Salim Rostam

Cyclotomic Yokonuma-Hecke algebras are cyclotomic quiver Hecke algebras

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Introduction

Iwahori-Hecke algebras appeared first in the context of finite Chevalley groups, as centralizer algebras of the induced representation from the trivial representation on a Borel subgroup. Since then, both their structure and their representation theory have been intensively studied. In particular, they have been defined independently as deformations of the group algebra of finite Coxeter groups. Further, connections with many other objects and theories have been established (this includes the theory of quantum groups, knot theory, etc.). Many variations and generalisations of the "classical" Iwahori-Hecke algebras have yet been defined. Among these, the following ones will catch our interest in this paper: Ariki-Koike algebras, Yokonuma-Hecke algebras and finally cyclotomic quiver Hecke algebras.

In their seminal paper, Ariki and Koike [ArKo] introduced and studied generalisations of Iwahori-Hecke algebras of type A and B: the so called Ariki-Koike algebras. It turns out that these algebras can be seen as cyclotomic quotients of affine Hecke algebras of type A, and also as deformations of the group algebra of the complex reflection group G(ℓ, 1, n). Such deformations, in the general case of complex reflection groups, have been defined by Broué, Malle and Rouquier [BMR]: in that sense, Ariki-Koike algebras are the Hecke algebras associated with G(ℓ, 1, n).

One of the most important results on the representation theory of Ariki-Koike algebras is Ariki's categorification theorem [Ar] (proving a conjecture of Lascoux, Leclerc and Thibon [LLT]). This result implies that the decomposition matrices of such algebras can be computed using the canonical bases for quantum groups in affine type A. Partially motivated by this work, Khovanov and Lauda [KhLau1,[START_REF] Khovanov | A diagrammatic approach to categorification of quantum groups II[END_REF] and Rouquier [Rou] have independently defined the same algebra, known as quiver Hecke algebra or KLR algebra, in order to categorify quantum groups. In fact, they have shown that we have the following algebra isomorphism:

U - A (g) ≃ [Proj(H)] = β∈Q + [Proj(H β )]
where U - A (g) is the integral form of the negative half of the quantum group Uq(g) associated with a symmetrizable Cartan datum, with A = Z[q, q -1 ], the set Q + is the positive root lattice associated with the Cartan datum, the algebra H = ⊕ β∈Q + H β is the quiver Hecke algebra corresponding to this Cartan datum and [Proj(H)] is the Grothendieck group of the additive category of finitely generated graded projective H-modules. A cyclotomic version of this theorem was conjectured in [START_REF] Khovanov | A diagrammatic approach to categorification of quantum groups I[END_REF], the cyclotomic categorification conjecture, which was later proved by Kang and Kashiwara [KanKa]. More specifically, for each dominant weight Λ the algebra H has a cyclotomic quotient H Λ which categorifies the corresponding highest weight module V (Λ).

A big step towards understanding these cyclotomic quiver Hecke algebras was made by Brundan and Kleshchev [BrKl] and independentely by Rouquier [Rou]. The first two authors proved that cyclotomic Hecke algebras of type A are particular cases of cyclotomic quiver Hecke algebras; a similar result in the affine case has also been proved by Rouquier. Brundan and Kleshchev also noticed that the cyclotomic Hecke algebra inherits the natural grading of the cyclotomic quiver Hecke algebra, whose grading allows in particular to study the graded representation theory of cyclotomic Hecke 1 algebras (see for example [START_REF] Brundan | Graded decomposition numbers for cyclotomic Hecke algebras[END_REF]). Moreover, they established a connection between the cyclotomic categorification theorem in type A for quiver Hecke algebras and Ariki's categorification theorem.

On the other hand, Yokonuma [Yo] defined the Yokonuma-Hecke algebras in the study of finite Chevalley groups: they arise once again as centralizer algebras of the induced representation from the trivial representation, but now on a maximal unipotent subgroup. Their natural presentation in type A has been transformed since (see [START_REF] Juyumaya | Sur les nouveaux générateurs de l'algèbre de Hecke H(G, U, 1)[END_REF][START_REF] Juyumaya | Markov traces on the Yokonuma-Hecke algebra[END_REF][START_REF] Juyumaya | Braid relations in the Yokonuma-Hecke algebra[END_REF][START_REF] Chlouveraki | Representation theory of the Yokonuma-Hecke algebra[END_REF]), and the one we use here is given in [ChPou]. Similarly to Ariki-Koike algebras, Yokonuma-Hecke algebras of type A can be viewed as deformations of the group algebra of G (d, 1, n). This deformation, unlike in the Ariki-Koike case, "respects" the wreath product structure G(d, 1, n) ≃ (Z/dZ) ≀ Sn. The representation theory of Yokonuma-Hecke algebras has been first studied by Thiem [Th1,[START_REF] Thiem | Unipotent Hecke algebras of GLn(Fq)[END_REF][START_REF] Thiem | A skein-like multiplication algorithm for unipotent Heke algebras[END_REF], while a combinatorial approach to this representation theory in type A has been given in [START_REF] Chlouveraki | Representation theory of the Yokonuma-Hecke algebra[END_REF][START_REF] Chlouveraki | Markov traces on affine and cyclotomic Yokonuma-Hecke algebras[END_REF]. In this latter paper [START_REF] Chlouveraki | Markov traces on affine and cyclotomic Yokonuma-Hecke algebras[END_REF], Chlouveraki and Poulain d'Andecy introduced and studied generalisations of these algebras: the affine Yokonuma-Hecke algebras and their cyclotomic quotients, which generalise affine Hecke algebras of type A and Ariki-Koike algebras respectively. The interest in Yokonuma-Hecke algebras has grown recently: in [CJKL], the authors defined a link invariant from Yokonuma-Hecke algebras which is stronger than the famous ones (such as the HOMFLYPT polynomial) obtained from classical Iwahori-Hecke algebras of type A and Ariki-Koike algebras.

The first aim of this paper is to show that cyclotomic Yokonuma-Hecke algebras are particular cases of cyclotomic quiver Hecke algebras, generalising thus the results of Brundan and Kleshchev [BrKl]; our goal will be achieved in Section 4 with Theorem 4.1. In fact, every known result on the cyclotomic quiver Hecke algebra can be applied to the cyclotomic Yokonuma-Hecke algebra: this includes the cyclotomic categorification theorem and the existence of a graded representation theory. In order to prove the main result Theorem 4.1, our strategy is to define inverse algebra homomorphisms, by constructing the images of the defining generators of the corresponding algebras; we proceed as in [BrKl]. In particular, we give an argument to avoid doing the calculations of [BrKl] again, see for instance Remark 2.7. In Section 5, as in [BrKl] we consider the degenerate case: we will define the degenerate cyclotomic Yokonuma-Hecke algebras and show that they are cyclotomic quiver Hecke algebras as well (Theorem 5.15). Finally, in Section 6 we relate our results to an isomorphism obtained in [Lu, JaPA, PA]. To that end, we first prove a general result on (cyclotomic) quiver Hecke algebras, when the quiver is the disjoint union of full subquivers. Although similar situations have already been studied in the literature (see for instance [START_REF] Shan | On the center of quiver-Hecke algebras[END_REF]Theorem 3.15] or [START_REF] Rouquier | Categorification and cyclotomic rational double affine Hecke algebras[END_REF]Lemma 5.33]), the result we obtain in our context seems to be new and of independent interest.

We give now a brief overview of this article. Given a base field F and d, n ∈ N * , we first define in Section 1 the cyclotomic Yokonuma-Hecke algebra Y Λ d,n (q) where q ∈ F \ {0, 1} has order e ∈ N ≥2 ∪ {∞} in F × and Λ is a finitely-supported e-tuple of non-negative integers. We also define the quiver Hecke algebra Hα(Q) in full generality, where α is a composition of n indexed by a set K and Q = (Q k,k ′ ) k,k ′ ∈K is a matrix satisfying some properties. Considering particular cases for the matrix Q = (Q k,k ′ ), we define the cyclotomic quiver Hecke algebra H Λ α (Γ) where Λ is now a finitely-supported tuple indexed by K and Γ is a loop-free quiver without any multiple edge; in particular, with the exception of Section 6 we consider the case where Λ is given by d copies of the previous e-tuple Λ and Γ is the quiver given by d disjoint copies of the (cyclic) quiver Γe with e vertices used in [BrKl]. We begin Section 2 by considering in Y Λ d,n (q) a natural system {e(α)} α|= ed n of pairwise orthogonal central idempotents. Then, we define the "quiver Hecke generators" of Y Λ α (q) := e(α)Y Λ d,n (q) and we check that they verify the defining relations of H Λ α (Γ). In Section 3 we define the "Yokonuma-Hecke generators" of H Λ α (Γ) and again check the corresponding defining relations. We conclude the proof of the main theorem in Section 4 by showing that we have defined inverse algebra homomorphisms. We justify in Section 5 that the isomorphism of Theorem 4.1 remains true for the degenerate cyclotomic Yokonuma-Hecke algebra Y Λ d,n (1) that we define in §5.1. We end the section with Corollary 5.17, which states that, under some conditions, the algebras Y Λ d,n (q) and Y Λ d,n (1) are isomorphic. Finally, we begin Section 6 by some quick calculations about the minimal length representatives of the cosets of a Young subgroup in the symmetric group on n letters Sn. The main results of the section are given in Theorems 6.26 and 6.30, where we prove an isomorphism about (cyclotomic) "disjoint quiver" Hecke algebra. We end the paper with Theorem 6.35: we show that we recover the isomorphism Y Λ d,n (q) ≃ ⊕ λ|= d n Matm λ H Λ λ (q) of [Lu, JaPA, PA], where m λ :=

n! λ 1 !•••λ d !
and the algebra H Λ λ (q) is a tensor product of cyclotomic Hecke algebras.

Cyclotomic Yokonuma-Hecke algebras

Let Λ = (Λi)i∈I ∈ N (I) be a weight; we assume that its level ℓ(Λ) := i∈I Λi verifies ℓ(Λ) > 0. The cyclotomic Yokonuma-Hecke algebra of type A, denoted by Y Λ d,n (q), is the unitary associative F -algebra generated by the elements g1, . . . , gn-1, t1, . . . , tn, X1 (1.1) subject to the following relations:

t d a = 1, (1.2) tat b = t b ta, (1.3) t b ga = gat sa(b) , (1.4)
g 2 a = q + (q -1)gaea, (1.5)

gag b = g b ga ∀|a -b| > 1, (1.6) ga+1gaga+1 = gaga+1ga, (1.7)
where sa is the transposition (a, a + 1) ∈ Sn and ea := 1 d j∈J t j a t -j a+1 , together with the following relations:

X1g1X1g1 = g1X1g1X1, (1.8) X1ga = gaX1 ∀a > 1, (1.9) X1t b = t b X1, (1.10)
and finally the cyclotomic one:

i∈I (X1 -q i ) Λ i = 0. (1.11)
Note that the presentation comes from [START_REF] Chlouveraki | Markov traces on affine and cyclotomic Yokonuma-Hecke algebras[END_REF], excepting the normalisation in (1.5) which was used in [ChPou]. In particular, it comes from (1.11) that X1 is invertible in Y Λ d,n (q). When d = 1, we recover the cyclotomic Hecke algebra H Λ n (q) of [BrKl]; it is the cyclotomic Yokonuma-Hecke algebra Y Λ 1,n (q). In particular, the element ea becomes 1. We write g H a (respectively X H 1 ) for the element ga (resp. X1) when d = 1, that is, considered in H Λ n (q). Following [START_REF] Chlouveraki | Markov traces on affine and cyclotomic Yokonuma-Hecke algebras[END_REF], we define inductively Xa+1 for a ∈ {1, . . . , n -1} by qXa+1 := gaXaga (1.12) (note that the q comes from our different normalisation in (1.5)). As for X1, we introduce the notation X H a to denote Xa in the case d = 1. The family {t1, . . . , tn, X1, . . . , Xn} is commutative and we have the following equalities:

gaX b = X b ga ∀b = a, a + 1, (1.13) gaXa+1 = Xaga + (q -1)Xa+1ea, (1.14) Xa+1ga = gaXa + (q -1)Xa+1ea. (1.15)
The proof of the following result is the same as in [START_REF] Chlouveraki | Markov traces on affine and cyclotomic Yokonuma-Hecke algebras[END_REF]Proposition 4.7], where we write gw := ga 1 • • • ga r for a reduced expression w = sa 1 • • • sa r ∈ Sn; by Matsumoto's theorem (see for instance [START_REF] Geck | Characters of finite Coxeter groups and Iwahori-Hecke algebras[END_REF]Theorem 1.2.2]) the value of gw does not depend on the choice of the reduced expression, since the generators ga satisfy the same braid relations as the sa ∈ Sn.

Proposition 1.1. The algebra Y Λ d,n (q) is a finite-dimensional F -vector space and a generating family is given by the elements

gwX u 1 1 • • • X un n t v 1 1 • • • t vn n for w ∈ Sn, ua ∈ {0, . . . , ℓ(Λ) -1} and va ∈ J. Remark 1.2. The above family is even an F -basis of Y Λ d,n (q)
, see [START_REF] Chlouveraki | Markov traces on affine and cyclotomic Yokonuma-Hecke algebras[END_REF]Theorem 4.15].

Cyclotomic quiver Hecke algebras

In their landmark paper [START_REF] Khovanov | A diagrammatic approach to categorification of quantum groups I[END_REF], starting from a quiver without loop and multiple edges Khovanov and Lauda have constructed the so-called "quiver Hecke algebra". Independently, Rouquier [Rou] made a similar construction, where the underlying object is a matrix

Q = (Q k,k ′ ) k,k ′ ∈K
which contains the case of quivers. Here, we will first give this definition of [Rou], which generality will be used in Section 6 only, and then specialise to the case of quivers.

General definition

Let K be a set, A a commutative ring, u and v two indeterminates and

Q = (Q k,k ′ ) k,k ′ ∈K a matrix
satisfying the following conditions:

• the polynomials Q k,k ′ ∈ A[u, v] verify Q k,k ′ (u, v) = Q k ′ ,k (v, u) for all k, k ′ ∈ K; • we have Q k,k = 0 for all k ∈ K.
Let α |=K n. The quiver Hecke algebra Hα(Q) associated with (Q k,k ′ ) k,k ′ ∈K at α is the unitary associative A-algebra with generating set {e(k)} k∈K α ∪ {y1, . . . , yn} ∪ {ψ1, . . . , ψn-1} (1.16) and the following relations:

k∈K α e(k) = 1, (1.17) e(k)e(k ′ ) = δ k,k ′ e(k), (1.18) yae(k) = e(k)ya, (1.19) ψae(k) = e(sa • k)ψa, (1.20) yay b = y b ya, (1.21) ψay b = y b ψa if b = a, a + 1, (1.22) ψaψ b = ψ b ψa if |a -b| > 1, (1.23) ψaya+1e(k) = (yaψa + 1)e(k) if ka = ka+1, yaψae(k) if ka = ka+1, (1.24) ya+1ψae(k) = (ψaya + 1)e(k) if ka = ka+1, ψayae(k) if ka = ka+1, (1.25) ψ 2 a e(k) = Q ka,k a+1 (ya, ya+1)e(k), (1.26) ψa+1ψaψa+1e(k) = ψaψa+1ψae(k) + Q ka ,k a+1 (ya,y a+1 )-Q k a+2 ,k a+1 (y a+2 ,y a+1 ) ya-y a+2 e(k) if ka = ka+2, ψaψa+1ψae(k) otherwise. (1.27) Remark 1.3. Let k ∈ K α , a ∈ {1, . . . , n -2} and let P := Q ka,k a+1 ; the relation (1.27) for ka = ka+2 is: ψa+1ψaψa+1e(k) = ψaψa+1ψae(k) + P (ya, ya+1) -P (ya+2, ya+1) ya -ya+2 e(k).
(1.28)

Writing P (u, v) = m≥0 u m Pm(v), we get that the right side of (1.28) is well-defined and is an element of A [ya, ya+1, ya+2]e(k). Remark 1.4. The generators in [Rou] are given by 1 k := e(k), x a,k := yae(k) and τ a,k := ψae(k).

When the set K is finite, in a similar way we can define the quiver Hecke algebra Hn(Q) as the unitary associative A-algebra with generating set {e(k)} k∈K n ∪ {y1, . . . , yn} ∪ {ψ1, . . . , ψn-1} (1.29) together with the same relations (1.18)-(1.27), where (1.17) is replaced by:

k∈K n e(k) = 1. (1.30)
Defining for α |=K n the central idempotent e(α) := k∈K α e(k) ∈ Hn(Q), we have:

e(α)Hn(Q) ≃ Hα(Q), thus: Hn(Q) ≃ α|= K n Hα(Q) (1.31)
(note that this equality can be seen as a definition of Hn(Q) is K is infinite).

For each w ∈ Sn, we now choose a reduced expression w = sa 1 • • • sa r and we set:

ψw := ψa 1 • • • ψa r ∈ Hn(Q). (1.32)
Although this reduced expression dependence differs from the usual case of Hecke algebras (for instance), we are still able to give a basis of Hn(Q). In fact, we have the following theorem ( [START_REF] Rouquier | 2-Kac-Moody algebras[END_REF]Theorem 3.7], [KhLau1, Theorem 2.5]).

Theorem 1.5. The family {ψwy

r 1 1 • • • y rn n e(k) : w ∈ Sn, ra ∈ N, k ∈ K α } is a basis of the free A-module Hα(Q).
We conclude this paragraph by introducing cyclotomic quotients of these quiver Hecke algebras; let α |=K n and Λ = (Λ k ) k∈K ∈ N (K) be a weight. Following [START_REF] Kang | Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras[END_REF]§4.1], we define the cyclotomic quiver Hecke algebra H Λ α (Q) at α as the quotient of the quiver Hecke algebra Hα(Q) by the following relations: (1.33) where cm ∈ A with c Λ k 1 = 1. Similarly, if K is finite we define the cyclotomic quiver Hecke algebra H Λ n (Q). Theorem 1.6 [START_REF] Kang | Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras[END_REF]Corollary 4.4]). The A-module H Λ α (Q) is finitely generated.

Λ k 1 m=0 cmy m 1 e(k) = 0 ∀k ∈ K α ,

Case of quivers

With the exception of Section 6, throughout this paper our matrix Q will always be associated with a loop-free quiver without multiple edges and with vertex set K. If Γ is such a quiver, following [START_REF] Rouquier | 2-Kac-Moody algebras[END_REF]§3.2.4] we associate the following matrix (1.34) where:

(Q k,k ′ ) k,k ′ ∈K : Q k,k ′ (u, v) :=              0 if k = k ′ , 1 if k -k ′ , v -u if k → k ′ , u -v if k ← k ′ , -(u -v) 2 if k ⇆ k ′ ,
• we write k -k ′ when k = k ′ and neither (k, k ′ ) or (k ′ , k) is an edge of Γ; • we write k → k ′ when (k, k ′ ) is an edge of Γ and (k ′ , k) is not; • we write k ← k ′ when (k ′ , k) is an edge of Γ and (k, k ′ ) is not; • we write k ⇆ k ′ when both (k, k ′ ) and (k ′ , k) are edges of Γ.
Moreover, we define: Hα(Γ) := Hα(Q), and if K is finite we also set Hn(Γ) := Hn(Q). Note that, in the setting of (1.34), the defining relations (1.26) and (1.27) become in Hα(Γ):

ψ 2 a e(k) =              0 if ka = ka+1, e(k) if ka -ka+1, (ya+1 -ya)e(k) if ka → ka+1, (ya -ya+1)e(k) if ka ← ka+1, (ya+1 -ya)(ya -ya+1)e(k) if ka ⇆ ka+1, (1.35) ψa+1ψaψa+1e(k) =        (ψaψa+1ψa -1)e(k) if ka+2 = ka → ka+1, (ψaψa+1ψa + 1)e(k) if ka+2 = ka ← ka+1, (ψaψa+1ψa + 2ya+1 -ya -ya+2)e(k) if ka+2 = ka ⇆ ka+1, ψaψa+1ψae(k)
otherwise.

(1.36)

We now give a remarkable fact about quiver Hecke algebras; its proof only requires a simple check of the different defining relations.

Proposition 1.7. Let Γ be a loop-free quiver without multiple edges with vertex set K. The quiver Hecke algebra Hα(Γ) is Z-graded through:

deg e(k) = 0, deg yae(k) = 2, deg ψae(k) = -c ka,k a+1 , where C = (c k,k ′ ) k,k ′ ∈K is the Cartan matrix of Γ, defined by: c k,k ′ :=        2 if k = k ′ , 0 if k -k ′ , -1 if k → k ′ or k ← k ′ , -2 if k ⇆ k ′ .
(1.37)

We now define the quivers which will be particularly important to us; we recall that I = Z/charq(F )Z and J = Z/dZ. We denote by Γe the following quiver:

• the vertices are the elements of I;

• for each i ∈ I there is a directed edge from i to i + 1.

In particular, for i, i ′ ∈ I:

• we have i → i ′ if and only if i ′ = i + 1 and i = i ′ + 1; • we have i ← i ′ if and only if i = i ′ + 1 and i ′ = i + 1;
• we have i ⇆ i ′ if and only if i = i ′ + 1 and i ′ = i + 1 (thus this only happens when e = 2);

• we have i -i ′ if and only if i = i ′ , i ′ ± 1.
We give some examples in Figure 1. We now define the quiver

Γ := j∈J Γe (1.38)
given by d disjoint copies of Γe. Hence, our quiver Γ is described in the following way:

• the vertices are the elements of K := K = I × J;

• for each (i, j) ∈ K there is a directed edge from (i, j) to (i + 1, j).

Quiver Γ 2 0 ⇆ 1 Quiver Γ 4 0 1 2 3 Quiver Γ ∞ • • • -2 -1 0 1 2 • • • Figure 1: Three examples of quivers Γ e
In particular, there is an arrow between (i, j) and (i ′ , j ′ ) in Γ if and only if there is an arrow between i and i ′ in Γe and j = j ′ . Moreover, the set K is finite if and only if e is finite.

We consider the diagonal action of Sn on

K n ≃ I n × J n , that is, σ • (i, j) := (σ • i, σ • j).
We will need the following notation:

I α := {i ∈ I n : ∃j ∈ J n , (i, j) ∈ K α }, J α := {j ∈ J n : ∃i ∈ I n , (i, j) ∈ K α }.
The sets I α and J α are finite and stable under the action of Sn; note that K α is included in I α × J α (we don't have the equality in general).

Let now Λ := (Λ k ) k∈K ∈ N (K) be a weight. The cyclotomic quiver Hecke algebra H Λ α (Γ) is given by the quotient of the quiver Hecke algebra Hα(Γ) by the relations:

y Λ k 1 1 e(k) = 0, ∀k ∈ K α ; (1.39)
note that this is indeed a particular case of (1.33). Note that the grading described in Proposition 1.7 is compatible with this quotient.

Remark 1.8. The cyclotomic Khovanov-Lauda algebra of [BrKl] is the quiver Hecke algebra H Λ α (Γe), that is, the algebra H Λ α (Γ) for d = 1. We write e H (i), y H a and ψ H a the generators of H Λ α (Γe); the reason for this notation will appear in §2.1.

The proof of the following result is the same as in [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Lemma 2.1].

Lemma 1.9. The elements ya ∈ H Λ α (Γ) are nilpotent for a ∈ {1, . . . , n}. As a corollary, together with Theorem 1.5 we recover a particular case of Theorem 1.6: in particular the algebra H Λ α (Γ) is a finite-dimensional F -vector space. Remark 1.10. Let us assume e < ∞; the algebra H Λ n (Γ) is hence finite-dimensional. However, comparing with Remark 1.2, it does not seem that easy to get its F -dimension; an answer will be given with Theorem 4.1 and (4.2).

2 Quiver Hecke generators of Y Λ α (q)

Let Λ = (Λ k ) k∈K ∈ N (K) be a weight; we assume that ℓ(Λ) = k∈K Λ k verifies ℓ(Λ) > 0. Moreover, we suppose that for any i ∈ I and j, j ′ ∈ J, we have: Λi,j = Λ i,j ′ =: Λi.

In particular, we will write Λ as well for the weight (Λi)i∈I .

In this section, our first task is to define some central idempotents e(α) ∈ Y Λ d,n (q) with α e(α) = 1 for α |=K n. We will then prove the following theorem.

Theorem 2.1. For any α |=K n, we can construct an explicit algebra homomorphism:

ρ : H Λ α (Γ) → Y Λ α (q)
, where Y Λ α (q) := e(α) Y Λ d,n (q). Note that Y Λ α (q) is a unitary algebra (if not reduced to {0}), with unit e(α). To define this algebra homomorphism, it suffices to define the images of the generators (1.16) and check that they verify the defining relations of the cyclotomic quiver Hecke algebra: the same strategy was used by Brundan and Kleshchev in [BrKl] for d = 1.

For a generator g of H Λ α (Γ), we will use as well the notation g for the corresponding element that we will define in Y Λ α (q). There will be no possible confusion since we will work with elements of Y Λ α (q).

Definition of the images of the generators

We define now our different "quiver Hecke generators".

Image of e(i, j)

Let M be a finite-dimensional Y Λ d,n (q)-module. Each Xa acts on M as an endomorphism of the finite-dimensional F -vector space (see Proposition 1.1); in particular, by (1.11) the eigenvalues of X1 can be written q i for i ∈ I. Hence, applying [START_REF] Cui | Modular representations and branching rules for affine and cyclotomic Yokonuma-Hecke algebras[END_REF]Lemma 5.2] we know that the eigenvalues of each Xa are of the form q i for i ∈ I. Concerning the ta, by (1.2) (they are diagonalizable and) their eigenvalues are dth roots of unity.

As the elements of the family {Xa, ta} 1≤a≤n pairwise commute, using Cayley-Hamilton theorem we can write M as the direct sum of its weight spaces (simultaneous generalized eigenspaces)

M (i, j) := v ∈ M : (Xa -q ia ) N v = (ta -ξ ja )v = 0 for all 1 ≤ a ≤ n (2.1)
for (i, j) ∈ I n × J n , where N ≫ 0 and ξ is the given primitive dth root of unity in F that we considered at the very beginning of Section 1. Observe that some M (i, j) may be reduced to zero; in fact, only a finite number of them are non-zero.

Remark 2.2. The element ea acts on M (i, j) as 0 if ja = ja+1 and as 1 if ja = ja+1.

We can now consider the family of projections {e(k)} k∈K n associated with the decomposition

M = ⊕ k∈K n M (k): the element e(k) is the projection onto M (k) along ⊕ k ′ =k M (k ′ ), in particular e(k) 2 = 0 and if k = k ′ then e(k)e(k ′ ) = 0.
Moreover, only a finite number of e(k) are non-zero.

As the e(k) are polynomials in X1, . . . , Xn,t1,. . . tn (in fact e(k) is the product of commuting projections onto the corresponding generalized eigenspaces of Xa and ta), they belong to Y Λ d,n (q). Remark 2.3. The above polynomials do not depend on the finite-dimensional Y Λ d,n (q)-module M . We are now able to define our central idempotents. We set, for α |=K n:

e(α) := k∈K α e(k)
(since K α is a Sn-orbit, the element e(α) is indeed central). Though we will not use this fact, we can notice that according to [START_REF] Poulain D'andecy | Invariants for links from classical and affine Yokonuma-Hecke algebras[END_REF]Corollary 3.2] and [LyMa], the subalgebras Y Λ α (q) := e(α) Y Λ d,n (q) which are not reduced to zero are the blocks of the Yokonuma-Hecke algebra Y Λ d,n (q); see also [START_REF] Cui | Modular representations and branching rules for affine and cyclotomic Yokonuma-Hecke algebras[END_REF]§6.3]. For d = 1, we recover the element eα of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF](1.3)].

We will sometimes need the following elements:

e(α)(i) := j∈J α e(α)e(i, j) = j∈J α (i,j)∈K α e(i, j), e(α)(j) := i∈I α e(α)e(i, j) = i∈I α (i,j)∈K α e(i, j).
(2.2) For d = 1, we recover with e(α)(i) the element e(i) of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§4.1]; we denote it by e H (i). Finally, note that:

e(α)(i) • e(α)(j) = e(α)(j) • e(α)(i) = e(i, j).
From now, unless mentioned otherwise, we always work in Y Λ α (q); every relation should be multiplied by e(α) and we write e(i) (respectively e(j)) for e(α)(i) (resp. e(α)(j)).

We give now a few useful lemmas.

Lemma 2.4. If 1 ≤ a < n and j ∈ J α is such that ja = ja+1 then we have:

g 2 a e(j) = qe(j), gaXa+1e(j) = Xagae(j), Xa+1gae(j) = gaXae(j).

Proof. We deduce it the from relations (1.5), (1.14) and (1.15) and from eae(j) = 0 (since ja = ja+1, see Remark 2.2).

For the next lemma, we should compare with [START_REF] Jacon | An isomorphism Theorem for Yokonuma-Hecke algebras and applications to link invariants[END_REF](15)].

Lemma 2.5. For 1 ≤ a < n and j ∈ J α we have gae(j) = e(sa • j)ga.

Proof. Let M := Y Λ α (q). Given the relation (1.4), we see that ga maps M (j) to M (sa • j). Fix now j ∈ J α and let j ′ ∈ J α and v ∈ M (j ′ ). If j ′ = j then we get: gae(j)v = e(sa • j)gav (= gav), and if j ′ = j, since gav ∈ M (sa • j ′ ) we have:

gae(j)v = e(sa • j)gav (= 0).
Hence, e(sa • j)ga and gae(j) coincide on each M (j ′ ) for j ′ ∈ J α thus coincide on M = ⊕ j ′ M (j ′ ) and we conclude since M = Y Λ α (q) is a unitary algebra.

Corollary 2.6. Let 1 ≤ a < n and j ∈ J α ; if ja = ja+1 then ga and e(j) commute.

Remark 2.7 (About Brundan and Kleshchev's proof -I). Let 1 ≤ a < n; if j ∈ J α verifies ja = ja+1, the following relations are verified between gae(j) and the X b e(j) for 1 ≤ b ≤ n in the unitary algebra e(j) Y Λ α (q)e(j):

g 2 a = q + (q -1)ga, qXa+1 = gaXaga, gaX b = X b ga ∀b = a, a + 1, gaXa+1 = Xaga + (q -1)Xa+1, Xa+1ga = gaXa + (q -1)Xa+1.
These are exactly the relations (1.5), (1.12)-(1.15) for H Λ n (q) (that is, for d = 1). Hence, when these only elements, together with the e H (i) for any i ∈ I α , and these only relations, together with those involving the e H (i), are used to prove any relation ( * ) in [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§4] (in H Λ α (q)), we claim that the same proof in Y Λ α (q) holds for ( * ), involving gae(j) instead of g H a , the elements X ±1 b e(j) instead of X H b ±1 and e(i, j) instead of e H (i). If j verifies in addition ja+1 = ja+2, we can add to the previous list the element ga+1e(j), which stands for g H a+1 . Lemma 2.8. For 1 ≤ a < n and (i, j) ∈ K α such that ja = ja+1 we have gae(i)e(j) = e(sa •i)gae(j), that is, gae(i, j) = e(sa • (i, j))ga.

Proof. Given Lemma 2.4, we show as in Lemma 2.5 that gae(i)e(j) = e(sa • i)gae(j); we get the final result applying Lemma 2.5.

Image of y a

We are now able to define the elements ya for 1 ≤ a ≤ n. We saw in Lemma 1.9 that these elements (in H Λ α (Γ)) are nilpotent, hence we define the following elements of Y Λ α (q) for 1 ≤ a ≤ n:

ya := i∈I α (1 -q -ia Xa)e(i) ∈ Y Λ α (q).
We can notice that i (q ia -Xa)e(i) is the nilpotent part of the Jordan-Chevalley decomposition of Xa; in particular, ya is nilpotent. As a consequence, we will be able to make calculations in the ring F [[y1, . . . , yn]] of power series in the commuting variables y1, . . . , yn. We will sometimes also need the following element: ya(i) := q ia (1 -ya), which verifies: ya(i)e(i) = Xae(i).

(2.3)

We end this paragraph with a lemma.

Lemma 2.9. For j ∈ J α such that ja = ja+1 we have:

gaya+1e(j) = yagae(j), ya+1gae(j) = gayae(j).
Proof. Indeed, gaya+1e(j) = i (ga -q -i a+1 gaXa+1)e(i, j) and applying Lemmas 2.4 and 2.8 we get:

gaya+1e(j) = i∈I α (1 -q -i a+1 Xa)e(sa • i)gae(j) = i∈I α
(1 -q -ia Xa)e(i)gae(j) gaya+1e(j) = yagae(j).

Image of ψ a

We first define some invertible elements Qa(i, j) ∈ F [[ya, ya+1]] × for 1 ≤ a < n and (i, j) ∈ K α by:

Qa(i, j) :=                               1 -q + qya+1 -ya if ia = ia+1, (ya(i) -qya+1(i))/(ya(i) -ya+1(i)) if ia -ia+1, (ya(i) -qya+1(i))/(ya(i) -ya+1(i)) 2 if ia → ia+1, q ia if ia ← ia+1, q ia /(ya(i) -ya+1(i)) if ia ⇆ ia+1, if ja = ja+1, f a,j if ja = ja+1,
where f a,j ∈ {1, q} is given for ja = ja+1 by:

f a,j := q if ja < ja+1, 1 if ja > ja+1,
with < being a total ordering on J = Z/dZ ≃ {1, . . . , d}. For d = 1 or for ja = ja+1, the power series Qa(i, j) coincides with the definition given at [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF](4.36)].

Remark 2.10. The elements Qa(i, j) depend only on (ia, ia+1) and (ja, ja+1). Moreover, as in [BrKl] the explicit expression of Qa(i, j) for ja = ja+1 does not really matter; only its properties are essential, for instance, those in Lemma 2.13. Remark 2.11. The scalar f a,j is only an artefact: if q admits a square root q 1/2 in F , we can simply set f a,j := q 1/2 . Finally, we give an easy lemma about these f a,j .

Lemma 2.12. If ja = ja+1 then f a,j f a,sa•j = q.

We introduce a notation for a power series

Q ∈ F [[y1, . . . , yn]]: if σ ∈ Sn is a permutation, we denote by Q σ the power series Q σ (y1, . . . , yn) := Q(y σ -1 (1) , . . . , y σ -1 (n) ); note that (Q σ ) ρ = Q σρ (we get a right action).
We will use later the following properties verified by Qa(i, j) (see [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF](4.35)]), where

Q σ (i, j) := Q(i, j) σ .
Lemma 2.13. We have:

Q sa a+1 (i, j) = Q s a+1 a (sa+1sa • (i, j)), Q s a+1 a (i, j) = Q sa a+1 (sasa+1 • (i, j)).
Proof. We check only the first equality, the second being straightforward considering (i ′ , j ′ ) := sasa+1 • (i, j). For i ∈ I, let us write ya(i) := q i (1 -ya); in particular, we have ya(ia) = ya(i).

Noticing that sa+1sa = (a, a + 2, a + 1), we get for ja+1 = ja+2:

Qa(sa+1sa • (i, j)) =              1 -q + qya+1 -ya if ia+1 = ia+2, (ya(ia+1) -qya+1(ia+2))/(ya(ia+1) -ya+1(ia+2)) if ia+1 -ia+2, (ya(ia+1) -qya+1(ia+2))/(ya(ia+1) -ya+1(ia+2)) 2 if ia+1 → ia+2, q i a+1 if ia+1 ← ia+2, q i a+1 /(ya(ia+1) -ya+1(ia+2)) if ia+1 ⇆ ia+2,
and we conclude using ya(ia+1) s a+1 = ya(ia+1) and ya+1(ia+2

) s a+1 = ya+2(ia+2). If ja+1 = ja+2, we have Qa(sa+1sa • (i, j)) = f a,(a,a+2,a+1)•j = q if ja+1 < ja+2, 1 if ja+1 > ja+2,
and this is exactly f a+1,j .

We now introduce the following element of Y Λ α (q):

Φa := ga + (1 -q) (i,j)∈K α ia =i a+1 ja=j a+1 1 -XaX -1 a+1 -1 e(i, j) + (i,j)∈K α ia=i a+1 ja=j a+1 e(i, j),
where (1 -XaX -1 a+1 )

-1 e(k) denotes the inverse of (1 -XaX -1 a+1 )e(k) in e(k) Y Λ α (q)e(k). Note that this element is indeed invertible, since for k = (i, j) with ia = ia+1 its only eigenvalue 1 -q ia-i a+1 is non-zero, thanks to the definition of I. In particular, we have:

Φae(j) = gae(j) if ja = ja+1.
For d = 1 we get the "intertwining element" defined in [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§4.2], and we write it Φ H a . Remark 2.14. Though we will not need this until Section 3, we define now the power series Pa(i, j) ∈ F [[ya, ya+1]] for 1 ≤ a < n and (i, j) ∈ K α by:

Pa(i, j) :=    1 if ia = ia+1, (1 -q)(1 -ya(i)ya+1(i) -1 ) -1 if ia = ia+1, if ja = ja+1, 0 if ja = ja+1.
For d = 1 or for ja = ja+1 we recover the definition given at [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF](4.27)]. Moreover, we have the following equality:

Φa = k∈K α (ga + Pa(k)) e(k).
(2.4) Indeed, the only non-obvious fact to check is Pa(i, j)e(i, j)

= (1-q)(1 -XaX -1 a+1 ) -1 e(i, j) if ia = ia+1
and ja = ja+1, but this is clear by (2.3). We will also use the following equality (the same one as in Lemma 2.13):

P sa a+1 (i, j) = P s a+1 a (sa+1sa • (i, j)).
(2.5)

Lemma 2.15. We have the following properties:

Φae(j) = e(sa • j)Φa, (2.6) Φae(i, j) = e(sa • (i, j))Φa, (2.7) ΦaX b = X b Φa ∀b = a, a + 1, (2.8) Φay b = y b Φa ∀b = a, a + 1, (2.9) ΦaQ b (k) = Q b (k)Φa ∀|b -a| > 1, (2.10) ΦaΦ b = Φ b Φa ∀|b -a| > 1.
(2.11)

Proof. We will use results from [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Lemma 4.1] (which is this lemma for d = 1).

(2.6) Using Lemma 2.5, it is clear if ja = ja+1 since then Φae(j) = gae(j) = e(sa • j)ga = e(sa • j)Φa. Using Corollary 2.6 it is clear if ja = ja+1 since e(j) commutes with every term in the definition of Φa.

(2.7) If ja = ja+1, we claim that the relation comes applying (2.6) and Remark 2.7 on the equality

Φ H a e H (i) = e H (sa • i)Φ H a .
If ja = ja+1 it follows directly from Lemma 2.8. (2.8) Straightforward using (1.13) and since e(i, j) are polynomials in X1, . . . , Xn.

(2.9) Using (2.2), (2.7) and (2.8) we get:

Φay b = i,j (1 -q -i b X b )e(sa • (i, j))Φa = i (1 -q -i b X b )e(sa • i)Φa = y b Φa, since (sa • i) b = i b . (2.10) Since Q b (k) ∈ F [[y b , y b+1 ]
] and b = a, a + 1 and b + 1 = a, a + 1 it follows from (2.9).

(2.11) Let us write Φ ′ a := Φa -ga. Using (1.13) and Lemma 2.8 we get:

Φ ′ a g b = g b     (1 -q) ia =i a+1 ja=j a+1 (1 -XaX -1 a+1 ) -1 e(s b • (i, j)) + ia=i a+1 ja=j a+1 e(s b • (i, j))     = g b Φ ′ a ,
and exchanging a and b we get

gaΦ ′ b = Φ ′ b ga. Noticing that Φ ′ a Φ ′ b = Φ ′ b Φ ′ a (we don't use here |b -a| > 1) and using (1.6), we get ΦaΦ b = (ga + Φ ′ a )(g b + Φ ′ b ) = gag b + Φ ′ a g b + gaΦ ′ b + Φ ′ a Φ ′ b = g b ga + g b Φ ′ a + Φ ′ b ga + Φ ′ b Φ ′ a = (g b + Φ ′ b )(ga + Φ ′ a ) = Φ b Φa.
We are now ready to define our elements ψa for 1 ≤ a < n:

ψa := k∈K α ΦaQa(k) -1 e(k) ∈ Y Λ α (q).
As usual, we write ψ H a for ψa when d = 1, and this element ψ H a corresponds with the ψa of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§4.3]. Note finally that for j ∈ J α we have:

ψae(j) = f -1 a,j gae(j) if ja = ja+1.

Check of the defining relations

We now check the defining relations (1.17)-(1.25), (1.35)-(1.36) and (1.39) for the elements we have just defined. The idea is the following: when an element e(i, j) lies in a relation to check, if ja = ja+1 then we get immediately the result by Remark 2.7 rewriting the same proof as [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Theorem 4.2], and if ja = ja+1 then it will be easy (at least, easier than in [BrKl]) to prove the relation. Recall that we always work in Y Λ α (q) (in particular every relation should be multiplied by e(α)) and we write e(i) and e(j) without any (α).

(1.39) We do exactly the same proof as in H Λ α (q). Let (i, j) ∈ K α and set M := Y Λ α (q); recall that the action of X1 on M is given by the action of e(α)X1. By (1.11) we have i∈I (X1 -q i ) Λ i = 0, hence:

i∈I (X1 -q i ) Λ i e(i) = 0.
(2.12)

As an endomorphism of M (i), the element (X1 -q i ) Λ i is invertible if i = i1 since its only eigenvalue (q i 1 -q i ) Λ i is non-zero (note that Λi may be equal to 0). This means that there exist elements

(X1 -q i ) -Λ i e(i) such that (X1 -q i ) Λ i e(i) • (X1 -q i ) -Λ i e(i) = e(i).
Hence, multiplying by all these inverses, the equation (2.12) becomes:

(X1 -q i 1 ) Λ i 1 e(i) = 0.
Finally, since y

Λ i 1 1 = i ′ ∈I α (1 -q -i ′ 1 X1) Λ i 1 e(i ′
) we obtain:

y Λ i 1 1 e(i, j) = (1 -q -i 1 X1) Λ i 1 e(i, j) = 0.
(1.17) Straightforward from the definition of the e(k) for k ∈ K α .

(1.18) Idem.

(1.19) Straightforward since ya and e(k) both lie in the commutative subalgebra generated by t1, . . . , tn and X1, . . . , Xn.

(1.20) Straightforward by (2.7) and since Qa(k ′ ) and e(k ′ ) commute with e(k).

(1.21) True since {Xa}a is commutative.

(1.22) True by (2.9).

(1.23) Let |a -b| > 1. We have, using (1.20), (2.10) and (2.11):

ψaψ b = k ΦaQa(k) -1 e(k)ψ b = k ΦaQa(k) -1 ψ b e(s b • k) = k ΦaQa(k) -1 Φ b Q b (s b • k) -1 e(s b • k) = k Φ b Q b (s b • k) -1 ΦaQa(k) -1 e(s b • k).
Hence, noticing that (see Remark 2.10)

Qa(k) = Qa(s b • k) and Q b (s b • k) = Q b (sas b • k) we get: ψaψ b = k Φ b Q b (s b • k) -1 ψae(s b • k) = k Φ b Q b (s b • k) -1 e(sas b • k)ψa ψaψ b = ψ b ψa.
(1.24) First, if k = (i, j) verifies ja = ja+1 then by Remark 2.7 we get from:

ψ H a y H a+1 e H (i) = (y H a ψ H a + 1)e H (i) if ia = ia+1, y H a ψ H a e H (i) if ia = ia+1,
the following equality:

ψaya+1e(i, j) = (yaψa + 1)e(i, j) if ia = ia+1 and ja = ja+1, yaψae(i, j) if ia = ia+1 and ja = ja+1.

Hence it remains to deal with the case ja = ja+1 (and no condition on i). Using Lemma 2.9 we get:

ψaya+1e(i, j) = ΦaQa(i, j) -1 ya+1e(i, j) = f -1 a,j gaya+1e(i, j) = f -1 a,j yagae(i, j) = yaΦaQa(i, j) -1 e(i, j) ψaya+1e(i, j) = yaψae(i, j).
Finally, we have proved:

ψaya+1e(k) = (yaψa + 1)e(k) if ka = ka+1, yaψae(k) if ka = ka+1,
which is exactly (1.24).

(1.25) Similar.

Remark 2.16. Thanks to relations (1.22), (1.24) and (1.25), given f ∈ F [[y1, . . . , yn]] and k ∈ K α such that ka = ka+1 we have f ψae(k) = ψaf sa e(k). In particular, this holds if ja = ja+1 with k = (i, j).

(1.35) Once again, the result is straightforward if ja = ja+1 using Remark 2.7. Let us then suppose ja = ja+1; hence, necessarily we have ka -ka+1 so we have to prove ψ 2 a e(k) = e(k). We have:

ψ 2 a e(k) = ψae(sa • k)ψa = ΦaQa(sa • k) -1 e(sa • k)ψa = f -1 a,sa•j gaΦaQa(k) -1 e(k) ψ 2 a e(k) = (f a,sa•j f a,j ) -1 g 2 a e(k).
Applying Lemmas 2.4 and 2.12 we find ψ 2 a e(k) = e(k) (recall e(k) = e(i)e(j) = e(j)e(i)) thus we are done.

(1.36) If ja = ja+1 = ja+2, we get the result using Remark 2.7. Let us then suppose that we are not in that case: we have to prove ψa+1ψaψa+1e(k) = ψaψa+1ψae(k). We will intensively use (1.20); note also that:

Φae(i, j) =      ga + (1 -q)(1 -XaX -1 a+1 )
-1 e(i, j) if ia = ia+1 and ja = ja+1, (ga + 1)e(i, j) if ia = ia+1 and ja = ja+1, gae (i, j) otherwise (ja = ja+1), and:

ψae(i, j) = ΦaQa(i, j) -1 e(i, j) if ja = ja+1, f -1 a,j gae(i, j) if ja = ja+1.
It is convenient to introduce some notation. The couple (i, j) shall only be modified by the action of sa or sa+1, hence we only write ((ia, ia+1, ia+2), (ja, ja+1, ja+2)) for (i, j). Moreover, for clarity we forget comas and only write the indexation, substituting 0 to a; thus, ((ia, ia+1, ia+2), (ja, ja+1, ja+2)) becomes (( 012), ( 012)). Finally, as Sn acts diagonally on I × J we can write (012) instead of (( 012), ( 012)). Because an example beats lines of explanation, here is one: ψ0e(102) stands for ψae(sa • k).

Case j 0 = j 1 = j 2 . Let us first compute ψ1ψ0ψ1e(012) and ψ0ψ1ψ0e(012). We have: Since Q1(201) -1 ∈ F [[y1, y2]] and recalling Remark 2.16 we get:

ψ1ψ0ψ1e(012) = Φ1e(201)ψ0Q s 0 1 (201) -1 ψ1 = Φ1e(201)ψ0ψ1Q s 0 s 1 1 (201) -1 .
By Lemma 2.13 we have

Q s 0 s 1 1 (201) = Q1(201) s 0 s 1 = (Q1(201) s 0 ) s 1 = (Q0(s1s0 • (201)) s 1 ) s 1 = Q0(012). Hence: ψ1ψ0ψ1e(012) = Φ1e(201)ψ0ψ1Q0(012) -1 .
As:

ψ0ψ1ψ0e(012) = ψ0ψ1Φ0e(012)Q0(012) -1 ,
to have (1.36) it suffices to prove:

Φ1e(201)ψ0ψ1 = ψ0ψ1Φ0e( 012).

(2.13)

Let us distinguish two subcases.

• If i0 = i1 then: Φ1e(201)ψ0ψ1 = g1 + (1 -q)(1 -X1X -1 2 ) -1 ψ0e(021)ψ1.
Recalling (1.13) and Lemma 2.4 we get:

Φ1e(201)ψ0ψ1 = f -1 0,(021) g1g0 + (1 -q)g0(1 -X0X -1 2 ) -1 ψ1e(012) = f -1 0,(021) f -1 1,(012) g1g0g1 + (1 -q)g0g1(1 -X0X -1 1 )
-1 e(012).

Using the braid relation (1.7) this becomes, recalling (2.7): 012) g0g1e( 102)Φ0, and then, noticing that f 1,(012) = f 1,(102) and f 0,(021) = f 0,(120) , we obtain: 120) g0e(120)ψ1Φ0 Φ1e(201)ψ0ψ1 = ψ0ψ1Φ0e(012), thus (2.13) is proved. 201), Φ0e(012) = (g0 + 1)e( 012), thus with the same calculation as above (even easier) we get: 012), so we got (2.13).

Φ1e(201)ψ0ψ1 = f -1 0,(021) f -1 1,(012) g0g1g0 + (1 -q)g0g1(1 -X0X -1 1 ) -1 e(012) = f -1 0,(021) f -1 1,(012) g0g1 g0 + (1 -q)(1 -X0X -1 1 ) -1 e(012) = f -1 0,(021) f -1 1,(012) g0g1Φ0e(012) Φ1e(201)ψ0ψ1 = f -1 0,(021) f -1 1,(
Φ1e(201)ψ0ψ1 = f -1 0,(120) f -1 1,(012) g0g1e(102)Φ0 = f -1 0,(
• If i0 = i1 then: Φ1e(201) = (g1 + 1)e(
Φ1e(201)ψ0ψ1 = f -1 0,(021) f -1 1,(012) (g1g0g1 + g0g1)e(012) = f -1 0,(120) f -1 1,(102) (g0g1g0 + g0g1)e(012) = ψ0ψ1Φ0e(
Until the end of the proof we use the same arguments as here, arguments which we will thus not recall.

Case j 0 = j 1 = j 2 . Similar.

Case j 0 = j 2 = j 1 . Once again we begin with the computation of ψ1ψ0ψ1e( 012) and ψ0ψ1ψ0e(012).

We have: ψ1ψ0ψ1e(012) = ψ1Φ0Q0(021) -1 e(021)ψ1 = ψ1Φ0e(021)ψ1Q s 1 0 (021) -1 , and:

ψ0ψ1ψ0e(012) = ψ0Φ1Q1(102) -1 e(102)ψ0 = ψ0Φ1e(102)ψ0Q s 0 1 (102) -1 . Since Q s 1 0 (021) -1 = Q s 0 1 (102) -1
, it suffices to prove: ψ1Φ0e(021)ψ1 = ψ0Φ1e(102)ψ0.

(2.14)

Once again we distinguish two subcases.

• If i0 = i2 then:

ψ1Φ0e(021)ψ1 = ψ1e(201)Φ0ψ1 = f -1 1,(201) g1 g0 + (1 -q)(1 -X0X -1 1 ) -1 ψ1e(012) = f -1 1,(201) f -1 1,(012) g1g0g1 + (1 -q)g 2 1 (1 -X0X -1 2 ) -1 e(012) ψ1Φ0e(021)ψ1 = f -1 1,(201) f -1 1,(012) g1g0g1 + (1 -q)q(1 -X0X -1 2 )
-1 e(012).

Similarly, we find:

ψ0Φ1e(102)ψ0 = ψ0e(120)Φ1ψ0 = f -1 0,(120) g0 g1 + (1 -q)(1 -X1X -1 2 ) -1 ψ0e(012) = f -1 0,(120) f -1 0,(012) g0g1g0 + (1 -q)g 2 0 (1 -X0X -1 2 ) -1 e(012) ψ0Φ1e(102)ψ0 = f -1 0,(120) f -1 0,(012) g0g1g0 + (1 -q)q(1 -X0X -1 2 )
-1 e( 012), thus we conclude since f 1,(201) = f 0,(012) and f 1,(012) = f 0,(120) (we see it on this particular case or we can use Lemma 2.13).

• If i0 = i2 we get as above, with α :

= f -1 1,(201) f -1 1,(012) = f -1 0,(120) f -1 0,(012) : ψ1Φ0e(021)ψ1 = α(g1g0g1 + g 2 1 )e(012) = α(g1g0g1 + q)e(012) = α(g0g1g0 + q)e(012) = α(g0g1g0 + g 2 0 )e(012) = ψ0Φ1e(102)ψ0e(012).
Case #{j a , j a+1 , j a+2 } = 3. We have ja = ja+1 and ja = ja+2 and ja+1 = ja+2 thus we get immediately:

ψ1ψ0ψ1e(012) = f -1 1,(201) f -1 0,(021) f -1 1,(012) g1g0g1e(012) = f -1 0,(120) f -1 1,(102) f -1 0,(012) g0g1g0e(012) = ψ0ψ1ψ0e(012), since f 1,(201) = f 0,(012) , f 0,(021) = f 1,(102) and f 1,(012) = f 0,(120) .
3 Yokonuma-Hecke generators of H Λ α (Γ)

Let Λ be a weight as in Section 2. The aim of this section is to prove the following theorem.

Theorem 3.1. For any α |=K n, we can construct an explicit algebra homomorphism:

σ : Y Λ d,n (q) → H Λ α (Γ).
Note that we do not consider yet Y Λ α (q). In particular, it suffices to define the images of the generators (1.1) and check if they verify the defining relations of the cyclotomic Yokonuma-Hecke algebra. As in Section 2, we use the same notation for a generator and its image.

Definition of the images of the generators

It is easier this time to define these images. First, since the elements y1, . . . , yn are nilpotent (Lemma 1.9), we can consider power series in these variables. Hence, the quantities Pa(k), Qa(k) and ya(i) that we defined in §2.1 are also well-defined as elements of H Λ α (Γ). We define finally as in (2.2) the elements e(i) and e(j) of H Λ α (Γ) for i ∈ I α and j ∈ J α . We recall that ξ is a primitive dth root of unity in F . Our "Yokonuma-Hecke generators" of H Λ α (Γ) are given below.

ga := k∈K α (ψaQa(k) -Pa(k)) e(k) for 1 ≤ a < n ta := j∈J α ξ ja e(j) for 1 ≤ a ≤ n Xa := i∈I α ya(i)e(i) for 1 ≤ a ≤ n
As usual, we write g H a and X H a for the corresponding elements when d = 1: we recover the elements of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§4.4].

Remark 3.2 (About Brundan and Kleshchev's proof -II). This remark is similar to Remark 2.7. If j ∈ J α verifies ja = ja+1 and if a relation in [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§4] involves only ψ H a , e H (i) for i ∈ I α and y H b for 1 ≤ b ≤ n, while its proof does not require any cyclotomic relation (1.39), then by the same proof, the same relation is satisfied between ψae(j), e(i, j) and y b e(j) in the unitary algebra e(j)H Λ α (Γ)e(j). If j verifies in addition ja+1 = ja+2, we will be able to add relations with ψ H a+1 , which we substitute by ψa+1e(j).

Check of the defining relations

As in §2.2, we will use Remark 3.2 when ja = ja+1 to get the result from the same corresponding proof of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Theorem 4.3], and when ja = ja+1 we will need a few calculations.

(1.2) Straightforward since e(j)e(j ′ ) = δ j,j ′ e(j) and ξ d = 1.

(1.3) Straightforward since e(j)e(j ′ ) = e(j ′ )e(j).

(1.4) According to (1.17), it suffices to prove t b gae(i, j) = gat sa(b) e(i, j) for every (i, j) ∈ K α . For (i, j) ∈ K α , we have, using (1.20):

t b gae(i, j) = t b (ψaQa(i, j) -Pa(i, j))e(i, j) = t b [e(sa • (i, j))ψaQa(i, j) -e(i, j)Pa(i, j)] = ξ (sa•j) b ψaQa(i, j)e(i, j) -ξ j b Pa(i, j)e(i, j) t b gae(i, j) = ψaQa(i, j)ξ (sa•j) b e(i, j) -Pa(i, j)ξ j b e(i, j),

and:

gat sa(b) e(i, j) = gaξ j sa (b) e(i, j) = ψaQa(i, j)ξ j sa(b) e(i, j) -Pa(i, j)ξ j sa(b) e(i, j).

As (sa • j) b = j sa(b) (by definition of the action of Sn on J n ), it suffices to prove the following:

Pa(i, j)ξ j b = Pa(i, j)ξ j sa (b) . • It is clear if b / ∈ {a, a + 1} since b = sa(b).
• If b ∈ {a, a + 1}, it is clear if ja = ja+1 and obvious if ja = ja+1 since then Pa(i, j) = 0.

(1.5). Let (i, j) ∈ K α and let us prove g 2 a e(i, j) = (q + (q -1)gaea)e(i, j); summing over all (i, j) ∈ K α will conclude. If ja = ja+1 then it is immediate applying Remark 3.2 on (g H a ) 2 = q + (q -1)g H a and left-multiplying by e(i), recalling eae(j) = e(j) and Corollary 2.6. If now ja = ja+1, since eae(j) = 0 it suffices to prove g 2 a e(i, j) = qe(i, j). But, recalling Qa(i, j) = f a,j and Pa(i, j) = 0:

g 2 a e(i, j) = ga(ψaQa(i, j) -Pa(i, j))e(i, j) = f a,j gaψae(i, j) = f a,j gae(sa • (i, j))ψa = f a,j (ψaQa(sa • (i, j)) -Pa(sa • (i, j)))ψae(i, j) g 2 a e(i, j) = f a,j f a,sa•j ψ 2 a e(i, j),
hence we conclude using Lemma 2.12 and (1.35), since ja = ja+1 implies (ia, ja) -(ia+1, ja+1).

(

gag b e(k) = ga(ψ b Q b (k) -P b (k))e(k) = gae(s b • k)ψ b Q b (k) -gae(k)P b (k) = (ψaQa(k) -Pa(k))ψ b Q b (k)e(k) -(ψaQa(k) -Pa(k))P b (k)e(k) = ψaψ b Qa(k)Q b (k)e(k) -ψ b Q b (k)Pa(k)e(k) -ψaQa(k)P b (k)e(k) + Pa(k)P b (k)e(k),
and we conclude since that expression is symmetric in a and b (recalling (1.23)).

(1.7). Again it suffices to prove ga+1gaga+1e(i, j) = gaga+1gae(i, j) for all (i, j) ∈ K α . If ja = ja+1 = ja+2 we get the result using Remark 3.2. Let us then suppose that we are not in that case. We will intensively use (1.20); recall the following fact:

gae(i, j) = (ψaQa(i, j) -Pa(i, j))e(i, j) if ja = ja+1, f a,j ψae(i, j) if ja = ja+1.
Finally, as during the proof of (1.36) in §2.2, we write for example g0e(102) instead of gae(sa • k). Thus, given our hypothesis on j0, j1 and j2 we have: ψ1ψ0ψ1e(012) = ψ0ψ1ψ0e(012).

(3.1)

Case j 0 = j 1 = j 2 . Let us first compute g1g0g1e(012) and g0g1g0e(012). We set α := f 1,(012) f 0,(021) ; we have:

g1g0g1e(012) = f 1,(012) g1g0e(021)ψ1
= f 1,(012) f 0,(021) g1e( 201)ψ0ψ1

= αψ1Q1( 201)ψ0ψ1e(012) -αP1( 201)ψ0ψ1e( 012)

g1g0g1e(012) = αψ1ψ0ψ1e(012)Q s 0 s 1 1 (201) -αψ0ψ1e(012)P s 0 s 1 1 (201).
We have already seen that Q s 0 s 1 1 (201) = Q0( 012) and similarly we have P s 0 s 1 1 (201) = P0(012) (see (2.5)). Hence we obtain, using (3.1) and noticing f 1,(012) = f 0,(120) and f 0,(021) = f 1,(102) : g1g0g1e(012) = αψ0ψ1ψ0e(012)Q0(012) -αψ0ψ1e( 012)P0(012) = αψ0ψ1e( 102)ψ0Q0( 012) -f 1,(012) f 0,(021) ψ0e( 021)ψ1P0(012) = f 1,(102) f 0,(120) ψ0e( 120)ψ1ψ0Q0( 012) -f 1,(012) g0ψ1e( 012)P0(012) = f 1,(102) g0ψ1e( 102)ψ0Q0( 012) -g0g1P0( 012)e(012) = g0g1(ψ0Q0(012) -P0( 012))e( 012) g1g0g1e(012) = g0g1g0e( 012), so we are done.

Case j 0 = j 1 = j 2 . Similar.

Case j 0 = j 2 = j 1 . Given these assumptions we have: 012).

ψ 2 0 e(012) = ψ 2 1 e(012) = e(
(3.2)

Hence, using (3.2), with α := f 1,(012) f 1,(201) :

g1g0g1e(012) = f 1,(012) g1(ψ0Q0(021) -P0( 021))e( 021)ψ1

= f 1,(012) g1e( 201)ψ0Q0( 021)ψ1 -f 1,(012) g1e( 021)P0( 021)ψ1 120 Case #{j 0 , j 1 , j 2 } = 3. We get immediately:

= αψ1ψ0ψ1e(012)Q s 1 0 (021) -αψ 2 1 e(012)P s 1 0 (021) = αψ0ψ1ψ0e(012)Q s 0 1 (102) -αψ 2 0 e(012)P s 0 1 (102) = αψ0e(
g1g0g1e(012) = f 1,(201) f 0,(021) f 1,(012) ψ1ψ0ψ1e(012) = f 0,(120) f 1,(102) f 0,(012) ψ0ψ1ψ0e(012) = g0g1g0e(012),
since f 1,(201) = f 0,(012) and f 0,(021) = f 1,(102) and f 1,(012) = f 0,(120) .

(1.8). Since for a ∈ {1, . . . , n -1} it is clear that Xa+1Xa = XaXa+1, it remains to prove that qXa+1 = gaXaga; we will conclude taking a = 1. As we proved (1.5), it suffices to prove (1.14). Let (i, j) ∈ K α and let us prove:

gaXa+1e(i, j) = Xagae(i, j) + (q -1)Xa+1e(i, j) if ja = ja+1, Xagae(i, j) if ja = ja+1.
Again, we deduce the case ja = ja+1 from Remark 3.2. If ja = ja+1 we have, using (1.20) and (1.24):

gaXa+1e(i, j) = q i a+1 gae(i, j)(1 -ya+1) = q i a+1 f a,j ψa(1 -ya+1)e(i, j) = q i a+1 f a,j (1 -ya)e(sa • (i, j))ψa = f a,j Xaψae(i, j) gaXa+1e(i, j) = Xagae(i, j).
(1.9). We prove in fact (1.13), that is, gaX b = X b ga for b = a, a + 1. As y b commutes with ψa by (1.22) we have, for any k ∈ K α (where ya(k) := ya(i) with k = (i, j)):

gaX b e(k) = gae(k)y b (k) = y b (k)(ψaQa(k) -Pa(k))e(k) = y b (k)e(sa • k)ψaQa(k) -y b (k)e(k)Pa(k) gaX b e(k) = X b gae(k), since y b (k)e(sa • k) = q (sa•i) b (1 -y b )e(sa • k) = q i b (1 -y b )e(sa • k) = X b e(sa • k).
(1.10). We prove in fact Xat b = t b Xa for every a, b; that is straightforward from (1.19).

(1.11). We have, using (1.17)-(1.19):

i∈I (X1 -q i ) Λ i = i∈I i∈I α q i 1 (1 -y1) -q i e(i) Λ i = i∈I i∈I α q i 1 (1 -y1) -q i Λ i e(i) i∈I (X1 -q i ) Λ i = i∈I α i∈I q i 1 (1 -y1) -q i Λ i e(i) .
Noticing that for each i ∈ I α the term for i = i1 vanishes by (1.39), we get the result.

Isomorphism theorem

We give now the main result of our paper; let Λ be a weight as in Sections 2 and 3.

Statement

Theorem 4.1. There is a presentation of the algebra Y Λ α (q) given by the generators (1.16) and the relations (1.18)-(1.25), (1.35)-(1.36) and (1.39), that is, we have an algebra isomorphism:

H Λ α (Γ) ∼ -→ Y Λ α (q).
As finitely many Y Λ α (q) are non-zero, we deduce from this isomorphism that only finitely many H Λ α (Γ) are non-zero. Hence, as Y Λ d,n (q) = ⊕ α|= K n Y Λ α (q), defining the cyclotomic quiver Hecke algebra of degree n by (recall (1.31)):

H Λ n (Γ) := α|= K n H Λ α (Γ), (4.1)
we get the unitary algebra isomorphism with the cyclotomic Yokonuma-Hecke algebra:

H Λ n (Γ) ≃ Y Λ d,n (q). (4.2)
Recalling that the cyclotomic quiver Hecke algebra is naturally graded (Proposition 1.7), we get the following corollary.

Corollary 4.2. The cyclotomic Yokonuma-Hecke algebra inherits the grading of the cyclotomic quiver Hecke algebra.

Moreover, as we obtain a presentation of Y Λ d,n (q) which does not depend on q, we also get another one (see Corollary 5.16 for a slight improvement).

Corollary 4.3. Let q ′ ∈ F \ {0, 1}. If char q ′ (F ) = charq(F ) then the algebras Y Λ d,n (q) and Y Λ d,n (q ′ ) are isomorphic.
Let us now prove Theorem 4.1. First, as we have a (non-unitary) algebra homomorphism Y Λ α (q) → Y Λ d,n (q), by Theorem 3.1 we get an algebra homomorphism Y Λ α (q) → H Λ α (Γ), that we still call σ. We will prove that σ : Y

Λ α (q) → H Λ α (Γ) and ρ : H Λ α (Γ) → Y Λ α (q) (from Theorem 2.1) verify σ •ρ = id H Λ α (Γ)
and ρ•σ = id Y Λ α (q) . Since these are algebra homomorphisms, it suffices to prove that they are identity on generators. To clarify the proof, let us add a Y on the quiver Hecke generators of Y Λ α (q) and a H on the Yokonuma-Hecke generators of H Λ α (Γ) (there isn't any confusion possible with the former notation referring to the case d = 1 since we won't use it any more).

Proof of σ

• ρ = id H Λ α (Γ)
We have to check that σ(ρ(e(k))) = e(k) for all k ∈ K α , that σ(ρ(ya)) = ya for all 1 ≤ a ≤ n and that σ(ρ(ψa)) = ψa for all 1 ≤ a < n.

Let us start by finding the image of e(k) by σ • ρ. By definition of ρ we have ρ(e(k)) = e Y (k), so we have to prove σ(e Y (k)) = e(k). Let M := H Λ α (Γ); the algebra homomorphism σ gives M a structure of Y Λ α (q)-module, finite-dimensional thanks to Theorem 1.6. If M (k) denotes the weight space as in (2.1), by Remark 2.3 we know that the projection onto M (k) along ⊕ k ′ =k M (k ′ ) is given by σ(e Y (k)). We prove that e(k) is this projection too.

Let (i, j) ∈ K α . For 1 ≤ a ≤ n, we have σ(Xa) = i ′ (q i ′ a -q i ′ a ya)e(i ′ ) so:

σ(Xa) -q ia = i ′ ∈I α (q i ′ a -q ia ) -q i ′ a ya e(i ′ ).
Since ya is nilpotent, thanks to (1.17)-(1.18) we have:

v ∈ M : (σ(Xa) -q ia ) N v = 0 =     i ′ ∈I α i ′ a =ia e(i ′ )     M,
hence, for N ≫ 0 we have:

M (i) := v ∈ M : (σ(Xa) -q ia ) N v = 0 ∀a = e(i)M.
In a similar way we have M (j) = e(j)M where M (j) := {v ∈ M : (σ(ta) -ξ ja )v = 0 ∀a}, thus:

M (k) = e(k)M.
Hence, as ⊕ k M (k) = M we conclude that e(k) is the desired projection and finally e(k) = σ(e Y (k)).

The end of the proof is without any difficulty. We have:

σ(ρ(ya)) = σ(y Y a ) = i∈I α [1 -q -ia σ(Xa)]σ(e Y (i)) = i∈I α [1 -q -ia X H a ]e(i) = i∈I α 1 -q -ia i ′ ∈I α ya(i ′ )e(i ′ ) e(i) = i∈I α [1 -q -ia ya(i)]e(i) = i∈I α [1 -q -ia q ia (1 -ya)]e(i) σ(ρ(ya)) = ya. Thus, we have σ(Q Y a (k)) = Qa(k) and σ(P Y a (k)) = Pa(k), hence, recalling (2.4): σ(ρ(ψa)) = σ(ψ Y a ) = k∈K α σ(Φa)σ(Q Y a (k)) -1 σ(e Y (k)) = k∈K α k ′ ∈K α σ(ga) + σ(P Y a (k ′ )) e(k ′ ) Qa(k) -1 e(k) = k∈K α (g H a + Pa(k))Qa(k) -1 e(k) = k∈K α k ′ ∈K α (ψaQa(k ′ ) -Pa(k ′ ))e(k ′ ) + Pa(k) Qa(k) -1 e(k) = k∈K α [(ψaQa(k) -Pa(k)) + Pa(k)]Qa(k) -1 e(k) σ(ρ(ψa)) = ψa. 4.3 Proof of ρ • σ = id Y Λ α (q)
This is even easier: we have to check ρ(σ(ga)) = ga for 1 ≤ a < n and ρ(σ(Xa)) = Xa and ρ(σ(ta)) = ta for 1 ≤ a ≤ n. We have:

ρ(σ(ga)) = k∈K α [ψ Y a Q Y a (k) -P Y a (k)]e Y (k) = k∈K α [ΦaQ Y a (k) -1 Q Y a (k) -P Y a (k)]e Y (k) = k∈K α [Φa -P Y a (k)]e Y (k) ρ(σ(ga)) = ga. Recalling (2.3): ρ(σ(Xa)) = i∈I α y Y a (i)e Y (i) = i∈I α Xae Y (i) = Xa.
Finally:

ρ(σ(ta)) = j∈J α ξ ja e Y (j) = j∈J α tae Y (j) = ta.
The proof of Theorem 4.1 is now over.

Degenerate case

In this section, we extend the previous results to the case q = 1. In particular, we need to define a new "degenerate" cyclotomic Yokonuma-Hecke algebra. Many calculations are not written, since they are entirely similar to the non-degenerate case. Note the following thing: since the cyclotomic quiver Hecke algebra has no q in its presentation, we do not need to define some new cyclotomic quiver Hecke algebra.

Let Λ = (Λ k ) k∈K ∈ N (K) be a weight; we assume that ℓ(Λ) = k∈K Λ k verifies ℓ(Λ) > 0. Moreover, as in Section 2 we suppose that for any i ∈ I and j, j ′ ∈ J, we have: Λi,j = Λ i,j ′ =: Λi.

In particular, we will write Λ as well for the weight (Λi)i∈I .

Degenerate cyclotomic Yokonuma-Hecke algebras

We introduce here the degenerate cyclotomic Yokonuma-Hecke algebra: this algebra can be seen as the rational degeneration of the cyclotomic Yokonuma-Hecke algebra Y Λ d,n (q). The degenerate cyclotomic Yokonuma-Hecke algebra of type A, denoted by Y Λ d,n (1), is the unitary associative F -algebra generated by the elements f1, . . . , fn-1, t1, . . . , tn, x1, . . . , xn (5.1) subject to the following relations:

t d a = 1, (5.2) tat b = t b ta, (5.3) t b fa = fat sa(b) , (5.4) f 2 a = 1 (5.5) faf b = f b fa ∀|a -b| > 1, (5.6) fa+1fafa+1 = fafa+1fa, (5.7)
where ea := 1 d j∈J t j a t -j a+1 , together with the following relations: and finally the cyclotomic one:

xax b = x b xa, ( 5
i∈I (x1 -i) Λ i = 0.
(5.12)

We obtained this presentation by setting Xa = 1 + (q -1)xa in Y Λ d,n (q), simplifying by (1 -q) as much as we can and then setting q = 1 (according to the transformation made by Drinfeld [Dr] to define degenerate Hecke algebras). As in the non-degenerate case, the element ea verifies e 2 a = ea and commutes with fa. Finally, note some consequences of (5.5) and (5.9):

xa+1 = faxafa + faea,
(5.13) xa+1fa = faxa + ea.

(5.14)

When d = 1, we recover the degenerate cyclotomic Hecke algebra H Λ n (1) of [BrKl]; it is the degenerate cyclotomic Yokonuma-Hecke algebra Y Λ 1,n (1). In particular, the element ea becomes 1, and fa (respectively x b ) is the element sa (resp. x b ) of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§3].

We will use the following lemma (see [START_REF] Chlouveraki | Markov traces on affine and cyclotomic Yokonuma-Hecke algebras[END_REF]Lemma 2.15] for the non-degenerate case).

Lemma 5.1. For u, v ∈ N we have the following equalities:

faxaxa+1 = xaxa+1fa, (5.15) fax v a+1 = x v a fa + ea v-1 m=0 x m a x v-1-m a+1 ,
(5.16)

fax u a = x u a+1 fa -ea u-1 m=0 x m a x u-1-m a+1 ,
(5.17)

fax u a x v a+1 =            x v a x u a+1 fa + ea v-u-1 m=0 x u+m a x v-1-m a+1 if u ≤ v, x v a x u a+1 fa -ea u-v-1 m=0 x u-1+m a x v-m a+1 if u ≥ v.
(5.18)

Proof. We deduce (5.15) from different previous relations. The relations (5.16) and (5.17) can be proved by an easy induction. The equality (5.18) follows finally from these previous equalities.

As the elements ga for 1 ≤ a < n verify the same braid relations as the sa ∈ Sn, for each w ∈ Sn there is a well-defined element gw :

= ga 1 • • • ga r ∈ Y Λ d,n
(1) which does not depend on the reduced expression w = sa 1 • • • sa r .

Proposition 5.2. The algebra Y Λ d,n (1) is a finite-dimensional F -vector space and a family of generators is given by the elements fwx

u 1 1 • • • x un n t v 1 1 • • • t vn n for w ∈ Sn, ua ∈ {0, . . . , ℓ(Λ) -1} and va ∈ J.
Proof. We use a similar method to [ArKo, OgPA]. As the unit element belongs to the above family, it suffices to prove that the F -vector space V spanned by these elements is stable under (right-)multiplication by the generators of Y Λ d,n (1).

Let us consider α

:= fwx u 1 1 • • • x un n t v 1 1 • • • t vn n
as in the proposition. By (5.2) and (5.3) the element αta remains in V . Moreover, writing (by (5.4) and (5.10)):

αfa = fwx u 1 1 • • • x u a-1 a-1 x ua a x u a+1 a+1 fa x u a+2 a+2 • • • x un n t v 1 1 • • • t vn n ,
and using (5.18) we conclude that αfa ∈ V , noticing that the element

x u 1 1 • • • x u a-1 a-1 eax u ′ a a x u ′ a+1 a+1 x u a+2 a+2 • • • x un n t v 1 1 • • • t vn n belongs to V for every 0 ≤ u ′ a , u ′ a+1 < ℓ(Λ).
Finally, according to (5.13), to prove that αxa remains in V it suffices now to prove that αx1 ∈ V , but this is clear by (5.8), (5.11) and (5.12).

Let now M be a finite-dimensional Y Λ d,n (1)-module; it is a finite-dimensional F -vector space thanks to Proposition 5.2. By (5.12), the eigenvalues of x1 on M belong to I. We prove in Lemma 5.4 that all the xa have in fact their eigenvalues in I: this is the degenerate analogue of [START_REF] Cui | Modular representations and branching rules for affine and cyclotomic Yokonuma-Hecke algebras[END_REF]Lemma 5.2], which we used in §2.1. Lemma 5.3. We have:

xaφa = φaxa+1, φ 2 a = (xa+1 -xa -ea)(xa -xa+1 -ea),
where φa is the "intertwining operator" defined by: φa := fa(xa -xa+1) + ea.

Proof. These are straightforward calculations. We have, using (5.9):

xaφa = (faxa+1 -ea)(xa -xa+1) + xaea = fa(xa -xa+1)xa+1 + xa+1ea xaφa = φaxa+1,
and:

φ 2 a = fa(xa -xa+1)fa(xa -xa+1) + 2fa(xa -xa+1)ea + ea = fa(faxa+1 -ea -faxa -ea)(xa -xa+1) + 2fa(xa -xa+1)ea + ea = (xa+1 -xa)(xa -xa+1) + ea φ 2 a = (xa+1 -xa -ea)(xa -xa+1 -ea).
Lemma 5.4. The eigenvalues of xa belong to I for every 1 ≤ a ≤ n.

Proof. We proceed by induction on a. The proposition is true for a = 1; we suppose that it is true for some 1 ≤ a < n. Let λ be an eigenvalue of xa+1 (in a algebraic closure of F ). As the family {xa, xa+1, ea} is commutative, we can find a common eigenvector v in the eigenspace of xa+1 associated with λ: we have xav = iv and eav = δv for some i ∈ I (by induction hypothesis) and δ ∈ {0, 1} (since e 2 a = ea). We distinguish now whether φav vanishes or not: • if φav = 0, we get by Lemma 5.3:

xa(φav) = φa(xa+1v) = λφav,
hence λ is an eigenvalue for xa and by induction hypothesis we get λ ∈ I;

• if φav = 0, by the same lemma we have:

φ 2 a v = (λ -i -δ)(i -λ -δ)v = 0, hence λ = i ± δ ∈ I.

Quiver Hecke generators of Y Λ d,n (1)

We proceed as in Section 2: we define some central idempotents, then some "quiver Hecke generators" on which we check the defining relations of H Λ α (Γ). The proofs are entirely similar to the nondegenerate case (even easier; note that once again the "hard work" has been made in [BrKl]), hence we won't write them down. However, we will still define the different involved elements.

Image of e(i, j)

Let M be a finite-dimensional Y Λ d,n (1)-module. We know that the ta are diagonalizable with eigenvalues in J. Hence, recalling Lemma 5.4, we can write (recall that the family {Xa, ta} 1≤a≤n is commutative):

M = (i,j)∈I n ×J n M (i, j),
with:

M (i, j) := v ∈ M : (xa -ia) N v = (ta -ξ ja )v = 0 for all 1 ≤ a ≤ n ,
Remark 5.9. As in [BrKl], the explicit expression of qa(i, j) does not really matter; we only need some properties verified by these power series.

Lemma 5.10. We have:

p sa a+1 (i, j) = p s a+1 a (sa+1sa • (i, j)),
q sa a+1 (i, j) = q s a+1 a (sa+1sa • (i, j)).

We now introduce the following element of Y Λ α (1):

ϕa := fa + (i,j)∈K α ia =i a+1 ja=j a+1 (xa -xa+1) -1 e(i, j) + (i,j)∈K α ia=i a+1 ja=j a+1 e(i),
where (xa -xa+1) -1 e(k) denotes the inverse of (xa -xa+1)e(k) in e(k) Y Λ α (1)e(k). In particular, we have:

ϕae(j) = fae(j) if ja = ja+1, ϕa = k∈K α (fa + pa(k))e(k).
Moreover, for d = 1 the element ϕa is the "intertwining element" defined in [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§3.2].

Lemma 5.11. We have the following properties: ϕae(j) = e(sa • j)ϕa, (5.20) ϕae(i, j) = e(sa • (i, j))ϕa, (5.21)

ϕax b = x b ϕa ∀b = a, a + 1, (5.22
)

ϕay b = y b ϕa ∀b = a, a + 1, (5.23) ϕaq b (k) = q b (k)ϕa ∀|b -a| > 1, (5.24) ϕaϕ b = ϕ b ϕa ∀|b -a| > 1.
(5.25)

Our element ψa is defined for 1 ≤ a < n by:

ψa := k∈K α φaqa(k) -1 e(k) ∈ Y Λ α (1).
When d = 1 this element ψa corresponds to the ψa of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§3.3]. Note finally that for j ∈ J α we have: ψae(j) = fae(j) if ja = ja+1.

Check of the defining relations

Theorem 5.12. The elements y1, . . . , yn, ψ1, . . . , ψn-1 and e(k) for k ∈ K α verify the defining relations (1.39)-(1.36) of H Λ α (Γ). The painstaking verification is exactly the same as in §2.2: we apply Remark 5.7 on the proof of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Theorem 3.2] for the cases ja = ja+1, and when ja = ja+1 then entirely similar (even the same) relations as in §2.2 are verified. Note two small differences with the proof in §2.2:

• we shall write (xa -x b ) instead of (1 -q)(1 -XaX -1
b ); • the elements f a,j are equal to 1.

Degenerate Yokonuma-Hecke generators of H Λ α (Γ)

We proceed as in Section 3. Once again, the proofs are entirely similar to the non-degenerate case, hence we do not write them down. First of all, since the elements y1, . . . , yn ∈ H Λ α (Γ) are nilpotent we can consider power series in these variables. Hence, the quantities pa(k), qa(k) that we defined in §5.2.3 are also well-defined as elements of H Λ α (Γ). We define finally as in (5.19) the elements e(i) and e(j) of H Λ α (Γ) for i ∈ I α and j ∈ J α .

We recall that ξ is a primitive dth root of unity in F . Our "degenerate Yokonuma-Hecke generators" of H Λ α (Γ) are given below.

fa := k∈K α (ψaqa(k) -pa(k)) e(k) for 1 ≤ a < n ta := j∈J α ξ ja e(j) for 1 ≤ a ≤ n xa := i∈I α (ya + ia)e(i) for 1 ≤ a ≤ n
When d = 1, the element fa (respectively xa) is the element sa (resp. xa) of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§3.4]. Remark 5.13 (About Brundan and Kleshchev's proof -IV). Let 1 ≤ a < n; if j ∈ J α verifies ja = ja+1, when a proof in [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§3.4] needs only the elements ψae(j), y b e(j), e(i, j) and the corresponding relations in H Λ α (Γe), we claim that the same proof holds in e(j)H Λ α (Γ)e(j). We extend this claim to the case ja = ja+1 = ja+2.

Finally, similarly to §5.2.4 we have the following theorem. Once again the check of the various relations is exactly the same as in §3.2.

Theorem 5.14. The elements f1, . . . , fn-1, t1, . . . , tn, x1, . . . , xn satisfy the defining relations (5.2)-( 5.12) of Y Λ d,n (1).

Isomorphism theorem

We give now the degenerate version of Theorem 4.1.

Theorem 5.15. There is a presentation of the degenerate cyclotomic Yokonuma-Hecke algebra Y Λ α (1) given by the generators (1.16) and the relations (1.17)-(1.25), (1.35)-(1.36) and (1.39), that is, we have an algebra isomorphism:

H Λ n (Γ) ∼ → Y Λ d,n
(1). The proof of this theorem is entirely similar to the one of Theorem 4.1. In particular, by Theorem 5.12 we can define an algebra homomorphism ρ : H Λ α (Γ) → Y Λ α (1) and by Theorem 5.14 we can define another algebra homomorphism σ :

Y Λ d,n (1) → H Λ α (Γ). From the inclusion Y Λ α (1) ⊆ Y Λ d,n
(1) we deduce an algebra homomorphism σ : Y Λ α (1) → H Λ α (Γ). We prove then that ρ and σ are inverse homomorphisms, taking the images of the different defining generators.

Together with Theorem 4.1 we get the following corollaries (cf. [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Corollary 1.3]).

Corollary 5.16. If q and q ′ are two elements of F × with charq(F ) = char q ′ (F ) then Y Λ d,n (q) and Y Λ d,n (q ′ ) are isomorphic algebras. Corollary 5.17. If F has characteristic charq(F ) then the cyclotomic Yokonuma-Hecke algebra Y Λ d,n (q) is isomorphic to its rational degeneration Y Λ d,n (1). This applies in particular when F has characteristic 0 and q is generic.

Another approach to the result

In [Lu, JaPA, PA], the authors have proved the following algebra isomorphism:

Y Λ d,n (q) ≃ λ|= d n Matm λ H Λ λ (q), (6.1) with F = C, e = ∞, H Λ λ (q) := H Λ λ 1 (q) ⊗ • • • ⊗ H Λ λ d (q)
, where we write |= d instead of |=J and where m λ are some integers (see (6.2)). We will see how we can relate this isomorphism to our previous work. To that extent, we prove in Theorem 6.26 a general result on quiver Hecke algebras in the case where the quiver is given by a disjoint union of full subquivers. The isomorphism is built from the following map (see (6.25)):

Ψ t ′ ,t : w → (ψπ t ′ wψ π -1 t )E t ′ ,t ,
for w ∈ e(t ′ )Hn(Q)e(t), where E t ′ ,t are elementary matrices. In particular:

• in §6.1.2 we introduce the elements πt, the minimal-length representatives of the right cosets of Sn under the action of the Young subgroup S λ where λ |= d n: this will lead to some calculations which will only be needed to explicit our homomorphism Ψ t ′ ,t ;

• in §6.2 we will study the elements ψπ t , and we will go on with the previous calculations.

Setting

Let K be a finite set; we recall that d, n ∈ N * and J = Z/dZ ≃ {1, . . . , d}. We consider a partition of K into d parts K = ⊔j∈J Kj . We recall that the left action of w ∈ Sn on tuples is given by w •(x1, . . . , xn) := (x w -1 (1) , . . . , x w -1 (n) ). We may use some elementary theory about Coxeter groups: we refer for instance to [GePf] or [Hum]. In particular, in that context we will denote by ℓ the usual length function Sn → N. Finally, let us mention that in this section, we will write t for the elements of J n .

Labellings and shapes

Let λ = (λj) 1≤j≤d |= d n be a d-composition of n: recall that it means λj ≥ 0 and λ1

+ • • • + λ d = n.
We define the integers λ1, . . . , λ d , given by λj := λ1 + • • • + λj for j ∈ J. In particular, λ1 = λ1 and λ d = n; we also set λ0 := 0. From now on, the letter λ always stands for a d-composition of n.

Definition 6.1. Let k ∈ K n and t ∈ J n .

• We say that k is a labelling of t when the following rule is satisfied:

∀a ∈ {1, . . . , n}, ka ∈ Kt a ,
that is: ∀a ∈ {1, . . . , n}, ∀j ∈ J, ka ∈ Kj ⇐⇒ ta = j.

We write K t for the elements K n which are labellings of t.

• We say that t has shape λ |= d n and we write [t] = λ if for all j ∈ J there are exactly λj components of t equal to j, that is:

∀j ∈ J, # a ∈ {1, . . . , n} : ta = j = λj .
We write J λ for the elements J n with shape λ.

The sets J λ are exactly the orbits of J n under the action of Sn, in particular [w • t] = [t] for every w ∈ Sn and t ∈ J n . Moreover, the cardinality of J λ is:

m λ := n! λ1! . . . λ d ! . (6.2)
We write t λ ∈ J λ for the trivial element of shape λ, given by: ∀a ∈ {1, . . . , n}, ∀j ∈ J, t λ a = j ⇐⇒ λj-1 < a ≤ λj , (6.3) that is: . . . , 1, . . . , d, . . . , d),

t λ := (1,
where each j ∈ J appears λj times. Note that

K t λ ≃ K λ 1 1 × • • • × K λ d d .

Young subgroups

Most results of this section are well-known; however, since in the literature they are stated either for a left or a right action (see Remark 6.9), for the convenience of the reader we state all of them with a left action. We remind the reader that some calculations made here will only be used in §6.4.3, namely with Lemmas 6.12 and 6.13. Let λ |= d n; the following group:

S λ := S λ 1 × • • • × S λ d ,
can be seen as a subgroup of Sn (the "Young subgroup"), where we consider that S λ j ≃ S({λj-1 + 1, . . . , λj}). Recall that:

• the group Sn (resp. S λ j ) is generated by s1, . . . , sn-1 (resp. s λ j-1 +1 , . . . , s λ j -1 );

• the subgroup S λ is generated by all the sa for a ∈ {1, . . . , n} \ {λ1, . . . , λ d }.

In particular:

∀j = j ′ , ∀(wj, w j ′ ) ∈ S λ j × S λ j ′ , wj w j ′ = w j ′ wj in Sn. (6.4) Remark 6.2. If w = sa 1 • • • sa r ∈ S λ
is a reduced expression, up to a reindexation we know by (6.4) that there is a sequence 0 =: r0 ≤ r1 ≤ • • • ≤ r d-1 ≤ r d := r such that for each j ∈ J, the word sa r j-1 +1 • • • sa r j is reduced and lies in S λ j . The converse is also true: if for each j ∈ J we have a reduced word sa r j-1

+1 • • • sa r j ∈ S λ j then their concatenation s1 • • • sr ∈ S λ is reduced.
The following proposition is straightforward.

Proposition 6.3. The stabiliser of t λ under the action of Sn is exactly S λ .

We now study the right cosets in Sn for the (left) action of S λ .

Lemma 6.4. Two words w, w ′ ∈ Sn are in the same right coset if and only if w

-1 • t λ = w ′-1 • t λ .
The proof is straightforward from Proposition 6.3. An element C ∈ S λ \Sn is thus determined by the constant value t := w -1 • t λ ∈ J λ for w ∈ C: we write Ct for the coset C (as each t ∈ J n has a unique shape, we do not need to precise the underlying composition in the indexation). Noticing that m λ = |Sn|/|S λ |, we conclude that the cosets are parametrised by the whole set J λ , that is, S λ \Sn = {Ct} t∈J λ .

We know by [START_REF] Geck | Characters of finite Coxeter groups and Iwahori-Hecke algebras[END_REF]Proposition 2.1.1] that each coset Ct has a unique minimal length element: we write πt ∈ Ct for this unique element. In particular, since Lemma 6.4 gives:

∀w ∈ Sn, w ∈ Ct ⇐⇒ w • t = t λ , (6.5)
we get the following proposition.

Proposition 6.5. The element πt is the unique minimal length element of Sn such that:

πt • t = t λ . (6.6)
Remark 6.6. The decomposition into right cosets is obtained in the following way. Given w ∈ Sn, we know that w belongs to the coset Ct with t := w -1 • t λ . The element w -1 := πtw -1 stabilises t λ , thus lies in S λ and we have w = wπt.

Proposition 6.7. The elements πt are given by:

∀a ∈ {1, . . . , n}, πt(a) = λt a -1 + #{b ≤ a : t b = ta}.
An example is given in Figure 2. To prove Proposition 6.7, we will use the vocabulary of "tableaux" {1, . . . , 6} 1 2 3 4 5 6

t 3 1 3 2 3 1 t λ 1 1 2 3 3 3 {1, . . . , 6} 1 2 3 4 5 6
Figure 2: The permutation π t for λ := (2, 1, 3) |= 3 6 and t := (3, 1, 3, 2, 3, 1).

(see for example [START_REF] Mathas | Iwahori-Hecke algebras and Schur algebras of the symmetric group[END_REF]§3.1]). As a quick reminder, a λ-tableau T is a bijection {(j, m) ∈ N 2 : 1 ≤ j ≤ d and 1 ≤ m ≤ λj} → {1, . . . , n}; the tableau T is row-standard if in each rows, its entries increase from left to right. Here are two examples of λ-tableaux, with λ := (2, 1, 3) |=3 6:

2 6 4 1 3 5 , 3 2 5 6 1 4
, the first only being row-standard.

To any t ∈ J λ , we associate the λ-tableau Tt given by the following rule: for j ∈ J and m ∈ {1, . . . , λj}, we label the node (j, m) by the index a of the mth occurrence of j in t, that is, by the integer a ∈ {1, . . . , n} determined by: ta = j and #{b ≤ a : t b = ta} = m.

(6.7)

In particular, the tableau Tt is row-standard; conversely, each row-standard λ-tableau is a Tt for a unique t ∈ J λ . With the notation of Figure 2, here are two examples of row-standard λ-tableaux:

Tt = 2 6 4 1 3 5 , T t λ = 1 2 3 4 5 6
.

We consider the natural left action of the symmetric group Sn on the set of λ-tableaux: if w ∈ Sn and T is a λ-tableau, the tableau w • T is obtained by applying w in each box of T . If now T and T ′ are two λ-tableaux, we write T ∼ T ′ if for all j ∈ J, the labels of the jth row of T are a permutation of the labels of the jth row of T ′ . Lemma 6.8. For w ∈ Sn and t ∈ J λ we have:

w • Tt ∼ T t λ ⇐⇒ w • t = t λ .
Proof. If j ∈ J and m ∈ {1, . . . , λj }, we denote by a [j, m] the label of the box (j, m) of Tt; by (6.7) we have t a[j,m] = j. We get:

w • Tt ∼ T t λ ⇐⇒ ∀j ∈ J, ∀m ∈ {1, . . . , λj }, w(a[j, m]) ∈ {λj-1 + 1, . . . , λj } ⇐⇒ ∀j ∈ J, ∀m ∈ {1, . . . , λj }, t λ w(a[j,m]) = j ⇐⇒ ∀j ∈ J, ∀m ∈ {1, . . . , λj }, t λ w(a[j,m]) = t a[j,m] ⇐⇒ ∀a ∈ {1, . . . , n}, t λ w(a) = ta ⇐⇒ ∀a ∈ {1, . . . , n}, t λ a = t w -1 (a) w • Tt ∼ T t λ ⇐⇒ t λ = w • t, as desired. Proof of Proposition 6.7. Let t ∈ J λ . There is a unique element d(t) ∈ Sn such that Tt = d(t) • T t λ , that is, d(t) -1 • Tt = T t λ .
By the equation of the coset Ct given at (6.5) and Lemma 6.8, we get that d(t) -1 ∈ Ct. Applying [START_REF] Mathas | Iwahori-Hecke algebras and Schur algebras of the symmetric group[END_REF]Proposition 3.3], we know that d(t) -1 is the unique minimal length element of Ct. As a consequence, we have d(t) -1 = πt and thus:

πt • Tt = T t λ .
(6.8)

Let j ∈ J and m ∈ {1, . . . , λj}, and let a (respectively α) be the label of the box (j, m) in Tt (resp. T t λ ). In particular, by (6.7) we have α = λj-1 + m. Moreover, by (6.8) we have πt(a) = α: we conclude that the announced formula is satisfied, since, by a last use of (6.7), we have j = ta and m = #{b ≤ a : t b = ta}.

Remark 6.9. In [Ma], the author considers the elements of Sn as acting on {1, . . . , n} from the right, by iw := w(i) where i ∈ {1, . . . , n} and w ∈ Sn is a permutation. This is the right action of S op n : in such a setting, we read products of permutations from left to right. Lemma 6.10. Let t ∈ J λ and let πt = sa 1 • • • sa r be a reduced expression. Then:

∀m ∈ {1, . . . , r}, sa m • (wm • t) = wm • t, where wm := sa m+1 • • • sa r (with wm = 1 if m = r). Proof. Let us suppose sa m • (wm • t) = wm • t and define πt := sa 1 • • • sa m-1 sa m+1 • • • sa r .
Using the assumption and the equality πt • t = t λ , we see that the element πt verifies πt • t = t λ too. As the element πt is strictly shorter that πt (since sa 1 • • • sa r is reduced), this is in contradiction with Proposition 6.5. Remark 6.11. Using t = π -1 t • t λ in Lemma 6.10, we get the following similar result for π

-1 t . If π -1 t = sa r • • • sa 1 is a reduced expression, then: ∀m ∈ {1, . . . , r}, w ′ m • t λ = sa m • (w ′ m • t λ ),
where

w ′ m := sa m-1 • • • sa 1 .
The next two lemmas are not essential to the proof of the main theorem of this section, Theorem 6.26; however, they will allow us to relate our construction to the one of [JaPA, PA]. Let t ∈ J n and a ∈ {1, . . . , n -1}. We give in the next lemma the decomposition of Remark 6.6 for the element πtsa; this is in fact a particular case of Deodhar's lemma (see, for instance, [START_REF] Geck | Characters of finite Coxeter groups and Iwahori-Hecke algebras[END_REF]Lemma 2.1.2]). Lemma 6.12. Let t ∈ J n and a ∈ {1, . . . , n -1}. The element πtsa belongs to the coset Cs a•t , more precisely we have:

πtsa = s π t (a) πt if ta = ta+1, πs a•t if ta = ta+1.

A "disjoint quiver" Hecke algebra

We consider the setting of §1.2.1. Note that since K is finite, we can consider the quiver Hecke algebra Hn(Q); recall that its generators are given at (1.29), which are subject to the relations (1.18)-(1.27) and (1.30). We already said that the elements ψa do not verify the same braid relations as the elements sa ∈ Sn: in particular, if s b 1 • • • s br is another reduced expression for w, we may have ψ b 1 • • • ψ br = ψw. However, according to Remark 6.2 we can assume that we chose the reduced expressions such that: (6.11) and in the sequel we do suppose that we did so. To that extent, we can first choose some reduced expressions for the elements of the subgroups S λ j for j ∈ J and then by product we obtain the reduced expressions of the element of S λ . Concerning the elements of Sn \ S λ , we can choose their reduced expressions arbitrary.

∀w = (w1, . . . , w d ) ∈ S λ , ψw = ψw 1 • • • ψw d ,
We now suppose that the matrix Q verifies:

∀j = j ′ , ∀(k, k ′ ) ∈ Kj × K j ′ , Q k,k ′ = 1. (6.12)
When the matrix Q is associated with a quiver Γ (recall §1.2.2), the condition (6.12) is satisfied when Γ is the disjoint union of d proper subquivers Γ 1 , . . . , Γ d . It means that:

• if v is a vertex in Γ then there is a unique 1 ≤ j ≤ d such that v is a vertex of Γ j ;
• if (v, w) is an edge in Γ then there is a (unique) 1 ≤ j ≤ d such that:

the vertices v and w are vertices of Γ j , -the edge (v, w) is an edge of Γ j .

Such a disjoint union in d proper subquivers was encountered at (1.38). Moreover, regarding the Cartan matrix of Γ we have:

∀j = j ′ , ∀(k, k ′ ) ∈ Kj × K j ′ , c k,k ′ = 0; (6.13)
that is, up to a permutation of the indexing set, the matrix is block diagonal. Finally, for j ∈ J we define: (6.14) in particular for each j ∈ J and for each n ′ ∈ N we have an associated quiver Hecke algebra H n ′ (Q j ).

∀k, k ′ ∈ Kj , Q j k,k ′ := Q k,k ′ ,

Useful idempotents

We define in this section some idempotents of Hn(Q) which are essential for our proof. Thanks to the defining relations (1.18)-(1.20) and (1.30), for each λ |= d n the following element:

e(λ) := t∈J λ k∈K t e(k), ( 6.15)
is a central idempotent in Hn(Q), that is, e(λ) = e(λ) 2 commutes with every element of Hn(Q). Moreover:

• if λ ′ |= d n is different from λ then e(λ)e(λ ′ ) = 0;
• we have λ|= d n e(λ) = 1; hence we have the following decomposition into subalgebras:

Hn(Q) = λ|= d n e(λ)Hn(Q).
(6.16)

For t ∈ J λ , we also define the following idempotent:

e(t) := k∈K t e(k).
We can note that e(λ) = t∈J λ e(t). Moreover, we have e(t)e(t ′ ) = 0 if t ′ ∈ J n \ {t}. We now give some lemmas which involve these elements e(t).

Lemma 6.15. Let t ∈ J n . We have the following relations:

ψaya+1e(t) = yaψae(t) if ta = ta+1, ψayae(t) = ya+1ψae(t) if ta = ta+1, ψ 2 a e(t) = e(t) if ta = ta+1, ψa+1ψaψa+1e(t) = ψaψa+1ψae(t) if ta = ta+2.
Proof. Let us first prove the first one. Let k ∈ K t ; we have ka ∈ Kt a and ka+1 ∈ Kt a+1 with ta = ta+1 thus ka = ka+1. Hence, we get the result using the defining relation (1.24) by summing over all k ∈ K t . The proofs of the second and the last equalities are similar.

Let us now prove ψ 2 a e(t) = e(t) if ta = ta+1. Let k ∈ K t ; we have ka ∈ Kt a and ka+1 ∈ Kt a+1 with ta = ta+1 thus Q ka,k a+1 = 1 (see (6.12)). Hence, the defining relation (1.26) gives ψ 2 a e(k) = e(k), and we again conclude by summing over all k ∈ K t .

About the ψ πt

Here we prove some identities which are satisfied by the elements we have just introduced; some of them will be essential to the proof of Theorem 6.26, while others will only be used in §6.4.3, namely with Lemmas 6.20 to 6.22. We first study some properties about the elements ψπ t for t ∈ J n . We begin by the most important one, which is mentioned in the proof of [START_REF] Shan | On the center of quiver-Hecke algebras[END_REF]Lemma 3.17

ψa 1 • • • ψa r e(t) = ψ b 1 • • • ψ br e(t).
In other words the element ψπ t e(t) ∈ Hn(Q) does not depend on the choice of a reduced expression for πt.

Proof. By Matsumoto's theorem, it suffices to check that every braid relation in sa 1 • • • sa r also occurs in ψa 1 • • • ψa r e(t). By (1.23), it is true for length 2-braids so it remains to check the case of the braids of length 3.

Suppose that we have a braid of length 3 in sa 1 • • • sa r , at rank m: we have am = am+2 = am+1 ±1. We set a := min(am, am+1). With w l := sa 1 • • • sa m-1 and wr := sa m+3 • • • sa r , we have: w l (sasa+1sa)wr = w l (sa+1sasa+1)wr, and we have to prove, with ψ l := ψa 1 • • • ψa m-1 and ψr := ψa m+3 • • • ψa r :

ψ l (ψaψa+1ψa)ψre(t) = ψ l (ψa+1ψaψa+1)ψre(t).
Using (1.20), this becomes, where s := wr • t: ψ l (ψaψa+1ψa)e(s)ψr = ψ l (ψa+1ψaψa+1)e(s)ψr.

(6.17) By Lemma 6.10, we have sa m+1 • (sa m+2 • s) = sa m+2 • s. Thus, we have either sa

• (sa+1 • s) = sa+1 • s or sa+1 • (sa • s) = sa • s;
both cases give sa = sa+2. Hence, applying Lemma 6.15 we know that (6.17) holds.

Remark 6.17. In particular, if k ∈ K t then ψπ t e(k) ∈ Hn(Q) does not depend on the choice of a reduced expression for πt (note that ψπ t e(k) = ψπ t e(t)e(k)).

Similarly to Lemma 6.16, using Remark 6.11 we prove that for t ∈ J λ the element:

e(t)ψ π -1 t = ψ π -1 t e(t λ ) ∈ Hn(Q), (6.18)
does not depend on the chosen reduced expression for π -1 t . We now give some analogues of the results of Lemma 6.15. Proposition 6.18. Let t ∈ J λ . We have:

ψ π -1 t ψπ t e(t) = e(t), ψπ t ψ π -1 t e(t λ ) = e(t λ ).
Remark 6.19. Both factors ψ π -1 t and ψπ t do not depend on the choices of reduced expressions: for instance, using (1.20) we have ψ π -1 t ψπ t e(t) = ψ π -1 t e(t λ )ψπ t thus we can apply Lemma 6.16 and (6.18).

Proof. We only prove the first equality, the proof of the second one being entirely similar. Let sa 1 • • • sa r be a reduced expression for πt. We prove by induction that for every m ∈ {1, . . . , r + 1} we have:

ψ π -1 t ψπ t e(t) = ψa r • • • ψa m ψa m • • • ψa r e(t). (6.19)
First, the case a = 1 comes with the definition of ψ π -1 t e(t λ ) and ψπ t e(t). Now,if (6.19) is true for some m ∈ {1, . . . , r} we have, using (1.20): (6.20) where wm := sa m+1 • • • sa r . By Lemma 6.10, we know that (wm • t)a m = (wm • t)a m+1 . Hence, by Lemma 6.15 we have ψ 2 am e(wm • t) = e(wm • t) thus (6.20) becomes:

ψ π -1 t ψπ t e(t) = ψa r • • • ψa m+1 ψ 2 am e(wm • t)ψa m+1 • • • ψa r ,
ψ π -1 t ψπ t e(t) = ψa r • • • ψa m+1 e(wm • t)ψa m+1 • • • ψa r ,
which becomes, with a last use of (1.20):

ψ π -1 t ψπ t e(t) = ψa r • • • ψa m+1 ψa m+1 • • • ψa r e(t).
Thus (6.19) holds for every m ∈ {1, . . . , r + 1}, in particular for m = r + 1 we get the statement of the Proposition.

Once again, what follows is not essential to the proof of the main result Theorem 6.26; however, it will allow us to relate our construction to the one of [JaPA, PA]. With a similar proof as Proposition 6.18, we obtain the following lemma. Lemma 6.20. Let a ∈ {1, . . . , n} and t ∈ J λ . We have:

yaψπ t e(t) = ψπ t y π -1 t (a) e(t), yaψ π -1 t e(t λ ) = ψ π -1 t y π t (a) e(t λ ).
We now want to see what is happening with Lemma 6.12 for the associated elements ψw. Lemma 6.21. Let t ∈ J n and a ∈ {1, . . . , n -1} such that ta = ta+1. We have:

e(t)ψ π -1 t ψπ sa •t = e(t)ψa.
Proof. By Lemma 6.12 we have ℓ(πs a•t ) = ℓ(πt) ± 1. We now simply distinguish cases.

• We first assume that ℓ(πs a•t ) = ℓ(πt) + 1. Hence, applying Lemma 6.16 for the elements ψπ sa•t and ψπ t we have e(t λ )ψπ sa•t = e(t λ )ψπ t ψa. Finally, using Proposition 6.18 we have:

e(t)ψ π -1 t ψπ sa•t = ψ π -1 t e(t λ )ψπ t ψa = ψ π -1 t
ψπ t e(t)ψa = e(t)ψa.

• We now suppose that ℓ(πs a•t ) = ℓ(πt)-1; hence, ℓ(π -1 t ) = ℓ(π -1 sa•t )+1. Recall that π -1 t = saπ -1 sa•t ; using the extension (6.18) of Lemma 6.16, we get e(t)ψ π -1 t = e(t)ψaψ π -1 sa•t and finally:

e(t)ψ π -1 t ψπ sa•t = ψae(sa • t)ψ π -1 sa•t ψπ sa•t = ψae(sa • t) = e(t)ψa.
Lemma 6.22. Let t ∈ J n and a ∈ {1, . . . , n-1} such that ta = ta+1. The element ψπ t sa e(t) ∈ Hn(Q) does not depend on the choice of a reduced expression for πtsa. In particular: ψπ t ψae(t) = ψ π t (a) ψπ t e(t).

Proof. As in the proof of Lemma 6.16, it suffices to prove that every 3-braid relation which occurs in a reduced expression of πtsa is also verified in the corresponding element of Hn(Q)e(t). By Lemma 6.13, we know that any reduced expression of πtsa can be written s b 0 • • • s br , where there is m ∈ {0, . . . r} such that:

• the word s b 0 • • • ŝbm • • • s br is a reduced expression of πt; • we have s bm = wmsaw -1 m with wm := s b m+1 • • • s br .
We suppose that a 3-braid appears in s b 0 • • • s br at index l, that is, we have b l = b l+2 = b l+1 ± 1; we set b := min(b l , b l+1 ). We want to prove that, with

ψ l := ψ b 0 • • • ψ b l-1 and ψr := ψ b l+3 • • • ψ br : ψ l (ψ b ψ b+1 ψ b )ψre(t) = ψ l (ψ b+1 ψ b ψ b+1 )ψre(t).
To that extent, as in the proof of Lemma 6.16, thanks to Lemma 6.15 it suffices to prove that s b = s b+2 where s := s b l+3 • • • s br • t.

• If m < l + 1 then applying Lemma 6.10 we have

s b l+1 • (s b l+2 • s) = s b l+2 • s thus s b = s b+2 .
• The case m = l + 1 is impossible: as b l = b l+2 , if m = l + 1 we would have bm-1 = bm+1 and this is nonsense since the expression s b m-1 s b m+1 is reduced (as a subexpression of the reduced expression

s b 0 • • • ŝbm • • • s br = πt).
• If m = l + 2 then by Lemma 6.10 we get

s b l • (s b l+1 • s) = s b l • s thus s b = s b+2 .
• Finally, if m > l + 2 then we can notice that:

s = s b l+3 • • • ŝbm • • • s br • t (since s bm • (wm • t) = wm • (sa • t) = wm • t;
recall that s bm = wmsaw -1 m and ta = ta+1). Hence, we deduce once again the result from Lemma 6.10.

The last statement of the lemma is now immediate. As ta = ta+1, we can use (6.9) hence ψπ t sa e(t) = ψπ t ψae(t). Moreover, applying Lemma 6.12 another consequence of (6.9) is ℓ(s π t (a) πt) = ℓ(πt) + 1, thus we get ψπ t sa e(t) = ψ π t (a) ψπ t e(t). Finally, we have ψπ t sa e(t) = ψπ t ψae(t) = ψ π t (a) ψπ t e(t).

Decomposition along the subquiver Hecke algebras

We are now ready to prove the main result of this section: we will give in Theorem 6.26 a decomposition of Hn(Q) involving the algebras Hn j (Q j ) for j ∈ J.

A distinguished subalgebra

In this paragraph, we prove the key of Theorem 6.26. Recall that we have set in (6.14):

∀k, k ′ ∈ Kj , Q j k,k ′ := Q k,k ′ ,
for any j ∈ J. Let λ |= d n be a d-composition of n. We define the following algebra:

H λ (Q) := H λ 1 (Q 1 ) ⊗ • • • ⊗ H λ d (Q d ).
With e λ := e(t λ ), we prove here that we can identify H λ (Q) with the subalgebra e λ Hn(Q)e λ (with unit e λ ). We reindex the generators ψ1, . . . , ψ λ j -1 and y1, . . . , y λ j of H λ j (Q j ), respectively by ψ λ j-1 +1 , . . . , ψ λ j -1 and y λ j-1 +1 , . . . , y λ j . In particular, we set:

ψ ⊗ w := ψw 1 ⊗ • • • ⊗ ψw d ∈ H λ (Q),
for w = (w1, . . . , w d ) ∈ S λ and:

e ⊗ (k 1 , . . . , k d ) := e(k 1 ) ⊗ • • • ⊗ e(k d ) ∈ H λ (Q), (6.21) for k = (k 1 , . . . , k d ) ∈ K λ 1 1 × • • • × K λ d d .
Lemma 6.23. The following family:

ψwy r 1 1 • • • y rn n e(k), w ∈ S λ , ra ∈ N, k ∈ K t λ , (6.22)
is an A-basis of e λ Hn(Q)e λ . Moreover, the algebra e λ Hn(Q)e λ is exactly the (non-unitary) subalgebra of Hn(Q) generated by:

• the ψae λ for a ∈ {1, . . . , n} \ {λ1, . . . , λ d };

• the yae λ for 1 ≤ a ≤ n;

• the e(k) for k ∈ K t λ .

Proof. The first part is a immediate application of Theorem 1.5, (1.19), (1.20) and Proposition 6.3. It remains to check that e λ Hn(Q)e λ is the described subalgebra, subalgebra that we temporarily denote by H. First, all the listed elements belong to e λ Hn(Q)e λ . In particular, for a = λj we have indeed ψae λ ∈ e λ Hn(Q)e λ : we can either use the above basis, or we can simply use (1.20) to get ψae λ = ψae 2 λ = e λ ψae λ . Hence, we have H ⊆ e λ Hn(Q)e λ . Finally, we conclude since every element of the basis (6.22) lies in H. Lemma 6.24. There is a unitary algebra homomorphism from H λ (Q) to e λ Hn(Q)e λ .

Proof. We define the algebra homomorphism from H λ (Q) to e λ Hn(Q)e λ by sending: • the generators e ⊗ (k) ∈ H λ (Q) for k ∈ K t λ to e(k) ∈ e λ Hn(Q)e λ .

• the generators ψ ⊗ a ∈ H λ (Q)
It suffices now to check the defining relations of H λ (Q). We will only check (1.24)-(1.27), the remaining ones being straightforward.

(1.24). Let a / ∈ {λ1, . . . , λ d } and k ∈ K t λ . If ka ∈ Kj , as a = λ j ′ for any j ′ we have ka+1 ∈ Kj (cf. (6.3)). Hence, in H λ (Q) the relation (1.24):

ψ ⊗ a ya+1e ⊗ (k) = (yaψ ⊗ a + 1)e ⊗ (k) if ka = ka+1, yaψ ⊗ a e ⊗ (k) if ka = ka+1,
comes from the corresponding relation in H λ j (Q j ). The same relation:

ψaya+1e(k) = (yaψa + 1)e(k) if ka = ka+1, yaψae(k) if ka = ka+1,
is verified in e λ Hn(Q)e λ , as a relation in Hn(Q).

(1.25). Similar.

(1.26). Similarly, the indices ka, ka+1 are in a same Kj . Hence, the relation (1.26) in H λ (Q) comes from a relation in H λ j (Q j ), and the same relation is verified in e λ Hn(Q)e λ .

(1.27). For j ∈ J and a ∈ {λj-1 + 1, . . . , λj -2}, the relation (1.27) is a relation from H λ j (Q j ), and this same relation is verified in e λ Hn(Q)e λ .

Proposition 6.25. The previous algebra homomorphism H λ (Q) → e λ Hn(Q)e λ is an isomorphism.

In particular, we can identify H λ (Q) to a (non-unitary) subalgebra of Hn(Q).

Proof. We know by Theorem 1.5 that H λ j (Q j ) has for basis:

ψw j y r λ j-1 +1 λ j-1 +1 • • • y r λ j λ j e(k j ) : wj ∈ S λ j , ra ∈ N, k j ∈ K λ j j
, hence, the following family:

ψ ⊗ w y r 1 1 • • • y rn n e ⊗ (k) : w ∈ S λ , ra ∈ N, k ∈ K λ 1 1 × • • • × K λ d d ,
is a basis of H λ (Q). We conclude since by (6.11) the homomorphism of Lemma 6.24 sends this basis onto the basis of e λ Hn(Q)e λ given in Lemma 6.23. (In particular, note that ψ ⊗ w ∈ H λ (Q) for w ∈ S λ is sent to ψwe λ ∈ e λ Hn(Q)e λ .)

Decomposition theorem

We recall the notation m λ for λ |= d n introduced at (6.2). Theorem 6.26. We have an A-algebra isomorphism:

Hn(Q) ≃ λ|= d n Matm λ H λ (Q).
The remaining part of this paragraph is devoted to the proof of Theorem 6.26. Due to (6.16), it suffices to prove that we have an A-algebra isomorphism:

e(λ)Hn(Q) ≃ Matm λ H λ (Q).
(6.23)

Let us label the rows and the columns of the elements of Matm λ H λ (Q) by (t ′ , t) ∈ (J λ ) 2 , and let us write E t ′ ,t for the elementary matrix with one 1 at position (t ′ , t) and 0 everywhere else. Recall the following property verified by the E t ′ ,t :

∀t, t ′ , s, s ′ ∈ J λ , E t ′ ,t E s ′ ,s = δ t,s ′ E t ′ ,s . (6.24)
We have the following A-module isomorphism, where t, t ′ ∈ J λ :

e(t ′ )Hn(Q)e(t) ≃ H λ (Q)E t ′ ,t .
Indeed, let us define: (6.25) by:

Φ t ′ ,t : H λ (Q)E t ′ ,t → e(t ′ )Hn(Q)e(t), Ψ t ′ ,t : e(t ′ )Hn(Q)e(t) → H λ (Q)E t ′ ,t ,
∀v ∈ H λ (Q), Φ t ′ ,t (vE t ′ ,t ) := ψ π -1 t ′ vψπ t , ∀w ∈ e(t ′ )Hn(Q)e(t), Ψ t ′ ,t (w) := (ψπ t ′ wψ π -1 t )E t ′ ,t .
The goal sets of (6.25) are respected, according to (1.20), (6.6) and Proposition 6.25. Indeed, for instance we have, for

v ∈ H λ (Q) ≃ e λ Hn(Q)e λ : Φ t ′ ,t (vE t ′ ,t ) = ψ π -1 t ′ vψπ t = ψ π -1 t ′ e(t λ )ve(t λ )ψπ t = e(π -1 t ′ • t λ )ψ π -1 t ′ vψπ t e(π -1 t • t λ ) Φ t ′ ,t (vE t ′ ,t ) = e(t ′ )ψ π -1 t ′
vψπ t e(t) ∈ e(t ′ )Hn(Q)e(t).

Remark 6.27. Our map Φ t ′ ,t is similar to [START_REF] Shan | On the center of quiver-Hecke algebras[END_REF](20)]. Furthermore, these two maps Φ t ′ ,t and Ψ t ′ ,t are clearly A-linear and by Proposition 6.18 these are inverse isomorphisms. We now set:

Φ λ := t,t ′ ∈J λ Φ t ′ ,t : Matm λ H λ (Q) → e(λ)Hn(Q), Ψ λ := t,t ′ ∈J λ Ψ t ′ ,t : e(λ)Hn(Q) → Matm λ H λ (Q). (6.26)
From the properties of Φ t ′ ,t and Ψ t ′ ,t , the above maps are inverse A-module isomorphisms; it now suffices to check that Ψ λ is an A-algebra homomorphism. This property comes from the following one:

Ψ t ′ ,t (w t ′ ,t )Ψ s ′ ,s (w s ′ ,s ) = Ψ t ′ ,s (w t ′ ,t w s ′ ,s ), (6.27)
where t, t ′ , s, s ′ ∈ J λ , w t ′ ,t ∈ e(t ′ )Hn(Q)e(t) and w s ′ ,s ∈ e(s ′ )Hn(Q)e(s). The equality (6.27) is obviously satisfied when t = s ′ since both sides are zero, thus we assume t = s ′ . We have, using Proposition 6.18 and noticing that wt,s = e(t)wt,s:

Ψ t ′ ,t (w t ′ ,t )Ψ s ′ ,s (w s ′ ,s ) = Ψ t ′ ,t (w t ′ ,t )Ψt,s(wt,s) = (ψπ t ′ w t ′ ,t ψ π -1 t )(ψπ t wt,sψ π -1 s )E t ′ ,t Et,s = ψπ t ′ w t ′ ,t [ψ π -1 t ψπ t e(t)]wt,sψ π -1 s E t ′ ,s = ψπ t ′ w t ′ ,t wt,sψ π -1 s E t ′ ,s Ψ t ′ ,t (w t ′ ,t )Ψ s ′ ,s (w s ′ ,s ) = Ψ t ′ ,s (w t ′ ,t w s ′ ,s ).
Finally, the maps Φ λ and Ψ λ are inverse A-algebra isomorphisms; we deduce the isomorphism (6.23) and thus Theorem 6.26.

Remark 6.28. For k ∈ K t , we write here k * := πt • k ∈ K t λ . Using Proposition 6.18 and Lemmas 6.20, 6.22, we can give the images of the generators of e(λ)Hn(Q) for each t ∈ J λ and k ∈ K t :

Ψ λ (e(k)) = e(k * )Et,t, ∀a ∈ {1, . . . , n}, Ψ λ (yae(k)) = y π t (a) e(k * )Et,t, ∀a ∈ {1, . . . , n -1}, Ψ λ (ψae(k)) = ψ π sa•t saπ -1 t e(k * )Es a•t,t = e(k * )Es a•t,t if ta = ta+1, ψ π t (a) e(k * )Et,t if ta = ta+1 (note that πs a•t saπ -1 t = id if ta = ta+1 s π t (a) if ta = ta+1
, cf. Lemma 6.12). We observe that these images look like those described in [START_REF] Jacon | An isomorphism Theorem for Yokonuma-Hecke algebras and applications to link invariants[END_REF](22)] and [START_REF] Poulain D'andecy | Invariants for links from classical and affine Yokonuma-Hecke algebras[END_REF](3.2)-(3.4)]. Remark 6.29. We consider the setting of Proposition 1.7. We can prove that the algebra isomorphism Hn(Q) ≃ ⊕ λ|= d n Matm λ H λ (Q) is a graded isomorphism (with the canonical gradings on the direct sum, matrix algebras and tensor products). In particular, we have (recall the notation k t of (6.10)):

deg ψ π -1 t ′ e(k) = deg ψπ t e(k t ) = 0,
as a consequence of Lemma 6.10 and Remark 6.11. Indeed, if for s ∈ J n and a ∈ {1, . . . , n -1} we have sa = sa+1 then for any k ∈ K s we have c ka,k a+1 = 0 (cf. (6.13)).

Cyclotomic version

Let us consider a weight Λ = (Λ k ) k∈K ∈ N K ; for j ∈ J, we write Λ j ∈ N K j the restriction of Λ to Kj . We show here how the isomorphism of Theorem 6.26 is compatible with cyclotomic quotients, as defined in (1.33).

Factorisation theorem

For λ |= d n, we define the cyclotomic quotient of H λ (Q) by: • Each element of I can be written as:

H Λ λ (Q) := H Λ 1 λ 1 (Q 1 ) ⊗ • • • ⊗ H Λ d λ d (Q d ), that is, H Λ λ (Q) is the quotient of H λ (Q)
H Λ n (Q) ≃ λ|= d n Matm λ H Λ λ (Q).
v,w∈Hn(Q) t∈J λ k t ∈K t v    Λ k t 1 m=0 cmy m 1 e(k t )    w.
By (1.20), Proposition 6.18 and Lemma 6.20, the previous element becomes:

v,w∈Hn(Q) t∈J λ k t ∈K t vψ π -1 t    Λ k t 1 m=0 cmy m π t (1) e(πt • k t )    ψπ t w.
As Ψ λ is an algebra homomorphism and Matm λ I λ is a two-sided ideal, it suffices to prove that for any k ∈ K t λ and t ∈ J λ we have, with a := πt(1):

Λ ka m=0 cmΨ λ (y m a e(k)) ∈ Matm λ (I λ ).
The above element is exactly:

Λ ka m=0 cmy m a e(k)E t λ ,t λ
(recall that π t λ = id), hence we are done since the element Λ ka m=0 cmy m a e(k) lies in I λ ; in particular, there is a j ∈ J such that a = λj-1 + 1, cf. Proposition 6.7.

• Each element of Matm λ I λ can be written:

t,t ′ ∈J λ v,w∈H λ (Q) k∈K t λ a∈{λ 0 +1,...,λ d-1 +1} v   Λ ka m=0 cmy m a e(k)E t ′ ,t   w.
As Φ λ is an algebra homomorphism and I is a two-sided ideal, it suffices to prove that for any t, t ′ ∈ J λ , k ∈ K t λ and a = λj-1 + 1 for j ∈ J we have:

Λ ka m=0 cmΦ λ (y m a e(k)E t ′ ,t ) ∈ I.
We consider an element s ∈ J λ which verifies s1 = j (we can take for instance s := (1, a) • t λ ); note that πs(1) = λj-1 + 1 = a. Using (6.24), we can write the above element:

Λ ka m=0 cmΦ λ (y m a e(k)E t ′ ,s )Φ λ (Es,t),
hence it suffices to prove that

α := Λ ka m=0 cmΦ λ (y m a e(k)E t ′ ,s ),
belongs to the ideal I. We get:

α = Λ ka m=0 cmψ π -1 t ′ y m a e(k)ψπ s = Λ ka m=0 cmψ π -1 t ′ ψπ s y m π -1 s (a) e(k s ) α = ψ π -1 t ′ ψπ s Λ ka m=0 cmy m 1 e(k s ),
where we recall the notation k s ∈ K s from (6.10). We have k s 1 = ka, thus we get:

α = ψ π -1 t ′ ψπ s Λ k s 1 m=0 cmy m 1 e(k s ) ∈I ,
and we are done.

An alternative proof

We explain here how we can get Theorem 6.30 from [START_REF] Shan | On the center of quiver-Hecke algebras[END_REF]§3.2.3]. We make the following assumption:

Q k,k ′ (u, v) = r k,k ′ (u -v) -c k,k ′ ,
where the r k,k ′ , c k,k ′ are some scalars; we refer to [START_REF] Shan | On the center of quiver-Hecke algebras[END_REF]§3.2.2] for more details about the setting.

In particular, we suppose that Λ = k∈K Λ k ω k where the ω k are the fundamental weights related to the ambient Cartan datum. We define the set K Λ by (cf. [START_REF] Shan | On the center of quiver-Hecke algebras[END_REF]§3.1.3]):

K Λ := (k, m) : k ∈ K, m ∈ {1, . . . , Λ k } ;
we will also use the sets K Λ j := (Kj ) Λ j for j ∈ J. For t = (k, m) ∈ K Λ , we write kt := k and ωt := ω k . Finally, for convenience we introduce some notation:

• we write |= Λ instead of |=K Λ ;

• if µ (respectively ν) is an r-composition (resp. s-composition), we set m ν µ := ν 1 !...νs! µ 1 !...µr ! ; in particular with the 1-partition ν := (µ1 + • • • + µr) we recover m ν µ = mµ. Theorem 6.31 [START_REF] Shan | On the center of quiver-Hecke algebras[END_REF]Theorem 3.15]). There is an algebra isomorphism

H Λ n (Q) ≃ (nt)t|= Λ n Matm (n t ) t∈K Λ H ωt nt (Q) .
If we look closer to the proof of [SVV], comparing to the proof of Theorem 6.26 the idea is still to consider some elements e(t) but with idempotents which "refine" the idempotents e(k) (these idempotents are indexed by K n Λ , which is much bigger than K n ). In particular, the following isomorphism for any λ |= d n, writing (n j t )t for the restriction of (nt)t∈K Λ to K Λ j :

Matm λ H Λ λ (Q) ≃ (nt)t|= Λ n s.t. ∀j,(n j t )t|= Λ j λ j Matm (n t ) t∈K Λ H ωt nt (Q) (6.29)
implies our Theorem 6.30 by summing over all λ |= d n. In order to prove (6.29), we can simply apply Theorem 6.31 to the factors H Λ j λ j (Q j ) of H Λ λ (Q). We have, for j ∈ J:

H Λ j λ j (Q j ) ≃ (n j t ) t |= Λ j λ j Matm (n j t )   t∈K Λ j H ωt n j t (Q j )   .
(6.30)

Before going further, we give the following lemma. Let us mention that we can find a noncyclotomic statement in [START_REF] Rouquier | 2-Kac-Moody algebras[END_REF]Corollary 3.8]; see also [START_REF] Mathas | Cyclotomic quiver Hecke algebras of type A, in Modular representation theory of finite and p-adic groups[END_REF]Proposition 2.4.6] and [START_REF] Boys | Quiver Hecke algebras for alternating groups[END_REF]Lemma 1.16

]. Lemma 6.32. Let j ∈ J. If k ∈ Kj then H ω k n (Q) ≃ H ω k n (Q j ). Proof. It suffices to prove that every e(k) ∈ Hn(Q) with k ∈ K n \ K n j vanishes in H ω k n (Q).
To that extent, we prove by induction on a ∈ {1, . . . , n} the following statement:

∀k ∈ K n , [∃b ∈ {1, . . . , a}, k b / ∈ Kj =⇒ e(k) = 0 in H ω k n (Q)]. (6.31)
First, we shall verify this proposition for a = 1: let k ∈ K n such that ∃b ∈ {1, . . . , 1}, k b / ∈ Kj . We obviously have k1 / ∈ Kj , in particular k1 = k thus it follows directly from the cyclotomic condition (1.33) that e(k) = 0 in H ω k n (Q). We now assume that (6.31) is verified for some a ∈ {1, . . . , n -1} and we let k ∈ K n such that ∃b ∈ {1, . . . , a + 1}, k b / ∈ Kj . We know from the induction hypothesis that e(k) = 0 in H

ω k n (Q) if ka /
∈ Kj or ka+1 ∈ Kj , hence it remains to deal with the case ka ∈ Kj and ka+1 / ∈ Kj . Recalling (6.12), this implies that Q ka,k a+1 = 1. In particular, the defining relation (1.26) becomes: ψ 2 a e(k) = e(k). (6.32)

Besides, using (1.20) and the induction hypothesis, we have:

ψae(k) = e(sa • k) =0 ψa = 0 in H ω k n (Q).
Thus, left-multiplying by ψa and using (6.32) we get e(k) = 0 in H ω k n (Q) which ends the induction. Finally, for a = n, we get that if k ∈ K n \ K n j then e(k) = 0 in H

ω k n (Q).
We now go back to the proof of (6.29). Using Lemma 6.32, the isomorphism (6.30) gives: Finally, we deduce (6.29) and thus Theorem 6.30 from the equality m λ m λ (nt) = m (nt) .

H Λ λ (Q) ≃ j∈J H Λ j λ j (Q j ) ≃ j∈J (

An application

Here we explain how our Theorem 6.30 is related to our previous result Theorem 4.1. We suppose that A = C and that all the Kj for j ∈ J ≃ {1, . . . , d} have the same finite cardinality e ≥ 2.

Identifying the sets Kj with I = Z/eZ, we have K ≃ K = I × J. We also assume that

Λ 1 = • • • = Λ d ;
in particular, we will simply write Λ for each of these parts. Finally, we consider q ∈ C × a primitive eth root of unity, and we write: BK : H Λ n (q)

∼ → H Λ n (Γe) for the C-algebra isomorphism of [BrKl].

We have an algebra isomorphism, constructed by Poulain d'Andecy [PA]:

JPA : Y Λ d,n (q) ∼ → λ|= d n
Matm λ H Λ λ (q), where H Λ λ (q) := H Λ λ 1 (q) ⊗ • • • ⊗ H Λ λ d (q). This isomorphism is a generalisation of the main result of [JaPA] (which is in fact a particular case of a result of Lusztig [Lu]), and is defined on the generators as follows: JPA(ta) = • our elements E t ′ ,t are written Eχ, where χ is a character of (Z/dZ) n = J n ;

• Poulain d'Andecy considers left cosets instead of our right ones, in particular his minimal length representatives πχ verify πχ = π -1 t . We recall from (1.38) that Γ = ∐j∈J Γe; in particular, its vertex set is exactly K ≃ K = I ×J. The two previous results, together with our Theorem 6.30, gives straightforwardly the following theorem. Theorem 6.34. We have an algebra isomorphism:

Φ Λ n • BK • JPA : Y Λ d,n (q) ≃ H Λ n (Γ),
where:

• the homomorphism BK : ⊕ λ Matm λ H Λ λ (q) → ⊕ λ Matm λ H Λ λ (Γ) is naturally induced by BK : H Λ n (q) → H Λ n (Γe); • the homomorphism Φ Λ n : ⊕ λ Matm λ H Λ λ (Γ) → H Λ n (Γ) is the isomorphism of Theorem 6.30, that is, induced by the homomorphism Φn : ⊕ λ Matm λ H λ (Γ) → Hn(Γ).

An algebra isomorphism Y Λ d,n (q) → H Λ n (Γ) was already constructed in Theorem 4.1; we shall denote it by BK. An interesting question is to know whether we recover the same isomorphism as above. In other words, does the diagram of Figure 3 As we deal with algebra homomorphisms, it suffices to check that the images of the generators of Y Λ d,n (q) are the same. We will use the following notation: for t ∈ J n we set t * := πt • t. (With λ := [t], we have of course t * = t λ .) Moreover, we will keep on using the notation t ∈ J n of Section 6 for the elements we denoted by j ∈ J n from Section 1 to Section 5. Image of X a . (It is in fact enough to study the case a = 1). Let 1 ≤ a ≤ n. Recall from §3.1 that: BK(Xa) = i∈I n q ia (1 -ya)e(i) = k∈K n q ka (1 -ya)e(k) ∈ H Λ n (Γ), (6.37)

Image of t

where we write q k := q i for k = (i, j) ∈ K = I × J. Recalling (6.34), we get:

BK • JPA(Xa) = t∈J n X π t (a) Et,t ∈ λ|= d n
Matm λ H Λ λ (Γ).

We write, where j := ta and λ := [t]: a) (1 -y π t (a) )e(k j ) ∈ H Λ λ j (Γe) ⊆ H Λ λ (Γ),

X π t (a) = k j ∈K λ j j q k j π t (
where k j ∈ K λ j j is indexed by (λj-1 + 1, . . . , λj ). Hence, we have: where we have used Lemma 6.20. Hence, using Proposition 6.18 we finally get:

X π t (a) = k∈K t λ q k π t (a) (
Φ Λ n • BK • JPA(Xa) =
t∈J n k∈K t q ka (1 -ya)e(k) = k∈K n q ka (1 -ya)e(k), which is (6.37).

Image of g a . Let 1 ≤ a < n. Recall from §3.1 that: BK(ga) = With j := ta and λ := [t], we have:

g π t (a) = k j ∈K λ j j (ψ π t (a) Q π t (a) (k j ) -P π t (a) (k j ))e(k j ) ∈ H Λ λ j (Γe) ⊆ H Λ λ (Γ)
(recall that k j ∈ K λ j j is indexed by (λj-1 + 1, . . . , λj )), hence:

g π t (a) = k∈K t λ (ψ π t (a) Q π t (a) (k) -P π t (a) (k))e(k) ∈ H Λ λ (Γ).
We obtain: We first focus on the first sum S1; let t ∈ J n such that ta = ta+1 and k ∈ K t * . We can notice that thanks to Proposition 6.7 we have πt(a + 1) = πt(a) + 1. Using Lemma 6.20 and the properties of Q we have (recalling the notation k t introduced at (6.10)): Q π t (a) (k)e(k)ψπ t = Q π t (a) (k)(y π t (a) , y π t (a)+1 )ψπ t e(k t ) = ψπ t Q π t (a) (k)(ya, ya+1)e(k t ) = ψπ t Qa(k t )(ya, ya+1)e(k t ) Q π t (a) (k)e(k)ψπ t = ψπ t Qa(k t )e(k t ).

Φ Λ n •
The same proof gives P π t (a) (k)e(k)ψπ t = ψπ t Pa(k t )e(k t ). We have thus: which is (6.38).

ψ π -1 t (ψ π t (a) Q π t (
To conclude, we have checked that the algebra homomorphisms BK and Φ Λ n • BK • JPA coincide on every generator of Y Λ d,n (q), hence we have the following theorem.

  ψ1ψ0ψ1e(012) = ψ1ψ0e(021)ψ1 = ψ1e(201)ψ0ψ1 ψ1ψ0ψ1e(012) = Φ1Q1(201) -1 e(201)ψ0ψ1.

  1.6). Let us prove gag b e(k) = g b gae(k) for every k ∈ K α . By (1.22) the element ψ b commutes with the elements Pa(k) and Qa(k) of F [[ya, ya+1]]. Moreover, Qa(s b • k) = Qa(k) and Pa(s b • k) = Pa(k), hence:

  )ψ1Q1(102)ψ0 -αψ0e(102)P1(102)ψ0.Noticing f 1,(012) = f 0,(120) = f 0,(102) and f 1,(201) = f 0,(012) we get finally: g1g0g1e(012) = f 0,(012) g0(ψ1Q1(102) -P1(102))e(102)ψ0 = f 0,(012) g0g1ψ0e(012) g1g0g1e(012) = g0g1g0e(012).

  .8) faxa+1 = xafa + ea, (5.9) fax b = x b fa ∀b = a, a + 1, (5.10) xat b = t b xa,(5.11)

  ]. Lemma 6.16. Let t ∈ J n . If sa 1 • • • sa r and s b 1 • • • s br are two reduced expressions of πt, then:

  for a ∈ {1, . . . , n} \ {λ1, . . . , λ d } to ψae λ ∈ e λ Hn(Q)e λ ; • the generators y b ∈ H λ (Q) for b ∈ {1, . . . , n} to y b e λ ∈ e λ Hn(Q)e λ ;

Proof.

  Let λ |= d n and let I (resp. I λ ) be the two-sided ideal of e(λ)Hn(Q) (resp. H λ (Q)) generated by the elements in (1.33) (resp. (6.28)). It suffices to prove that Ψ λ (I) = Matm λ I λ : we will prove Ψ λ (I) ⊆ Matm λ I λ and I ⊇ Φ λ (Matm λ I λ ).

  Figure 3: A commutative diagram?

  k), Pa(k) ∈ C[[ya, ya+1]] are some power series. For convenience, we write Qa(k), Pa(k) ∈ C[[Ya, Ya+1]] the underlying power series, which verify Qa(k)(ya, ya+1) = Qa(k) and Pa(k)(ya, ya+1) = Pa(k). These power series depend only on ka and ka+1, that is,Qa(k)(Y, Y ′ ) = Q a ′ (k ′ )(Y, Y ′ ) and Pa(k)(Y, Y ′ ) = P a ′ (k ′ )(Y, Y ′ ) if ka = k ′ a ′ and ka+1 = k ′ a ′ +1. Moreover, recall that if k and sa • k are labellings of two different t ∈ J n , we have Qa(k) = √ q and Pa(k) = 0 (cf. Remark 2.11).Recalling (6.35), we get:BK • JPA(ga) = t∈J n ta=t a+1 g π t (a) Et,t + t∈J n ta =t a+1 √ qEt,s a•t ∈ λ|= d n Matm λ H Λ λ (Γ).

  k) -Pa(k))e(k) = t∈J n k∈K t (ψaQa(k) -Pa(k))e(k) Φ Λ n • BK • JPA(ga) = k∈K n (ψaQa(k) -Pa(k))e(k),

  by the two-sided ideal generated by the elements:

	Λ ka
	m=0	cmy m a e(k) = 0	∀k ∈ K t λ	, ∀a ∈ {λ0 + 1, . . . , λ d-1 + 1}.	(6.28)
	Theorem 6.30. The isomorphism of Theorem 6.26 factors through the cyclotomic quotients, in
	other words we have:

  n j t ) t |= Λ j λ j

											
					Matm (n j t )		t∈K Λ j	H ωt t n j	(Q)	 .
	We obtain:										
	H Λ λ (Q) ≃	(nt)t|= Λ n s.t. ∀j,(n j t )t|= Λ j λ j	j∈J	Matm (n j t )		t∈K Λ j 	H ωt j n t	(Q)		
	≃	(nt)t|= Λ n s.t. ∀j,(n t )t|= Λ j λ j j	Matm (n 1 t )	•••m (n d t )		j∈J t∈K Λ j	H ωt t n j	(Q)	
	H Λ λ (Q) ≃	(nt)t|= Λ n s.t. ∀j,(n t )t|= Λ j λ j j	Mat m λ (n t )	t∈K Λ	H ωt nt (Q) .

  Remark 6.33. Note two slight differences with[PA]:

		ξ ta Et,t,		(6.33)
	t∈J n
	JPA(Xa) =	X π t (a) Et,t,		(6.34)
	t∈J n
	JPA(ga) =	g π t (a) Et,t +	√ qEt,s a•t .	(6.35)
	t∈J n	t∈J n
	ta=t a+1	ta =t a+1

  Let 1 ≤ a ≤ n. Recall from §3.1 that: BK(ta) =

	j∈J n	e(j)ξ ja =	t∈J n	e(t)ξ ta ∈ H Λ n (Γ).	(6.36)
	Recalling (6.33), we obtain:
	BK • JPA(ta) =					λ|= d n	Matm λ H Λ λ (Γ).
	Hence, with the usual manipulations, we get:
	Φ Λ n • BK • JPA(ta) =	t∈J n k∈K t *	ξ ta ψ π -1 t	ψπ t e(π -1 t	• k)
		=			ξ ta e(k)
			t∈J n k∈K t
	Φ Λ n • BK • JPA(ta) =	t∈J n	ξ ta e(t) ∈ H Λ n (Γ),
	thus it coincides with (6.36).

a .

t∈J n k∈K t * ξ ta e(k)Et,t ∈

  1 -y π t (a) )e(k) ∈ H Λ -y π t (a) )e(k)ψπ t = t∈J n k∈K t * q k π t (a) ψ π -1 t (1 -y π t (a) )ψπ t e(k t ) Φ Λ n • BK • JPA(Xa) = t∈J n k t ∈K t

			λ (Γ),
	and thus:
	Φ Λ n • BK • JPA(Xa) =	t∈J n k∈K t *	q k π t (a) Φ Λ n (1 -y π t (a) )e(k)Et,t
	=	t∈J n k∈K t *	q k π t (a) ψ π -1 t (1 q k t a ψ π -1 t ψπ t (1 -ya)e(k t ),

  BK • JPA(ga) = (ψ π t (a) Q π t (a) (k) -P π t (a) (k))e(k)ψπ t

		ψ π -1 t
	t∈J n ta=t a+1	k∈K t *
		S 1
		+	√ qψ π -1 t	e(k)ψπ sa •t .	(6.39)
		t∈J n ta =t a+1	k∈K t *
			S 2

  a) (k) -P π t (a) (k))e(k)ψπ t = ψ π -1 t (ψ π t (a) ψπ t Qa(k t ) -ψπ t Pa(k t ))e(k t ).Using (1.19) and Lemma 6.22, we obtain:ψ π -1 t (ψ π t (a) Q π t (a) (k) -P π t (a) (k))e(k)ψπ t = ψ π -1 t ψπ t (ψaQa(k t ) -Pa(k t ))e(k t ).Finally, since sa • t = t, we get by Proposition 6.18:ψ π -1 t (ψ π t (a) Q π t (a) (k) -P π t (a) (k))e(k)ψπ t = (ψaQa(k t ) -Pa(k t ))e(k t ),so the first sum becomes:(ψaQa(k t ) -Pa(k t ))e(k t ). (6.40)We now focus on the second sum S2; let t ∈ J n with ta = ta+1 and let k ∈ K t * . By Lemma 6.21 we have directly:ψ π -1(Qa(k t )ψa -Pa(k t ))e(k t ).

		ψ π -1
	t∈J n ta=t a+1	k∈K t *
						=
						t∈J n ta=t a+1	k t ∈K t
				√ qψ π -1 t	e(k)ψπ sa•t =	√ qe(k t )ψa
		t∈J n ta =t a+1	k∈K t *		t∈J n ta =t a+1	k t ∈K t
						=	t∈J n ta =t a+1	k t ∈K t	√ qψae(sa • k t )
						=	√ qψae(k sa•t )
						t∈J n ta =t a+1	k sa•t ∈K sa•t
						=	√ qψae(k t )
						t∈J n ta =t a+1	k t ∈K t
				√ qψ π -1 t	e(k)ψπ sa•t =	(6.41)
		t∈J n ta =t a+1	k∈K t *		t∈J n ta =t a+1	k t ∈K t
	Finally, by (6.39)-(6.41) we get:
	Φ Λ n • BK • JPA(ga) =	t∈J n ta=t a+1	k∈K t	(ψaQa(k) -Pa(k))e(k) +	t∈J n ta =t a+1

t (ψ π t (a) Q π t (a) (k) -P π t (a) (k))e(k)ψπ t t e(k)ψπ sa•t = e(k t )ψ π -1 t ψπ sa •t = e(k t )ψa, thus:

Acknowledgements I am grateful to Maria Chlouveraki for many discussions about the proof and a careful reading of the paper, to Nicolas Jacon for his corrections and to Loïc Poulain d'Andecy for useful conversations around this work.

with N ≫ 0; since M is a finite-dimensional F -vector space, only finitely many M (i, j) are non-zero. Considering once again the family of projections {e(k)} k∈K n (5.19)

In particular, with e(α)(i) for d = 1 we recover the element e(i) of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§3.1].

From now on, unless mentioned otherwise we always work in Y Λ α (1); every relation should be multiplied by e(α) and we write e(i) and e(j) without any (α).

Lemma 5.5. If 1 ≤ a < n and j ∈ J α is such that ja = ja+1 then we have:

Lemma 5.6. For 1 ≤ a < n and j ∈ J α we have fae(j) = e(sa • j)fa. In particular, if ja = ja+1 then fa and e(j) commute. Moreover, if ja = ja+1 then fae(i, j) = e(sa • (i, j))fa.

Remark 5.7 (About Brundan and Kleshchev's proof -III). Let 1 ≤ a < n; if j ∈ J α verifies ja = ja+1, when a proof in [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§3.3] needs only the elements fa, x b , e(i) and the corresponding relations in H Λ α (1), we claim that the same proof holds in e(j) Y Λ α (1)e(j). We extend this claim to the case ja = ja+1 = ja+2.

Image of y a

We define the following elements of Y Λ α (1) for 1 ≤ a ≤ n:

When d = 1 we recover the elements defined in [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]§3.3]. These elements are nilpotent: we will be able to make calculations in the ring F [[y1, . . . , yn]].

Lemma 5.8. For j ∈ J α such that ja = ja+1 we have:

Image of ψ a

We first define some elements pa(i, j) ∈ F [[ya, ya+1]] for 1 ≤ a < n and (i, j) ∈ K α by:

and then some invertible elements qa(i, j) ∈ F [[ya, ya+1]] × for 1 ≤ a < n and (i, j) ∈ K α by:

Proof. First, from (6.5) we have πtsa • (sa • t) = t λ (where λ |= d n is the shape of t) thus πtsa lies in the coset Cs a •t.

We suppose that ta = ta+1. We have πtsaπ -1 t = (πt(a), πt(a + 1)), and we conclude since πt(a + 1) = πt(a) + 1 by Proposition 6.7.

We now suppose that ta = ta+1. Using the same Proposition 6.7, we know that the permutation w := π -1 t πs a •t ∈ Sn is supported in {a, a + 1}; thus either w = sa or w = id. Since t = sa • t we have πt = πs a•t , hence w = id. Hence, we get w = sa, that is, πtsa = πs a•t .

We now generalise the result of Lemma 6.12 in the case ta = ta+1. m for some m ∈ {0, . . . , r}, then:

is a reduced expression. Moreover, every reduced expression of πtsa is as above.

Proof. We first make an observation. As ta = ta+1, we deduce from (6.5) that the element πtsa remains in Ct. Hence, by minimality of πt we have:

(6.9)

Let now s b 1 • • • s br be a reduced expression of πt and let b ∈ {1, . . . , n -1} and m ∈ {0, . . . r} such that s b = wmsaw -1 m . We have:

and this expression is reduced since ℓ(πtsa

r be a reduced expression of πtsa. Since ℓ((πtsa)sa) < ℓ(πtsa), we can apply [START_REF] Humphreys | Reflection groups and Coxeter groups[END_REF]§5.8 Theorem]: we know that there is some m ∈ {0, . . . r} such that

r is a reduced expression of πt, where the hat denotes the omission. We have: Moreover:

m if and only if wm(a + 1) = wm(a) ± 1. Moreover, as Lemma 6.10 ensures that wm(a + 1) > wm(a), we have wm(a + 1) = wm(a) ± 1 ⇐⇒ wm(a + 1) = wm(a) + 1.

We end this subsection by introducing a notation. If t ∈ J λ and k ∈ K t λ , we define: (6.10) in particular, we may denote by k t the elements of K t .