t}
*.*

The formula by Geck, Iancu and Malle is more symmetric, and describes the Schur
*elements in terms of beta numbers. If λ*=

*(λ*

^{0}

*, λ*

^{1}

*, . . . , λ*

^{l}^{−}

^{1}

*)*is an

*l-partition ofn,*

*then the length of*

*is*

**λ***(λ)*=max{

*(λ*

^{s}*)*|0≤

*s*≤

*l*−1}. Fix an integer

*L*such that

*L*≥

*(λ). TheL-beta numbers forλ*

*are the integers*

^{s}*β*

_{i}*=*

^{s}*λ*

^{s}*+*

_{i}*L*−

*i*for

*i*= 1, . . . , L. Set

*B*

*= {*

^{s}*β*

_{1}

^{s}*, . . . , β*

_{L}*}for*

^{s}*s*=0, . . . , l−

**1. The matrix B**=

*(B*

^{s}*)*0≤

*s*≤

*l*−1is called the

*L-symbol of*

**λ.****Theorem 2.9 Let*** λ*=

*(λ*

^{0}

*, . . . , 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)^{a}^{L}*q*^{b}^{L}*(q*−1)^{−}^{n}*(Q*_{0}*Q*_{1}*. . . Q*_{l}_{−}_{1}*)*^{−}^{n}*ν*_{λ}*/δ*_{λ}*,*
*where*

*ν** λ*=

0≤*s<t*≤*l*−1

*(Q** _{s}*−

*Q*

_{t}*)*

^{L}0≤*s, t*≤*l*−1

*b**s*∈*B*^{s}

1≤*k*≤*b**s*

*q*^{k}*Q** _{s}*−

*Q*

_{t}*and*

*δ** λ*=

0≤s<t≤l−1

*(b**s**,b**t**)*∈*B** ^{s}*×

*B*

^{t}*q*^{b}^{s}*Q**s*−*q*^{b}^{t}*Q**t*

0≤s≤l−1

1≤i<j≤L

*q*^{β}^{i}^{s}*Q**s*−*q*^{β}^{s}^{j}*Q**s*

*.*

As the reader may see, in both formulas above, the factors of*s*_{λ}*(q)*are not obvious.

**Hence, it is not obvious for which values of q the Schur element***s***λ***(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 of*H*^{q}*n*.
This formula is also symmetric.

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

* Theorem 3.2 Letλ*=

*(λ*

^{0}

*, λ*

^{1}

*, . . . , λ*

^{l}^{−}

^{1}

*)be anl-partition ofn. Set*¯:=

**λ**0≤*s*≤*l*−1*λ*^{s}*.*
*Then*

*s***λ***(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*_{s}*Q*^{−}_{t}^{1}−1
*.*

(1)
*Since the total number of nodes in λis equal ton, the above formula can be rewritten*

*as follows:*

*s**λ**(q)*=*(*−1)^{n(l}^{−}^{1)}*q*^{−}^{n(}^{λ)}^{¯}

·

0≤*s*≤*l*−1

*(i,j )*∈[*λ** ^{s}*]

*h*^{λ}_{i,j}^{s}^{,λ}^{s}

*q*

0≤*t*≤*l*−1, t=*s*

*q*^{h}

*λs ,λt*

*i,j* *Q*_{s}*Q*^{−}_{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 terms*q*^{−}^{n(}^{λ)}^{¯} and*q*^{−}* ^{α(λ)}*.

**Lemma 3.4 Let****λ**be anl-partition ofn. We have

*α(λ)*+

0≤*s<t*≤*l*−1

*i*≥1

*λ*^{s}_{i}^{}*λ*^{t}_{i}^{}=*n( λ).*¯

*Proof Following the definition of the conjugate partition, we have λ*¯

^{}

*=*

_{i}0≤*s*≤*l*−1*λ*^{s}_{i}^{}*,*
for all*i*≥1. Therefore,

*n( λ)*¯ =

*i*≥1

* λ*¯

^{}

*2*

_{i}

=

*i≥*1

0≤*s*≤*l*−1*λ*^{s}_{i}^{}
2

=

*i*≥1 0≤*s*≤*l*−1

*λ*^{s}_{i}^{}
2

+

0≤*s<t*≤*l*−1

*λ*^{s}_{i}^{}*λ*^{t}_{i}^{}

=*α(λ)*+

0≤*s<t*≤*l*−1

*i*≥1

*λ*^{s}_{i}^{}*λ*^{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}^{≥}

^{1}

^{λ}

^{s}

^{i}^{}

^{λ}

^{t}

^{i}^{}

*Q*

^{|}

_{s}

^{λ}

^{t}^{|}

*Q*

^{|}

_{t}

^{λ}

^{s}^{|}

*(i,j )*∈[*λ** ^{s}*]

*q*^{h}

*λs ,λt*

*i,j* *Q*_{s}*Q*^{−}_{t}^{1}−1

·

*(i,j )*∈[*λ** ^{t}*]

*q*^{h}

*λt ,λs*

*i,j* *Q*_{t}*Q*^{−}_{s}^{1}−1

*.* (3)

**3.5 We will proceed by induction on the number of nodes of***λ** ^{s}*. We do not need
to do the same for

*λ*

*, because the symmetric formula for the Schur elements given by Theorem2.9implies the following: if*

^{t}*is the multipartition obtained from*

**μ***λ*by exchanging

*λ*

*and*

^{s}*λ*

*, then*

^{t}*X*^{λ}_{st}*(Q*_{s}*, Q*_{t}*)*=*X*^{μ}_{st}*(Q*_{t}*, Q*_{s}*).*

If*λ** ^{s}*= ∅, then

*X*_{st}*^{λ}* =

*(i,j )*∈[*λ** ^{t}*]

*q*^{j}^{−}^{i}*Q** _{t}*−

*Q*

_{s}=*Q*^{|λ}_{s}^{t}^{|}

*(i,j )*∈[*λ** ^{t}*]

*q*^{j−i}*Q*_{t}*Q*^{−}_{s}^{1}−1

=*Q*^{|}_{s}^{λ}^{t}^{|}

1≤i≤λ^{t}_{1}

1≤*j*≤*λ*^{t}_{i}

*q*^{j}^{−}^{i}*Q**t**Q*^{−}_{s}^{1}−1

=*Q*^{|}_{s}^{λ}^{t}^{|}

1≤*i*≤*λ*^{t}_{1}

1≤j≤λ^{t}_{i}

*q*^{λ}^{t}^{i}^{−}^{j}^{+}^{1}^{−}^{i}*Q*_{t}*Q*^{−}_{s}^{1}−1

=*Q*^{|}_{s}^{λ}^{t}^{|}

*(i,j )*∈[*λ** ^{t}*]

*q*^{h}

*λt ,λs*

*i,j* *Q**t**Q*^{−}_{s}^{1}−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. If

*λ*

*= ∅, then there exists*

^{s}*i*such that

*(i, λ*

^{s}

_{i}*)*is a removable node of

*λ*

*. Let*

^{s}*be the multipartition defined by*

**ν***ν*_{i}* ^{s}*:=

*λ*

^{s}*−1,*

_{i}*ν*

_{j}*:=*

^{s}*λ*

^{s}*for all*

_{j}*j*=

*i,*

*ν*

*:=*

^{t}*λ*

*for all*

^{t}*t*=

*s.*

Then[*λ** ^{s}*] = [

*ν*

*] ∪ {*

^{s}*(i, λ*

^{s}

_{i}*)*}. Since (3) holds for

*X*

_{st}*and*

^{ν}*X*^{λ}* _{st}*=

*X*

^{ν}*·*

_{st}*q*^{λ}^{s}^{i}^{−}^{i}*Q**s*−*q*^{λ}^{t}^{1}*Q**t*
1≤*k*≤*λ*^{t}_{1}

*q*^{λ}^{s}^{i}^{−}^{i}*Q** _{s}*−

*q*

^{k}^{−}

^{1}

^{−}

^{λ}

^{t}

^{k}^{}

*Q*

_{t}*q*

^{λ}

^{s}

^{i}^{−}

^{i}*Q*

*−*

_{s}*q*

^{k}^{−}

^{λ}

^{t}

^{k}^{}

*Q*

_{t}
*,*

it is enough to show that (to simplify notation, from now on set*λ*:=*λ** ^{s}*and

*μ*:=

*λ*

*):*

^{t}*q*^{λ}^{i}^{−}^{i}*Q** _{s}*−

*q*

^{μ}^{1}

*Q*

_{t}1≤*k*≤*μ*_{1}

*q*^{λ}^{i}^{−}^{i}*Q** _{s}*−

*q*

^{k}^{−}

^{1}

^{−}

^{μ}^{}

^{k}*Q*

_{t}*q*

^{λ}

^{i}^{−}

^{i}*Q*

*−*

_{s}*q*

^{k−μ}^{}

^{k}*Q*

_{t}=*q*^{−}^{μ}^{}^{λi}*Q**t*

*q*^{λ}^{i}^{−}^{i}^{+}^{μ}^{}^{λi}^{−}^{λ}^{i}^{+}^{1}*Q**s**Q*^{−}_{t}^{1}−1

·*A*·*B,* (4)
where

*A*:=

1≤k≤λ*i*−1

*q*^{λ}^{i}^{−}^{i}^{+}^{μ}^{}^{k}^{−}^{k}^{+}^{1}*Q*_{s}*Q*^{−}_{t}^{1}−1
*q*^{λ}^{i}^{−i+μ}^{}^{k}^{−k}*Q*_{s}*Q*^{−}_{t}^{1}−1
and

*B*:=

1≤*k*≤*μ*^{}

*λi*

*q*^{μ}^{k}^{−}^{k}^{+}^{λ}^{}^{λi}^{−}^{λ}^{i}^{+}^{1}*Q*_{t}*Q*^{−}_{s}^{1}−1
*q*^{μ}^{k}^{−}^{k}^{+}^{λ}^{}^{λi}^{−}^{λ}^{i}*Q*_{t}*Q*^{−}_{s}^{1}−1

*.*

Note that, since*(i, λ*_{i}*)*is a removable node of*λ, we haveλ*^{}_{λ}

*i*=*i. We have*

*A*=*q*^{λ}^{i}^{−}^{1}

1≤*k*≤*λ**i*−1

*q*^{λ}^{i}^{−}^{i}*Q** _{s}*−

*q*

^{k}^{−}

^{1}

^{−}

^{μ}^{}

^{k}*Q*

_{t}*q*

^{λ}

^{i}^{−}

^{i}*Q*

*−*

_{s}*q*

^{k}^{−}

^{μ}^{}

^{k}*Q*

_{t}*.*

Moreover, by Lemma2.2, for*y*=*q*^{i}^{−}^{λ}^{i}*Q*_{t}*Q*^{−}_{s}^{1}*,*we obtain
*B*= *(q*^{μ}^{1}^{+}^{i}^{−}^{λ}^{i}*Q**t**Q*^{−}_{s}^{1}−1)

*(q*^{−}^{μ}^{}^{λi}^{+}^{λ}^{i}^{−}^{1}^{+}^{i}^{−}^{λ}^{i}*Q*_{t}*Q*^{−}_{s}^{1}−1)

·

*λ** _{i}*≤

*k*≤

*μ*

_{1}

*q*^{−}^{μ}^{}^{k}^{+}^{k}^{−}^{1}^{+}^{i}^{−}^{λ}^{i}*Q**t**Q*^{−}_{s}^{1}−1
*q*^{−}^{μ}^{}^{k}^{+}^{k}^{+}^{i}^{−}^{λ}^{i}*Q**t**Q*^{−}_{s}^{1}−1

*,*

i.e.,

*B*=*Q*^{−}_{t}^{1}*q*^{μ}^{}^{λi}^{−}^{λ}^{i}^{+}^{1} *(q*^{λ}^{i}^{−}^{i}*Q**s*−*q*^{μ}^{1}*Q**t**)*
*(q*^{μ}^{}^{λi}^{−}^{λ}^{i}^{+}^{1}^{+}^{λ}^{i}^{−}^{i}*Q*_{s}*Q*^{−}_{t}^{1}−1)

·

*λ**i*≤k≤μ1

*q*^{λ}^{i}^{−}^{i}*Q** _{s}*−

*q*

^{k}^{−}

^{1}

^{−}

^{μ}^{}

^{k}*Q*

_{t}*q*

^{λ}

^{i}^{−}

^{i}*Q*

*−*

_{s}*q*

^{k}^{−}

^{μ}^{}

^{k}*Q*

_{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 that*q, Q*_{0}*, . . . , Q*_{l}_{−}_{1}are indeterminates and*R*=Q*(q, Q*_{0}*, . . . , Q*_{l}_{−}_{1}*).*

Then the resulting “generic” Ariki–Koike algebra*H*^{q}*n* is split semisimple. Now as-
sume that*θ*:Z[*q*^{±}^{1}*, Q*^{±}_{0}^{1}*, . . . , Q*^{±}_{l}_{−}^{1}_{1}] →Kis a specialization and letKH^{q}*n* be the
specialized algebra, whereKis any field. Note that for all* λ*∈

*Π*

_{n}*, we have*

^{l}*s*

**λ***(q)*∈ Z[

*q*

^{±}

^{1}

*, Q*

^{±}

_{0}

^{1}

*, . . . , Q*

^{±}

_{l}_{−}

^{1}

_{1}]. Then by [12, Theorem 7.2.6],KH

^{q}*n*is (split) semisimple if and only if, for all

*∈*

**λ***Π*

_{n}*, we have*

^{l}*θ (s*

_{λ}*(q))*=0. From this, we can deduce the following:

* Theorem 4.2 (Ariki) Assume that*K

*is a field. The algebra*KH

^{q}*n*

*is (split) semisimple*

*if and only ifθ (P (q))*=

*0, where*

*P (q)*=

1≤*i*≤*n*

1+*q*+ · · · +*q*^{i}^{−}^{1}

0≤*s<t*≤*l*−1

−*n<k<n*

*q*^{k}*Q** _{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*+ · · · +*q*^{i}^{−}^{1}*)*=0, then we have
*θ (*[*h*^{λ}_{1,n−i+}^{0}^{,λ}^{0} _{1}]*q**)*=0 for * λ*=

*((n),*∅

*, . . . ,*∅

*)*∈

*Π*

_{n}*. Thus, for this*

^{l}*l-partition, we*have

*θ (s*

_{λ}*(q))*=0, which implies thatKH

*n*

**is not semisimple.**

^{q}(b) If there exist 0≤*s < t*≤*l*−1 and 0≤*k < n*such that*θ (q*^{k}*Q** _{s}*−

*Q*

_{t}*)*=0, then we have

*θ (q*

^{h}*λs ,λt*

1,n−k*Q**s**Q*^{−}_{t}^{1}−1)=0 for* λ*∈

*Π*

_{n}*such that*

^{l}*λ*

*=*

^{s}*(n),λ*

*= ∅. We have*

^{t}*θ (s*

**λ***(q))*=0 andKH

^{q}*n*is not semisimple.

(c) If there exist 0≤*s < t*≤*l*−1 and−*n < k <*0 such that*θ (q*^{k}*Q**s*−*Q**t**)*=0, then
we have*θ (q*^{h}

*λt ,λs*

1,n+k*Q**t**Q*^{−}_{s}^{1}−1)=0 for* λ*∈

*Π*

_{n}*such that*

^{l}*λ*

*= ∅,*

^{s}*λ*

*=*

^{t}*(n). Again,*we have

*θ (s*

**λ***(q))*=0 andKH

^{q}*n*is not semisimple.

Conversely, ifKH^{q}*n*is not semisimple, then there exists* λ*∈

*Π*

_{n}*such that*

^{l}*θ (s*

_{λ}*(q))*= 0. As for all 0≤

*s, t*≤

*l*−1 and

*(i, j )*∈ [

*λ*

*], we have−*

^{s}*n < h*

^{λ}

_{i,j}

^{s}

^{,λ}

^{t}*< n, we conclude*

that*θ (P (q))*=0.

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

Let*u*be an indeterminate. Let*r*∈Z*>0*and let*r*_{0}*, . . . , r*_{l}_{−}_{1}**be any integers. Set r**:=

*(r*_{0}*, . . . , r*_{l}_{−}_{1}*)*and*η** _{l}*:=exp(2√

−1π/ l). For all*i*=0, . . . , l−1, we set*m** _{i}*:=

*r*

_{i}*/r*

**and we define m**:=

*(m*

_{0}

*, . . . , m*

_{l}_{−}

_{1}

*)*∈Q

*. Assume that*

^{l}*R*=Z[

*q*

^{±}

^{1}

*, Q*

^{±}

_{0}

^{1}

*, . . . , Q*

^{±}

_{l}_{−}

^{1}

_{1}] and consider the morphism

*θ*:*R*→Z[*η** _{l}*]

*u*

^{±}

^{1}

such that*θ (q)*=*u** ^{r}* and

*θ (Q*

_{j}*)*=

*η*

^{j}

_{l}*u*

^{r}*for*

^{j}*j*=0,1, . . . , l−1. We will denote by

*H*

^{m,r}*n*the specialization of the Ariki–Koike algebra

*H*

^{q}*n*via

*θ. The algebraH*

*n*

**is**

^{m,r}*called a cyclotomic Ariki–Koike algebra. It is defined over*Z[*η** _{l}*][

*u*

^{±}

^{1}]and has a pre- sentation as follows:

• generators:*T*_{0}*, T*_{1}*, . . . , T*_{n}_{−}_{1},

• relations:

*T*_{0}−*u*^{r}^{0}

*T*_{0}−*η*_{l}*u*^{r}^{1}

· · ·

*T*_{0}−*η*_{l}^{l}^{−}^{1}*u*^{r}^{l−1}

=0,
*T** _{j}*−

*u*

^{r}*(T** _{j}*+1)=0 for

*j*=1, . . . , n−1 and the braid relations symbolized by the diagram

^{4} · · ·

*T*_{0} *T*_{1} *T*_{2} *T*_{n}_{−}_{1}

.

We set*K*:=Q*(η**l**). The algebra* *K(u)H*^{m,r}*n* , which is obtained by extension of
scalars to*K(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,r}*n* -modules Irr(K(u)*H*^{m,r}*n* *)*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 simple*K(u)H*^{m,r}*n* -module*V*^{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 of*V**^{λ}*in

*u, that is, the negative of the valuation ofθ (s*

**λ***(q)). We*

**call this number the a-value of**

*combinatorially: Let*

**λ. By [11, §5.5], this value may be easily computed***∈*

**λ***Π*

_{n}*and let*

^{l}*s*∈Z

*>0*such that

*s > (λ). Let*Bbe the shifted

**m-symbol of**

*of size*

**λ***s*∈Z

*>0*. This is the

*l-tuple(B*

^{0}

*, . . . ,B*

^{l}^{−}

^{1}

*)*where, for all

*j*=0, . . . , l−1 and for all

*i*=1, . . . , s+ [

*m*

*](where[*

_{j}*m*

*]denotes the integer part of*

_{j}*m*

*), we have*

_{j}B^{j}* _{i}* =

*λ*

^{j}*−*

_{i}*i*+

*s*+

*m*

*and B*

_{j}*= B*

^{j}

^{j}

_{s}_{+[}

_{m}*j*]*, . . . ,B*^{j}_{1}
*.*

Write

*κ*1*(λ)*≥*κ*2*(λ)*≥ · · · ≥*κ**h**(λ)*

for the elements ofBwritten in decreasing order (allowing repetitions), where*h*=
*ls*+

0≤*j*≤*l*−1[*m** _{j}*]. Let

*κ*

_{m}*(λ)*=

*(κ*

_{1}

*(λ), . . . , κ*

_{h}*(λ))*∈Q

^{h}_{≥}

_{0}and define

*n*

_{m}*(λ)*:=

1≤*i*≤*h*

*(i*−1)κ*i**(λ).*

Then, by [11, Proposition 5.5.11], the a-value of* λ*is

**a**

^{(m,r)}*(λ)*=

*r*

*n***m***(λ)*−*n***m***(***∅***)*
*,*
where**∅**denotes the empty multipartition.

Generalizing the dominance order for partitions, we will write*κ***m***(λ)κ***m***(μ)*if
*κ*_{m}*(λ)*=*κ*_{m}*(μ)*and

1≤*i*≤*t**κ*_{i}*(λ)*≥

1≤*i*≤*t**κ*_{i}*(μ)*for all*t*≥1. The following result
[11, Proposition 5.5.16] will be useful in then next sections:

**Proposition 4.4 Assume that****λ**and**μ**are 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 of**λ:**

* Proposition 4.5 Letλ*∈

*Π*

_{n}

^{l}

**. The a-value of****λ**is**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}^{,λ}* ^{t}* +

*m*

*s*−

*m*

*t*

*,*0

*.*

**4.6 We now consider another type of specialization. Let***v*_{0}*, . . . , v*_{l}_{−}_{1}be any integers.

Let*k*be a subfield ofCand let*η*be a primitive root of unity of order*e >*1. Assume
that*R*=Z[*q*^{±}^{1}*, Q*^{±}_{0}^{1}*, . . . , Q*^{±}_{l}_{−}^{1}_{1}]and consider the morphism

*θ*:*R*→*k(η)*

such that*θ (q)*=*η* and*θ (Q*_{j}*)*=*η*^{v}* ^{j}* for

*j*=0,1, . . . , l−1. By Theorem 4.2, the specialized algebra

*k(η)H*

^{q}*n*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*∈Π

*n*

^{l}*, M∈*Irr(k(η)

*H*

^{q}*n*

*)*

be the associated decomposition matrix (see [12, §7.4]), which relates the irreducible
representations of the split semisimple Ariki–Koike algebra*H*^{q}*n* and the specialized
Ariki–Koike algebra*k(η)H*^{q}*n*. We are interested in the classification of the blocks
of defect 0. That is, we want to classify the *l-partitions* * λ*∈

*Π*

_{n}*which are alone in their blocks in the decomposition matrix. These correspond to the modules*

^{l}*V*

*which remain projective and irreducible after the specialization*

^{λ}*θ. By [17, Lemme*2.6] (see also [12, Theorem 7.2.6]), these elements are characterized by the property that

*θ (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}

^{,λ}*+*

^{t}*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 algebra***H*^{m,r}*n* defined in Sect.4.3, re-
placing from now on the indeterminate*u*by the indeterminate*q*(following the usual
notation). Let*θ*:Z[*η** _{l}*][

*q*

^{±}

^{1}] →

*K(η)*be a specialization such that

*θ (q)*=

*η*∈C

^{∗}. We obtain a specialized Ariki–Koike algebra

*K(η)H*

^{m,r}*n*. The relations between the generators are the usual braid relations together with the following ones:

*T*0−*η*^{r}^{0}

*T*0−*η*_{l}*η*^{r}^{1}

· · ·

*T*0−*η*^{l}_{l}^{−}^{1}*η*^{r}^{l}^{−}^{1}

=0,
*T** _{j}*−

*η*

^{r}*(T** _{j}*+1)=0 for

*j*=1, . . . , n−1.

Let

*D*= *V**^{λ}*:

*M*

* λ*∈Π

*n*

^{l}*, M∈*Irr(K(η)

*H*

^{m,r}*n*

*)*

be the associated decomposition matrix (see [12, §7.4]). The matrix*D* relates the
irreducible representations of the split semisimple Ariki–Koike algebra*K(q)H*^{m,r}*n*

and the specialized Ariki–Koike algebra*K(η)H*^{m,r}*n* . The goal of this section is to
study the form of this matrix in full generality.

First assume that*η*is not a root of unity. Then, for all 0≤*i*=*j* ≤*l*−1, we have
*η*^{i}_{l}^{−}^{j}*η*^{r}^{i}^{−}^{r}* ^{j}*=

*η*

^{rd}for all*d*∈Z*>0*. By the criterion of semisimplicity due to Ariki (Theorem4.2), this
implies that the algebra*K(η)H*^{m,r}*n* is split semisimple, and thus *D* is the identity
matrix. Hence, from now, one may assume that*η*is a primitive root of unity of order
*e >*1. Then there exists*k*∈Z*>0*such that gcd(k, e)=1 and*η*=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 of*D. Set I*:= {0,1, . . . , l−1}. There is a partition

**I**=**I**1 **I**2 · · · **I***p*

such that

• for all 1≤*α < β*≤*p,(i, j )*∈**I***α*×**I***β* and*d*∈Z*>0*, we have
*η** ^{rd}*−

*η*

^{i}

_{l}^{−}

^{j}*η*

^{r}

^{i}^{−}

^{r}*=0*

^{j}• for all 1≤*α*≤*p*and*(i, j )*∈**I***α*×**I***α*, there exists*d*∈Z*>0*such that
*η** ^{rd}*−

*η*

^{i}

_{l}^{−}

^{j}*η*

^{r}

^{i}^{−r}

*=0.*

^{j}For all*j*=1, . . . , p, we set*l*[*j*] := |I*j*|**and we consider I***j*as an ordered set
**I***j*=*(i*_{j,1}*, i*_{j,2}*, . . . , i*_{j,l}_{[}_{j}_{]}*)* with*i*_{j,1}*< i*_{j,2}*<*· · ·*< i*_{j,l}_{[}_{j}_{]}*.*
We define

*π**j*: Q* ^{l}* → Q

^{l}

^{i}*(x*0*, x*1*, . . . , x**l*−1*)*→*(x**i*_{j,1}*, x**i*_{j,2}*, . . . , x**i*_{j,l[j]}*)*