Salim Rostam

Cyclotomic quiver Hecke algebras and Hecke algebra of G(r, p, n)

Given a quiver automorphism with nice properties, we give a presentation of the fixed point subalgebra of the associated cyclotomic quiver Hecke algebra. Generalising an isomorphism of Brundan and Kleshchev between the cyclotomic Hecke algebra of type G(r, 1, n) and the cyclotomic quiver Hecke algebra of type A, we apply the previous result to find a presentation of the cyclotomic Hecke algebra of type G(r, p, n) which looks very similar to the one of a cyclotomic quiver Hecke algebra. In addition, we give an explicit isomorphism which realises a well-known Morita equivalence between Ariki-Koike algebras.

Introduction

Generalising real reflection groups, also known as finite Coxeter groups, complex reflection groups are finite groups generated by complex reflections, that is, by endomorphisms of C n which fix a hyperplane. As for Coxeter groups, there is a classification of irreducible complex reflection groups ( [ShTo]). This classification is given by an infinite series {G(r, p, n)} where r, p, n are positive integers with r = dp for d ∈ N * , together with 34 exceptional groups. More precisely, the group G(r, p, n) can be seen as the group consisting of all n×n monomial matrices such that each non-zero entry is a complex rth root of unity, and the product of all non-zero entries is a dth root of unity. If ξ ∈ C × is a primitive rth root of unity, the latter group is generated by the elements:

s := ξ p E 1,1 + n k=2 E k,k , t 1 := ξE 1,2 + ξ -1 E 2,1 + n k=3 E k,k , t a := E a,a+1 + E a+1,a + 1≤k≤n k =a,a+1
E k,k , for all a ∈ {1, . . . , n -1}, where E k,ℓ is the elementary n × n matrix with 1 as the (k, ℓ)-entry and 0 everywhere else. Inspired by work in the context of finite Chevalley groups, an algebra was associated with each real reflection group W : the Iwahori-Hecke algebra H(W ). This deformation of the group algebra of W was the starting point of many connections with other objects and theories, for instance, the theory of quantum groups or knot theory. Aiming at generalising this construction, Broué, Malle and Rouquier [BMR] defined such a deformation for every complex reflection group, also known as Hecke algebra. Besides, Ariki and Koike [ArKo] defined such a Hecke algebra H n (q, u) for G(r, 1, n) where q and u = (u 1 , . . . , u r ) are some parameters, followed by Ariki [START_REF] Ariki | Representation theory of a Hecke algebra of G(r, p, n)[END_REF] who did the same thing for G (r, p, n). In particular, for a suitable choice of parameters q and u, this Hecke algebra H Λ p,n (q) of G(r, p, n) can be seen as a subalgebra of H n (q, u).

In the semisimple case, Ariki and Koike have determined all irreducible modules for H n (q, u). The modular case was treated by Ariki and Mathas [START_REF] Ariki | The number of simple modules of the Hecke algebra of type G(r, 1, n)[END_REF][START_REF] Ariki | On the classification of simple modules for cyclotomic Hecke algebras of type G(m, 1, n) and Kleshchev multipartitions[END_REF], and also by Graham and Lehrer [GrLe] and Dipper, James and Mathas [DJM], using the theory of cellular algebras. Moreover, in [START_REF] Ariki | On the decomposition number of the Hecke algebra of G(m, 1, n)[END_REF], Ariki proved a conjecture of Lascoux, Leclerc and Thibon [LLT]. The theorem has the following consequence in characteristic 0. If all u k for 1 ≤ k ≤ r are powers of q, then determining the decomposition matrix of H n (q, u) or the canonical basis of a certain integrable highest weight sl e -module L(Λ) are equivalent problems, where sl e denotes the Kac-Moody algebra of type A

(1) e-1 . Together with the work of Uglov [Ug] which computes this canonical basis, we are thus able to explicitly describe the decomposition matrix of H n (q, u).

Once again in the semisimple case, Ariki [START_REF] Ariki | Representation theory of a Hecke algebra of G(r, p, n)[END_REF] used Clifford theory to determine all irreducible modules for H Λ p,n (q). In the modular case, Genet and Jacon [GeJa] and Chlouveraki and Jacon [ChJa] gave a parametrisation of the simple modules of H Λ p,n (q) over C, and Hu [START_REF] Hu | Modular representations of Hecke algebras of type G(p, p, n)[END_REF][START_REF] Hu | Crystal bases and simple modules for Hecke algebras of type G(p, p, n)[END_REF] classified them over a field containing a primitive pth root of unity. Furthermore, Hu and Mathas [HuMa09,[START_REF] Hu | Decomposition numbers for Hecke algebras of type G(r, p, n): the (ε, q)-separated case[END_REF] gave a procedure to compute the decomposition matrix of H Λ p,n (q) in characteristic 0, under a separation condition (where the Hecke algebra is not semisimple in general). Let us also mention the work of Geck [Ge], who deals with the case of type D (corresponding to r = p = 2).

Partially motivated by Ariki's theorem, Khovanov and Lauda [KhLau09,[START_REF] Khovanov | A diagrammatic approach to categorification of quantum groups II[END_REF] and Rouquier [Rou] independently introduced the algebra R n (Γ), known as a quiver Hecke algebra or KLR algebra. This led to a categorification result:

U - q (g Γ ) ≃ n≥0 [Proj(R n (Γ))] ,
where U - q (g Γ ) is the negative part of the quantum group of g Γ , the Kac-Moody algebra associated with the quiver Γ, and [Proj(R n (Γ))] denotes the Grothendieck group of the additive category of finitely generated graded projective R n (Γ)-modules. Moreover, considering some cyclotomic quotients R Λ n (Γ) of the quiver Hecke algebra, Kang and Kashiwara [KanKa] also proved a categorification result for the highest weight U q (g Γ )-modules.

With Γ a quiver of type A

(1)

e-1 and a suitable choice of u, we thus obtain a connection between the Ariki-Koike algebra H n (q, u) and R Λ n (Γ). Brundan and Kleshchev [BrKl] (and independently Rouquier [Rou]) gave an explicit isomorphism between these two algebras. In particular, the Ariki-Koike algebra inherits the natural Z-grading of R Λ n (Γ). Further, Hu and Mathas [START_REF] Hu | Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A[END_REF] constructed a homogeneous (cellular) basis of the Ariki-Koike algebra.

We aim to generalise the previous results concerning G (r, 1, n) to the remaining elements of the infinite series {G(r, p, n)} (and thus for all but finitely many complex reflection groups). In this paper, we restrict the isomorphism of [BrKl] to the subalgebra H Λ p,n (q) of H n (q, u) (for a suitable choice of u), and then study its image in the cyclotomic quiver Hecke algebra R Λ n (Γ). Our two main results are the following.

• We obtain a cyclotomic quiver Hecke-like presentation for H Λ p,n (q) (see Corollary 4.16). • We find an isomorphism which makes explicit the Morita equivalence of Ariki-Koike algebras of [DiMa] (see §3.4).

We will first need to generalise the main result of [BrKl]. Surprisingly, combined with a theorem of [START_REF] Rostam | Cyclotomic Yokonuma-Hecke algebras are cyclotomic quiver Hecke algebras[END_REF], this leads to the isomorphism realising the above Morita equivalence.

We then exploit the fact that H Λ p,n (q) is the fixed point subalgebra of H n (q, u) for a particular automorphism σ, the shift automorphism. After some technical work, we obtain the cyclotomic quiver Hecke-like presentation for H Λ p,n (q). We deduce that H Λ p,n (q) depends only on the quantum characteristic and is a graded subalgebra of H n (q, u). We note that Boys and Mathas [Bo, BoMa] already studied a restriction of the isomorphism of [BrKl] to the fixed point subalgebra of an automorphism, in order to study alternating (cyclotomic) Hecke algebras. The approach taken here is globally similar, however, we are here able to use a trick of Stroppel and Webster [StWe] to simplify the final proof (see §4.1).

We now give a brief overview of this article. Let r, p, d, n ∈ N * be some integers with r = dp, an element q = 0, 1 of a field F , a primitive pth root of unity ζ ∈ F × and J := Z/pZ ≃ ζ . We set e ∈ N ≥2 ∪ {∞} the order of q in F × and we define I := Z/eZ (with I := Z if e = ∞). Finally, we consider a tuple Λ = (Λ i,j ) where (i, j) ∈ I × J and Λ i,j ∈ N. and the referee for their numerous corrections. I also thank Andrew Mathas for his comments on a preliminary version of this paper, and Ivan Marin for discussions concerning the presentation of G (r, p, n).

Notation Let d, p, n ∈ N * with n ≥ 2 and set r := pd. We work over a field F that contains a primitive pth root of unity ζ, in particular, the characteristic of F does not divide p. We consider an element q ∈ F \ {0, 1} and we write e ∈ N ≥2 ∪ {∞} its order in F × . We define:

I := Z/eZ if e < ∞, Z otherwise (e = ∞),
and

J := Z/pZ ≃ ζ .
1 The Hecke algebra of G(r, p, n)

Here we review two constructions of Ariki and Koike. Let u = (u 1 , . . . , u r ) be a tuple of elements of F × .

The Hecke algebra of G(r, 1, n)

We recall here the definition of the cyclotomic Hecke algebra H n (q, u) of type G(r, 1, n), also known as Ariki-Koike algebra.

Definition 1.1 ( [ArKo]). The algebra H n (q, u) is the unitary associative F -algebra generated by the elements S, T 1 , . . . , T n-1 , subject to the following relations:

r k=1 (S -u k ) = 0, (1.2a) (T a + 1)(T a -q) = 0, (1.2b) ST 1 ST 1 = T 1 ST 1 S, (1.2c) ST a = T a S if a > 1, (1.2d) T a T b = T b T a if |a -b| > 1, (1.2e) T a T a+1 T a = T a+1 T a T a+1 . (1.2f)
According to [BMR], the algebra H n (q, u) is a Hecke algebra of the complex reflection group G(r, 1, n). Let X 1 := S and define for a ∈ {1, . . . , n-1} the elements X a+1 ∈ H n (q, u) by: qX a+1 := T a X a T a .

(1.3) [START_REF] Ariki | A Hecke algebra of Z/nZ ≀ S n and construction of its irreducible representations[END_REF]Theorem 3.10]). The elements

X m1 1 • • • X mn n T w (1.5)
for m 1 , . . . , m n ∈ {0, . . . , r -1} and w ∈ S n form a basis of the F -vector space H n (q, u).

Let Λ = (Λ i,j ) ∈ N (I×J) be a finitely-supported tuple of non-negative integers with i∈I j∈J Λ i,j = r. We say that Λ is a weight of level r. In particular, we can choose the parameters u 1 , . . . , u r such that the relation (1.2a) in H n (q, u) becomes the following one:

i∈I j∈J (S -ζ j q i )
Λi,j = 0.

(1.6) Definition 1.7. In the above setting, we define H Λ n (q, ζ) := H n (q, u). We will often need the following condition on Λ:

Λ i,j = Λ i,j ′ =: Λ i
for all i ∈ I and j, j ′ ∈ J.

(1.8)

In this case, the weight (Λ i ) i∈I has level d = r p . Moreover, we can write (1.6) as:

i∈I j∈J (S -ζ j q i ) Λi = i∈I (S p -q pi ) Λi = 0.
Thus, we get the following result.

Proposition 1.9. Suppose that Λ satisfies (1.8). There is a well-defined algebra homomorphism σ :

H Λ n (q, ζ) → H Λ n (q, ζ)
given by: σ(S) := ζS, σ(T a ) := T a , for all a ∈ {1, . . . , n -1}.

Remark 1.10. The homomorphism σ has order p, in particular σ is bijective. We will refer to σ as the shift automorphism of H Λ n (q, ζ). In the remaining part of this subsection, we assume that (1.8) is satisfied, so that the shift automorphism is defined. The following lemma is an easy induction.

Lemma 1.11. For every a ∈ {1, . . . , n} we have σ(X a ) = ζX a .

Proposition 1.12. The elements of H Λ n (q, ζ) fixed by σ are exactly the elements in the

F -span of X m1 1 • • • X mn n T w for m 1 , . . . , m n ∈ {0, .
. . , r -1} and w ∈ S n , with the additional following condition:

m 1 + • • • + m n = 0 (mod p).
Proof. Let h be an arbitrary element of H Λ n (q, ζ). By Theorem 1.4, we can write, with m = (m a ) a :

h = m∈N n ,w∈Sn 0≤ma<r h m,w X m1 1 • • • X mn n T w ,
for some h m,w ∈ F . Applying Lemma 1.11, we have:

σ(h) = m∈N n ,w∈Sn 0≤ma<r h m,w ζ m1+•••+mn X m1 1 • • • X mn n T w , thus σ(h) = h if and only if ζ m1+•••+mn = 1 when h m,w = 0.
We conclude since ζ is a primitive pth root of unity.

Note that the family in Proposition 1.12 is a basis, since it is free (by Theorem 1.4).

The Hecke algebra of G(r, p, n)

We continue to assume here that the weight Λ satisfies the condition (1.8). In particular, for any i ∈ I and j ∈ J we have Λ i,j = Λ i . We will first define the algebra that Ariki [START_REF] Ariki | Representation theory of a Hecke algebra of G(r, p, n)[END_REF] associated with G(r, p, n), and then relate this algebra to §1.1.

Definition 1.13 [START_REF] Ariki | Representation theory of a Hecke algebra of G(r, p, n)[END_REF]). We denote by H Λ p,n (q) the unitary associative F -algebra generated by s, t ′ 1 , t 1 , . . . , t n-1 , subject to the following relations:

i∈I (s -q pi ) Λi = 0, (1.14a) (t ′ 1 + 1)(t ′ 1 -q) = (t a + 1)(t a -q) = 0, (1.14b) t ′ 1 t 2 t ′ 1 = t 2 t ′ 1 t 2 , t a t a+1 t a = t a+1 t a t a+1 , (1.14c) (t ′ 1 t 1 t 2 ) 2 = (t 2 t ′ 1 t 1 ) 2 , (1.14d) t ′ 1 t a = t a t ′ 1 if a ∈ {3, . . . , n -1}, (1.14e) t a t b = t b t a if |a -b| > 1, (1.14f) st a = t a s if a ∈ {2, . . . , n -1}, (1.14g) st ′ 1 t 1 = t ′ 1 t 1 s, (1.14h) st ′ 1 t 1 t ′ 1 t 1 . . . p+1 factors = t 1 st ′ 1 t 1 t ′ 1 . . . p+1 factors
.

(1.14i)

According to [BMR], the algebra H Λ p,n (q) is a Hecke algebra of the complex reflection group G(r, p, n).

Remark 1.15. We prove in §A.1 that the above presentation considered for p = 1 yields the Ariki-Koike algebra H Λ n (q, 1) as defined in §1.1. Remark 1.16. Assume here that p ≥ 2. The reader may have noticed that the above presentation is not the one given by Ariki [START_REF] Ariki | Representation theory of a Hecke algebra of G(r, p, n)[END_REF]. Instead of (1.14i) Ariki gives the following relation:

st ′ 1 t 1 = (q -1 t ′ 1 t 1 ) 2-p t 1 st ′ 1 + (q -1) p-2 k=1 (q -1 t ′ 1 t 1 ) 1-k st ′ 1 . (1.14i')
We claim that these two presentations define isomorphic algebras. Using (1.14b), we can show by induction the following equality:

(q -1 t ′ 1 t 1 ) 2-p t 1 st ′ 1 + (q -1) p-2 k=1 (q -1 t ′ 1 t 1 ) 1-k st ′ 1 = (t -1 1 t ′ 1 -1 t -1 1 t ′ 1 -1 . . . p-2 factors )(. . . t 1 t ′ 1 t 1 t ′ 1 p-2 factors )t 1 st ′ 1 .
Hence, using (1.14h) we get that (1.14i) and (1.14i') are equivalent, which proves the claim (we refer to §A.2 for more details). We conclude this remark by mentioning that the generators of [START_REF] Ariki | Representation theory of a Hecke algebra of G(r, p, n)[END_REF] are given by a 0 = s, a 1 = t ′ 1 and a k = t k-1 for k ∈ {2, . . . , n}. We state the main result of this section (note that the case p = 1 is proved by Theorems 1.4 and A.2).

Theorem 1.17 [START_REF] Ariki | Representation theory of a Hecke algebra of G(r, p, n)[END_REF]Proposition 1.6]). The algebra homomorphism φ : H Λ p,n (q) → H Λ n (q, ζ) given by:

φ(s) := S p , φ(t ′ 1 ) := S -1 T 1 S, φ(t a ) := T a ,
for all a ∈ {1, . . . , n -1}.

is well-defined and one-to-one. Moreover, the elements

X m1 1 • • • X mn n T w of Theorem 1.4 such that m 1 + • • • + m n = 0 (mod p) form an F -basis of φ(H Λ p,n (q)
). In particular, using Proposition 1.12 we get the following one.

Corollary 1.18. The algebra H Λ p,n (q) is isomorphic via φ to H Λ n (q, ζ) σ , the fixed point subalgebra of H Λ n (q, ζ) under the shift automorphism σ.

Removing repetitions

The following map:

I × J -→ F × (i, j) -→ ζ j q i ,
is not one-to-one. The first aim of this subsection is to find a subset J ′ ⊆ {1, . . . , p} ≃ J such that the restriction of the previous map to I × J ′ has the same image and is one-to-one. Moreover, for our purposes, we would like relation (1.6) to be of the form:

i∈I j∈J ′ (S -ζ j q i ) Λi,j = 0, (1.19) where Λ = (Λ i,j ) i∈I,j∈J ′ ∈ N (I×J ′ )
is a weight of level r. The second aim of this subsection is to know for which tuples Λ ∈ N (I×J ′ ) of level r there is some Λ ∈ N (I×J) such that the relation (1.6) in H Λ n (q, ζ) is exactly (1.19). We will be particularly interested in the case where Λ satisfies (1.8). This will require some quite long but easy computations.

Let us define the following integer: . . . , p}, (1.20) together with the following set:

p ′ := min{m ∈ N * : ζ m ∈ q } ∈ {1,
J ′ := {1, . . . , p ′ }.
Lemma 1.21. The integer p ′ is given by:

• if e = ∞ then p ′ = p;

• if e < ∞ then p ′ = p gcd(p,e) . In particular, the integer p ′ divides p and depends only on p and e.

Proof. The statement for e = ∞ is obvious since each element of q \ {1} has infinite order. Thus, we now assume that e < ∞. For any m ∈ N * , the order of ζ m in F × is p gcd(p,m) . Since q is a primitive eth root of unity, the set q is precisely the set of elements of F × of order dividing e. Hence: We conclude that the minimal m

ζ m ∈ q ⇐⇒ p gcd(p,
∈ N * such that ζ m ∈ q is p ′ = p gcd(p,e) .
The first aim of this subsection is achieved thanks to the next lemma, which is a immediate consequence of the minimality of p ′ .

Lemma 1.22. The elements ζ j q i for i ∈ I and j ∈ J ′ are pairwise distinct.

Let us denote by η the (unique) element of I such that:

ζ p ′ = q η .
(1.23)

Note that p ′ = p ⇐⇒ η = 0 ⇐⇒ q ∩ ζ = {1}. In particular, if η = 0 then e < ∞.
In that case, we are not necessarily in the setting of [START_REF] Hu | Decomposition numbers for Hecke algebras of type G(r, p, n): the (ε, q)-separated case[END_REF] (see [Lemma 2.6.(a),loc. cit.]).

We now consider the following map: .24) where ω := p p ′ . It is well-defined and surjective, hence bijective by a counting argument. Equation (1.6) becomes:

J ′ × Z/ωZ -→ J (j, a) -→ j + p ′ a , ( 1
i∈I j∈J (S -ζ j q i ) Λi,j = i∈I j∈J ′ a∈Z/ωZ S -ζ j (ζ p ′ ) a q i Λ i,j+p ′ a = i∈I j∈J ′ a∈Z/ωZ S -ζ j q i+ηa Λ i,j+p ′ a . (1.25)
For each (i, j) ∈ I × J ′ , we define:

Λ i,j := i ′ ∈I a∈Z/ωZ i ′ +ηa=i Λ i ′ ,j+p ′ a , (1.26) so that, by (1.25): i∈I j∈J (S -ζ j q i ) Λi,j = i∈I j∈J ′ (S -ζ j q i ) Λi,j . Hence, relation (1.6) transforms to the desired one (1.19). Conversely, it is clear that each weight Λ ∈ N (I×J ′ ) comes from some Λ ∈ N (I×J) through (1.26), that is, for each Λ ∈ N (I×J ′ )
there is some Λ ∈ N (I×J) such that (1.26) is satisfied. Indeed, given any Λ ∈ N (I×J ′ ) it suffices to set:

Λ i,j := Λ i, if j is the image of (, 0
) by the bijection of (1.24), 0 otherwise, for all (i, j) ∈ I × J.

Definition 1.27. Let Λ ∈ N (I×J ′ ) be a weight of level r. We consider Λ ∈ N (I×J) a weight of level r which gives Λ through (1.26). We write

H Λ n (q, ζ) := H Λ n (q, ζ).
In particular, the relation (1.6) becomes (1.19).

We now assume that the weight Λ ∈ N (I×J) satisfies the condition (1.8), that is, factors to Λ ∈ N (I) : we want to know which condition we recover on Λ. The defining equality (1.26) becomes:

Λ i,j = i ′ ∈I a∈Z/ωZ i ′ +ηa=i Λ i ′ ,
for i ∈ I and j ∈ J ′ . In particular, for any i ∈ I and j, j ′ ∈ J ′ we have Λ i,j = Λ i,j ′ =: Λ i , so that Λ ∈ N (I) is a weight of level ωd, and:

Λ i = i ′ ∈I a∈Z/ωZ i ′ +ηa=i Λ i ′ , (1.28)
for all i ∈ I.

Lemma 1.29. For any i ∈ I we have:

#{a ∈ Z/ωZ : ηa = i} = 0 if i / ∈ ηI, (1.30) 1 if i ∈ ηI. (1.31)
Proof. The result is straightforward if η = 0, in particular in that case we have ω = 1. Thus we assume η = 0, in particular e < ∞ and I = Z/eZ. Let us compute the cardinality of the fibre of i under the following group homomorphism:

φ : Z -→ I a -→ ηa.
First, the image of φ is ηI; this proves (1.30). The element ω lies in ker φ. Indeed, we have order(ζ p ′ ) = order(q η ), hence: ω = e gcd(e, η) , (1.32) thus e = gcd(e, η)ω | ηω. As a consequence, we have a well-defined surjective map:

φ : Z/ωZ -→ ηI a -→ ηa .
We have, using (1.32):

ηI = (ηZ + eZ)/eZ ≃ Z/ e gcd(e,η) Z = Z/ωZ,
thus, by a counting argument we get that the map φ is bijective. This concludes the proof.

The second aim of this subsection is achieved thanks to the following proposition.

Proposition 1.33. A weight Λ ∈ N (I) of level ωd comes from a weight Λ ∈ N (I) of level d through (1.28) if and only if for all i ∈ I,

Λ i = Λ i+η ,
that is, if and only if the weight Λ factors to a weight Λ ∈ N (I/ηI) of level d.

Proof. First, by applying Lemma 1.29 to (1.28), we obtain the equivalent equality:

Λ i = i ′ ∈I i ′ -i∈ηI Λ i ′ = i ′ ∈i+ηI Λ i ′ , (1.34)
for all i ∈ I. The necessary condition is hence straightforward. We now suppose that Λ ∈ N (I) factors to a weight Λ ∈ N (I/ηI) of level d. For any γ ∈ I/ηI, we choose any ω non-negative integers Λ i for i ∈ γ such that i∈γ Λ i = Λ γ . We conclude that (1.34) and thus (1.26) hold since ) of level ωd and Λ ∈ N (I) of level d are as in Proposition 1.33. In particular, the cyclotomic relation

Λ i = Λ γ if i ∈ γ. Definition 1.35. We write H Λ n (q, ζ) := H Λ n (q, ζ) and H Λ p,n (q) := H Λ p,n (q) if Λ ∈ N (I
(1.6) in H Λ n (q, ζ) is: i∈I j∈J ′ (S -ζ j q i ) Λi = 0.

Cyclotomic quiver Hecke algebras

Let K be a set and Γ be a loop-free quiver without multiple (directed) edges, with vertex set K. For k, k ′ ∈ K:

• we write k -k ′ if neither (k, k ′ ) nor (k ′ , k) is an edge of Γ; • we write k → k ′ if (k, k ′ ) is an edge of Γ and (k ′ , k) is not; • we write k ← k ′ if k ′ → k; • we write k ⇆ k ′ if both (k, k ′ ) and (k ′ , k) are edges of Γ.
If the next section we will define the (cyclotomic) quiver Hecke algebra associated with Γ. Furthermore, given an automorphism of this algebra built from an automorphism of the quiver, we will give a presentation of the fixed point subalgebra. Note that what follows has roughly the same outline as [Bo, BoMa].

Definition

Let n ∈ N * and α = (α k ) k∈K be a (finitely supported) tuple of non-negative integers, whose sum is equal to n. We say that α is a K-composition of n and we write α |= K n. Let K α be the subset of K n consisting of the elements of K n which have, for any k ∈ K, exactly α k components equal to k. The sets K α are the S n -orbits of K n , in particular they are finite. Following [START_REF] Khovanov | A diagrammatic approach to categorification of quantum groups I[END_REF][START_REF] Khovanov | A diagrammatic approach to categorification of quantum groups II[END_REF] and [Rou], we denote by R α (Γ) the quiver Hecke algebra at α associated with Γ. It is the unitary associative F -algebra with generating set

{e(k)} k∈K α ∪ {y 1 , . . . , y n } ∪ {ψ 1 , . . . , ψ n-1 } together with the relations k∈K α e(k) = 1, (2.1a) e(k)e(k ′ ) = δ k,k ′ e(k), (2.1b) y a e(k) = e(k)y a , (2.1c) ψ a e(k) = e(s a • k)ψ a , (2.1d) y a y b = y b y a ,
(2.1e)

ψ a y b = y b ψ a if b = a, a + 1, (2.1f) ψ a ψ b = ψ b ψ a if |a -b| > 1,
(2.1g)

ψ a y a+1 e(k) = (y a ψ a + 1)e(k) if k a = k a+1 , y a ψ a e(k) if k a = k a+1 , (2.1h) y a+1 ψ a e(k) = (ψ a y a + 1)e(k) if k a = k a+1 , ψ a y a e(k) if k a = k a+1 , (2.1i) ψ 2 a e(k) =                0 if k a = k a+1 , e(k) if k a -k a+1 , (y a+1 -y a )e(k) if k a → k a+1 , (y a -y a+1 )e(k) if k a ← k a+1 , (y a+1 -y a )(y a -y a+1 )e(k) if k a ⇆ k a+1 , (2.1j) ψ a+1 ψ a ψ a+1 e(k) =          (ψ a ψ a+1 ψ a -1)e(k) if k a+2 = k a → k a+1 , (ψ a ψ a+1 ψ a + 1)e(k) if k a+2 = k a ← k a+1 , (ψ a ψ a+1 ψ a + 2y a+1 -y a -y a+2 )e(k) if k a+2 = k a ⇆ k a+1 , ψ a ψ a+1 ψ a e(k)
otherwise.

(2.1k)

The following proposition is easy to check (see [START_REF] Khovanov | A diagrammatic approach to categorification of quantum groups I[END_REF][START_REF] Khovanov | A diagrammatic approach to categorification of quantum groups II[END_REF] or [Rou]).

Proposition 2.2.

There is a unique Z-grading on R α (Γ) such that:

• for all k ∈ K α , deg e(k) := 0;

• for all k ∈ K α and a ∈ {1, . . . , n}, deg y a e(k) := 2;

• for all k ∈ K α and a ∈ {1, . . . , n -1}:

deg ψ a e(k) :=          -2 if k a = k a+1 , 0 if k a -k a+1 , 1 if k a → k a+1 or k a ← k a+1 , 2 if k a ⇆ k a+1 .
If K is finite, we can define the quiver Hecke algebra R n (Γ) as above, with generators

{e(k)} k∈K n ∪ {y 1 , . . . , y n } ∪ {ψ 1 , . . . , ψ n-1 }
and the relations (2.1), where the sum over

K α in (2.1a) is now over K n . In that case, define for α |= K n: e(α) := k∈K α e(k) ∈ R n (Γ). (2.3)
The element e(α) is central and R n (Γ)e(α) ≃ R α (Γ). We have:

R n (Γ) ≃ α|=K n R α (Γ),
and this isomorphism can be considered as a definition of R n (Γ) when K is infinite (note that in that case, the algebra R n (Γ) is not unitary).

Now let Λ = (Λ k ) k∈K ∈ N (K) be a finitely-supported tuple of non-negative integers. We define a particular case of cyclotomic quotient of R α (Γ) (see [KanKa] for the general case).

Definition 2.4. The cyclotomic quiver Hecke algebra R Λ α (Γ) is the quotient of the quiver Hecke algebra R α (Γ) by the two-sided ideal I Λ α generated by the following relations:

y Λ k 1 1 e(k) = 0, for all k ∈ K α . (2.5)
Note that the grading on R α (Γ) gives a grading on R Λ α (Γ). If K is finite, the cyclotomic quotient R Λ n (Γ) is obtained by quotienting R n (Γ) by the relations:

y Λ k 1 1 e(k) = 0, for all k ∈ K n .
Moreover, we have e(α)R Λ n (Γ) ≃ R Λ α (Γ) and:

R Λ n (Γ) ≃ α|=K n R Λ α (Γ),
and this can be considered as a definition of

R Λ n (Γ) if K is infinite.

Properties of the underlying vector spaces

For each w ∈ S n , we choose a reduced expression w = s a1 • • • s ar and we set:

ψ w := ψ a1 • • • ψ ar ∈ R n (Γ). (2.6) Let α |= K n.
We have the following theorem ([Rou, Theorem 3.7], [KhLau09, Theorem 2.5]).

Theorem 2.7. The family

B α := {ψ w y m1 1 • • • y mn n e(k) : w ∈ S n , m a ∈ N, k ∈ K α } is a basis of the F -vector space R α (Γ).
Let Λ ∈ N (K) be a weight. It is not obvious that we can deduce a basis of the cyclotomic quiver Hecke algebra from the basis of Theorem 2.7. With this in mind, let us give the following lemma. Remark 2.9. The proof in [BrKl] is given for the quiver Γ e . However, we can see immediately that their proof is valid for the quivers we use here.

We obtain the following theorem (see also [START_REF] Kang | Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras[END_REF]§4.1]).

Theorem 2.10. The family:

B Λ α := ψ w y m1 1 • • • y mn n e(k) : w ∈ S n , m a ∈ N, k ∈ K α , is finite and spans R Λ α (Γ) over F . In particular, the F -vector space R Λ α (Γ) is finite-dimensional.

Fixed point subalgebra

Let σ be an automorphism of Γ, that is:

• the map σ : K → K is a bijection; • if (k, k ′ ) ∈ K 2 is an edge of Γ then (σ(k), σ(k ′ )) ∈ K 2 is also an edge of Γ.
We assume that for any p 1 ∈ {1, . . . , p-1} and for any k ∈ K we have σ p1 (k) = k = σ p (k), in particular, we have σ p = id K . Remark 2.11. Everything in this subsection §2.3 remains true if we only assume that σ p = id K : we just have to modify the sentence containing (2.22) and equalities (2.24), (2.41). Note that the automorphism involved in the proof of the main theorem (the automorphism which is defined in (4.8)) satisfies the above stronger condition.

Since σ p = id K , we have, for k, k ′ ∈ K:

(k, k ′ ) ∈ K 2 is an edge of Γ ⇐⇒ (σ(k), σ(k ′ )) ∈ K 2 is an edge of Γ,
thus we deduce the following lemma.

Lemma 2.12. Let k, k ′ ∈ K. We have:

k → k ′ ⇐⇒ σ(k) → σ(k),
and there are similar equivalences for ←, ⇆, = and =.

The map σ naturally induces a map σ : K n → K n , defined by σ(k) := (σ(k 1 ), . . . , σ(k n )). For α |= K n, the following lemma explains how σ : K n → K n restricts to K α (compare to [START_REF] Boys | Alternating quiver Hecke algebras[END_REF]after Lemma 5.3.2]).

Lemma 2.13. For α |= K n, the map σ :

K n → K n maps K α onto K σ•α , where σ • α is the K-composition of n given by: (σ • α) k := α σ -1 (k) ,
for all k ∈ K.

Proof. Let k ∈ K n . We have:

for all k ∈ K, k has α k components equal to k ⇐⇒ for all k ∈ K, σ(k) has α k components equal to σ(k) ⇐⇒ for all k ∈ K, σ(k) has α σ -1 (k) components equal to k ⇐⇒ for all k ∈ K, σ(k) has (σ • α) k components equal to k. We conclude that k ∈ K α ⇐⇒ σ(k) ∈ K σ•α .
We can now explain how σ induces an isomorphism between (cyclotomic) quiver Hecke algebras. We will also give a presentation for the fixed point subalgebras.

Affine case

In the affine case, we will be able to give a basis for the subalgebra of the fixed points of σ. As an easy consequence, we will give a presentation of this subalgebra.

Theorem 2.14. Let α |= K n. There is a well-defined algebra homomorphism σ : R α (Γ) → R σ•α (Γ) given by:

σ(e(k)) := e(σ(k)), for all k ∈ K α , σ(y a ) := y a ,
for all a ∈ {1, . . . , n}, σ(ψ a ) := ψ a , for all a ∈ {1, . . . , n -1}.

Proof. We check the different relations (2.1), thanks to Lemma 2.12 and the following fact:

σ(k) a = σ(k a ),
for all a ∈ {1, . . . , n} and k ∈ K n . Note that to prove (2.1a) we use the additional fact that σ :

K α → K σ•α is a bijection.
Remark 2.15. By Lemma 2.12, the homomorphism σ : R α (Γ) → R σ•α (Γ) is homogeneous with respect to the grading given in Proposition 2.2. As in Section 1, we want to study the fixed points of σ. To that extent, we first need to find an algebra which is stable under σ. Let [α] be the orbit of α under the action of σ . Note that since σ p = id K , the cardinality of [α] is at most p. For α |= K n we define the following finite subset of K n :

K [α] := β∈[α] K β , (2.16)
and similarly we define the following unitary algebra:

R [α] (Γ) := β∈[α]
R β (Γ).

(2.17)

We obtain an automorphism σ :

R [α] (Γ) → R [α] (Γ). Remark 2.18. We have R n (Γ) ≃ ⊕ [α] R [α] (Γ), in particular, for k ∈ K n the idempotent e(k) of R n (Γ) belongs to R [α] (Γ) if and only if k ∈ K [α] .
We consider the equivalence relation ∼ on K generated by:

k ∼ σ(k), for all k ∈ K. (2.19)
We extend it to K [α] by:

k ∼ σ(k), for all k ∈ K [α] .
(2.20)

Definition 2.21. We write K

[α] σ for the quotient set K [α] /∼.

In particular, each element γ ∈ K

[α] σ has cardinality p and is of the form:

γ = {k, σ(k), . . . , σ p-1 (k)}, (2.22) with k ∈ K [α] . Definition 2.23. For γ ∈ K [α]
σ , we define:

e(γ) := k∈K [α] k∈γ e(k).

These elements e(γ) have the property of being fixed by σ. Note that for any k ∈ γ, by (2.22) we have:

e(γ) = p-1 m=0 e(σ m (k)).
(2.24)

We now give the analogue of Proposition 1.12, by describing all the fixed points of σ.

Theorem 2.25. The following family:

B σ [α] := ψ w y m1 1 • • • y mn n e(γ) : w ∈ S n , m a ∈ N, γ ∈ K [α] σ , is an F -basis of R [α] (Γ) σ , the vector space of σ-fixed points of R [α] (Γ).
Proof. First, by Theorem 2.7 we know that

B [α] := ⊔ β∈[α] B β is a linear basis of R [α] (Γ).
Hence, the family B σ [α] is linearly independent. Moreover, each element of B σ [α] is fixed by σ. Now let h ∈ R [α] (Γ) be fixed by σ: we want to prove that h lies in span F (B σ [α] ). Using Theorem 2.7, we can write, with m = (m a ) a :

h = w∈Sn m∈N n k∈K [α] h w,m,k ψ w y m1 1 • • • y mn n e(k),
where h w,m,k ∈ F . We have:

h = σ(h) = w∈Sn m∈N n k∈K [α] h w,m,k ψ w y m1 1 • • • y mn n e(σ(k)),
and thus, since B [α] is linearly independent:

h w,m,k = h w,m,σ(k) ,
for all w ∈ S n , m ∈ N n and k ∈ K [α] . In particular, for each γ ∈ K

[α]
σ there is a well-defined scalar h w,m,γ . We obtain:

h = w∈Sn m∈N n γ∈K [α] σ h w,m,γ ψ w y m1 1 • • • y mn n k∈γ e(k) = w∈Sn m∈N n γ∈K [α] σ h w,m,γ ψ w y m1 1 • • • y mn n e(γ), thus h lies in span F (B σ [α] ).
Theorem 2.25 allows us to give a presentation of the algebra R [α] (Γ) σ . First, let us note that for any a, b, c ∈ {1, . . . , n} with c < n and γ ∈ K

[α]
σ , the expressions:

s c • γ, γ a = γ b , γ a = γ b , γ a → γ b , γ a ← γ b , γ a ⇆ γ b ,
are well-defined, thanks to Lemma 2.12.

Remark 2.26. In contrast, if γ ′ is another element of K [α]
σ then the expression γ a = γ ′ a (for instance) is not well-defined. In particular, recalling (2.19) and (2.20), instead of subsets of K n σ = K n /∼ we may want to consider subsets of (K/∼) n . This set (K/∼) n has much worse properties. For instance, denoting by κ

∈ (K/∼) n the image of k ∈ K n (or γ ∈ K n σ ), if κ a = κ a+1 then we may have k a = k a+1 (or γ a = γ a+1 ).
Corollary 2.27. The algebra R [α] (Γ) σ has the following presentation. The generating set is:

{e(γ)} γ∈K [α] σ ∪ {y 1 , . . . , y n } ∪ {ψ 1 , . . . , ψ n-1 }, (2.28)
and the relations are:

γ∈K [α] σ e(γ) = 1, (2.29a) e(γ)e(γ ′ ) = δ γ,γ ′ e(γ),
(2.29b) y a e(γ) = e(γ)y a ,

(2.29c)

ψ a e(γ) = e(s a • γ)ψ a , (2.29d)
y a y b = y b y a , (2.29e) ψ a y b = y b ψ a if b = a, a + 1, (2.29f) ψ a ψ b = ψ b ψ a if |a -b| > 1, (2.29g) ψ a y a+1 e(γ) = (y a ψ a + 1)e(γ) if γ a = γ a+1 , y a ψ a e(γ) if γ a = γ a+1 , (2.29h) y a+1 ψ a e(γ) = (ψ a y a + 1)e(γ) if γ a = γ a+1 , ψ a y a e(γ) if γ a = γ a+1 , (2.29i) ψ 2 a e(γ) =                0 if γ a = γ a+1 , e(γ) if γ a -γ a+1 , (y a+1 -y a )e(γ) if γ a → γ a+1 , (y a -y a+1 )e(γ) if γ a ← γ a+1 , (y a+1 -y a )(y a -y a+1 )e(γ) if γ a ⇆ γ a+1 , (2.29j) ψ a+1 ψ a ψ a+1 e(γ) =          (ψ a ψ a+1 ψ a -1)e(γ) if γ a+2 = γ a → γ a+1 , (ψ a ψ a+1 ψ a + 1)e(γ) if γ a+2 = γ a ← γ a+1 , (ψ a ψ a+1 ψ a + 2y a+1 -y a -y a+2 )e(γ) if γ a+2 = γ a ⇆ γ a+1 , ψ a ψ a+1 ψ a e(γ)
otherwise.

(2.29k)

Proof. Let us temporary write e(γ) σ , y σ a and ψ σ a for the generators of Corollary 2.27, and write R σ for the algebra which admits this presentation. Given the defining relations (2.1) of R [α] (Γ) and Lemma 2.12, there is a well-defined algebra homomorphism f : R σ → R [α] (Γ) given by: f (e(γ) σ ) := e(γ), for all γ ∈ K [α] σ , f (y σ a ) := y a , for all a ∈ {1, . . . , n}, f (ψ σ a ) := ψ a , for all a ∈ {1, . . . , n -1}.

We can notice that the family:

B σ := ψ σ w (y σ 1 ) m1 • • • (y σ n ) mn e(γ) σ : w ∈ S n , m a ∈ N, γ ∈ K [α] σ ,
spans R σ over F , where the elements ψ σ w are defined as in (2.6), with the same reduced expressions. We recall from Theorem 2.25 that the family

B σ [α] = ψ w y m1 1 • • • y mn n e(γ) : w ∈ S n , m a ∈ N, γ ∈ K [α] σ is an F -basis of R σ [α] (Γ). Noticing that the algebra homomorphism f maps B σ onto B σ [α]
, we deduce that:

• the family B σ is linearly independent;

• the map f surjects onto R [α] (Γ) σ .
Finally, the family B σ is a basis of R σ . In particular, the homomorphism f sends a basis to a basis hence f is an isomorphism.

The reader may have noticed the similarity between the relations (2.29) defining R [α] (Γ) σ and the relations (2.1) defining R α (Γ). However, now the indexing set for the idempotents is generally not an S n -stable subset of I n for I an indexing set. Remark 2.30. Since σ : R

[α] (Γ) → R [α] (Γ) is homogeneous (cf. Remark 2.15), the subalgebra R [α] (Γ) σ is a graded subalgebra of R [α] (Γ).
More precisely, as in Proposition 2.2 there is a unique Z-grading on R [α] (Γ) σ such that e(γ) is of degree 0, the element y a is of degree 2 and ψ a e(γ) is of degree -c γa,γa+1 where:

c γa,γa+1 :=          2 if γ a = γ a+1 , 0 if γ a -γ a+1 , -1 if γ a → γ a+1 or γ a+1 → γ a , -2 if γ a ⇆ γ a+1 .

Cyclotomic case

Recall the Definition 2.4 of a cyclotomic quiver Hecke algebra. For α |= K n, we want the algebra homomorphism σ : R α (Γ) → R σ•α (Γ) to factor through cyclotomic quotients. Contrary to the affine case, it will be more difficult to get a presentation for the fixed point subalgebra of the cyclotomic quiver Hecke algebra (recall that we do not have an analogue of Theorem 2.7). In particular, the whole proof relies on the map µ which will be introduced in (2.37).

Let Λ ∈ N (K) be a weight. As for K-compositions, we define the weight σ • Λ ∈ N (K) by:

(σ • Λ) k := Λ σ -1 (k) , for all k ∈ K.
Lemma 2.31. We have σ(

I Λ α ) = I σ•Λ σ•α . In particular, the algebra homomorphism σ : R α (Γ) → R σ•α (Γ) induces an algebra homomorphism σ Λ : R Λ α (Γ) → R σ•Λ σ•α (Γ).
Proof. We notice that for k ∈ K α we have:

σ(y Λ k 1 1 e(k)) = y Λ k 1 1 e(σ(k)) ∈ I σ•Λ σ•α , since Λ k1 = (σ • Λ) σ(k)1 . Hence σ(I Λ α ) ⊆ I σ•Λ σ•α
and we have equality by repeating the argument with σ -1 .

Until the end of this section, we make the following σ-stability assumption on our weight

Λ ∈ N (K) : Λ k = Λ σ(k) , for all k ∈ K. (2.32)
that is, we assume that Λ = σ • Λ. Equivalently, the weight Λ factors to an element of N (K/∼) (with the notation of (2.19)). The reader may have noticed the similarity with the equation of Proposition 1.33. In §4.2 we will explicitly make the link between these two assumptions.

Similarly to (2.17), we define:

R Λ [α] (Γ) := β∈[α] R Λ β (Γ).
This algebra is the quotient of R [α] (Γ) by the two sided ideal

I Λ [α] := β∈[α]
I Λ β generated by the elements y

Λ k 1
1 e(k) for k ∈ K [α] . We deduce from Lemma 2.31 the following statement.

Lemma 2.33. We have σ(I Λ

[α] ) = I Λ [α] . Moreover, σ : R [α] (Γ) → R [α] (Γ) induces an algebra homomorphism σ Λ : R Λ [α] (Γ) → R Λ [α] (Γ). If π [α] : R [α] (Γ) ։ R Λ [α] (Γ)
is the canonical projection, by definition the induced automorphism σ Λ satisfies:

σ Λ • π [α] = π [α] • σ.
(2.34)

We will often write σ as well for the automorphism σ Λ .

Definition 2.35. We define R Λ [α] (Γ) σ as the F -algebra of the fixed points of R Λ [α] (Γ) under the automorphism σ Λ .

We recall the notation of §2.3.1. Since Λ satisfies the σ-stability assumption (2.32) and considering the canonical map K n σ = K n /∼ → (K/∼) n , we may also consider the algebra (R [α] (Γ) σ ) Λ , the quotient of R [α] (Γ) σ by the two-sided ideal I Λ [α],σ generated by the following relations: y

Λγ 1

1 e(γ) = 0, for all γ ∈ K [α] σ .

(2.36)

In order to give a presentation of R Λ [α] (Γ) σ , we want to prove that this algebra is isomorphic to the following one:

(

R [α] (Γ) σ ) Λ = R [α] (Γ) σ I Λ [α],
σ , for which we know a presentation. Recalling that the characteristic of F does not divide p, we can define the following linear map:

µ : R [α] (Γ) -→ R [α] (Γ) h -→ 1 p p-1 m=0 σ m (h)
.

(2.37)

We now give a succession of lemmas involving this map µ.

Lemma 2.38. The following properties are satisfied by the linear map µ:

µ(R [α] (Γ)) = R [α] (Γ) σ , µ(h) = h, for all h ∈ R [α] (Γ) σ .
Moreover, we have:

µ(I Λ [α] ) = I Λ [α] ∩ R [α] (Γ) σ .
Proof. The first two statements follow from σ p = id. We deduce the last one using Lemma 2.33.

Proof. Recalling Corollary 2.27, we begin by noticing that the given presentation is a presentation of (R [α] (Γ) σ ) Λ . In particular, we can define a homomorphism of algebras

f : (R [α] (Γ) σ ) Λ → R Λ [α] (Γ) σ by:
f (e(γ)) := e(γ), for all γ ∈ K [α] σ , f (y a ) := y a , for all a ∈ {1, . . . , n}, f (ψ a ) := ψ a , for all a ∈ {1, . . . , n -1}.

If π σ [α] : R [α] (Γ) σ ։ (R [α] (Γ) σ )
Λ is the canonical projection, we have:

f • π σ [α] (h) = π [α] (h) (2.44) for all h ∈ R [α] (Γ) σ (it suffices to check this equality for each generator of R [α] (Γ) σ ). We now want to construct an inverse R Λ [α] (Γ) σ → (R [α] (Γ) σ
) Λ for f . Note that it is not obvious since we do not have any presentation of the starting algebra yet. By Lemma 2.38, we have a well-defined linear map µ 1 :

R [α] (Γ) → (R [α] (Γ) σ ) Λ given by µ 1 = π σ [α] • µ. By Lemma 2.40 we have ker µ 1 ⊇ I Λ [α]
, hence we have a well-defined linear map:

µ 2 : R Λ [α] (Γ) → (R [α] (Γ) σ ) Λ ,
and by restriction we get a linear map µ : R Λ

[α] (Γ) σ → (R [α] (Γ) σ ) Λ .
To summarise, we have

a commutative diagram (2.45). R [α] (Γ) R [α] (Γ) σ R Λ [α] (Γ) (R [α] (Γ) σ ) Λ R Λ [α] (Γ) σ µ π [α] π σ [α] µ 2 µ 1 µ (2.45) Let h ∈ R [α] (Γ) σ .
We want to prove the following equality:

µ • π [α] (h) = π σ [α] (h). ( 2
µ • π [α] (h) = µ 2 • π [α] (h) = µ 1 (h) = π σ [α]
• µ(h), and we conclude since µ(h) = h by Lemma 2.38.

Finally, let us prove that f and µ are mutual inverses.

• Let h ∈ (R [α] (Γ) σ ) Λ . If h ∈ R [α] (Γ) σ is such that π σ [α] (h) = h, we have, using (2.44) and (2.46): µ • f (h) = µ • f • π σ [α] (h) = µ • π [α] (h) = π σ [α] (h) = h, hence µ • f is the identity of (R [α] (Γ) σ ) Λ . • Let h ∈ R Λ [α] (Γ) σ . By Lemma 2.42, we can find h ∈ R [α] (Γ) σ such that π [α] (h) = h.
Using once again (2.44) and (2.46) we get:

f • µ(h) = f • µ • π [α] (h) = f • π σ [α] (h) = π [α] (h) = h, thus f • µ is the identity of R Λ [α] (Γ) σ .
In particular, the algebra homomorphism f is bijective, hence is an algebra isomorphism between R Λ

[α] (Γ) σ and (R [α] (Γ) σ ) Λ .
Remark 2.47. The grading of Remark 2.30 thus gives a grading on R Λ [α] (Γ) σ , for which σ Λ is homogeneous (recall Remark 2.15). Moreover, the algebra R Λ

[α] (Γ) σ is a graded subalgebra of R Λ

[α] (Γ). Remark 2.48. The proof of Theorem 2.43 uses the assumption that p is invertible in the base field. A characteristic-free (and in a slightly more general setting) version can be found in the author's PhD thesis [START_REF] Rostam | Quiver Hecke algebras and generalisations of Iwahori-Hecke algebras[END_REF]§1.4].

The isomorphism of Brundan and Kleshchev

In this section, we generalise an isomorphism of Brundan and Kleshchev [BrKl] involving H Λ n (q, 1) to the case of the algebra H Λ n (q, ζ).

Statement

We consider the quiver Γ e defined as follows:

• the vertex set is {q i } i∈I ;

• there is a directed edge from v to qv for each vertex v of Γ e .

We will often identify the vertex set with I in the canonical way. In particular, if i is a vertex then there is a directed arrow from i to i + 1. For i, i ′ ∈ I, with the notation of Section 2 we thus have:

i → i ′ ⇐⇒ [i ′ = i + 1 and i = i ′ + 1], i ← i ′ ⇐⇒ [i = i ′ + 1 and i ′ = i + 1], i ⇆ i ′ ⇐⇒ [i = i ′ + 1 and i ′ = i + 1], i -i ′ ⇐⇒ i = i ′ , i ′ ± 1.
The quiver Γ e is the cyclic quiver with e vertices if e < ∞, and a two-sided infinite line if e = ∞: we give some examples in Figure 1, where we used the identification between the vertex set of Γ e and I.

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
We recall the notation p ′ and J ′ introduced at §1.3. We set K := I × J ′ . Let us consider p ′ non-zero elements v 1 , . . . , v p ′ of F which lie in distinct orbits under the action of q on F × , that is, for any k = l we have:

v k v l / ∈ q . (3.1)
We then consider the quiver Γ defined as follows:

• the vertex set is V := {v j q i } i∈I,j∈J ′ ;

• there is a directed edge from v to qv for each vertex v of Γ.

Since the v k lie in different q-orbits, the vertex set V of Γ can be identified with K = I×J ′ . More precisely, we have the following decomposition:

V = j∈J ′ {v j q i } i∈I .
(3.2)

Since:

• the subquiver of Γ with vertex set {v j q i } i∈I is a copy of Γ e ;

• for j = j ′ ∈ J ′ , there is no arrow between any element of {v j q i } i∈I and {v j ′ q i } i∈I ;

• the set J ′ has cardinality p ′ ; we conclude from (3.2) that Γ is exactly p ′ disjoint copies of Γ e . In particular, the quiver Γ is loop-free and has no multiple edges.

As a consequence, we will often write (i, j) ∈ I × J ′ for the vertex v j q i ∈ V of Γ. For any i, i ′ ∈ I and j, j ′ ∈ J ′ , what precedes ensures that the vertices (i, j) and (i ′ , j ′ ) are a same copy of Γ e if and only if j = j ′ . Further, there is a directed edge from (i, j) to (i ′ , j ′ ) if and only if j = j ′ and there is a directed edge in Γ e from i to i ′ . We give some examples of quivers Γ in Figure 2, where, for aesthetic reasons, we write i j instead of v j q i . We also recall from Lemma 1.21 that

p ′ = p gcd(p,e) if e < ∞ and p ′ = p if e = ∞.
Case (e, p) = (2, 3)

0 1 ⇆ 1 1 0 2 ⇆ 1 2 0 3 ⇆ 1 3 Case (e, p) = (2, 6) 0 1 ⇆ 1 1 0 2 ⇆ 1 2 0 3 ⇆ 1 3
Case (e, p) = (∞, 2) ) be a weight of level r. Mimicking the definition of H Λ n (q, ζ), let us choose a tuple u ∈ (F × ) r which is given by exactly Λ i,j copies of v j q i for each (i, j) ∈ I × J ′ and set H Λ n (q, v) := H n (q, u). As a result, the relation (1.2a) in H n (q, u) is:

• • • -2 1 -1 1 0 1 1 1 2 1 • • • • • • -2 2 -1 2 0 2 1 2 2 2 • • • Figure 2: Three examples of quivers Γ Now let Λ = (Λ k ) k∈K ∈ N (K
i∈I j∈J ′ (S -v j q i ) Λi,j = 0. (3.3)
The remaining part of this section is devoted to the proof of the following theorem.

Theorem 3.4.

There is an explicit F -algebra isomorphism:

H Λ n (q, v) ≃ R Λ n (Γ).
Brundan and Kleshchev [BrKl] proved Theorem 3.4 for p = 1: in that case, we have p ′ = 1, the tuple v has only one component (that can be taken equal to 1) and Γ = Γ e . We will see that the same argument proves the general case. Such an isomorphism, for e < ∞, was already obtained by Rouquier [START_REF] Rouquier | 2-Kac-Moody algebras[END_REF]Corollary 3.20].

Candidate homomorphisms

We recall that V ≃ K = I × J ′ , together with the definitions X 1 := S and qX a+1 := T a X a T a from (1.3). To prove Theorem 3.4, it suffices to give an isomorphism between H Λ α (q, v) (see (3.7)) and R Λ α (Γ) for any α |= K n. Let M be a finite-dimensional H Λ n (q, v)-module. Lemma 3.5. For any a ∈ {1, . . . , n}, the eigenvalues of X a on M are of the form v j q i for i ∈ I and j ∈ J ′ .

Proof. The statement is of course true for a = 1 by (1.6). By induction, using [ArKo] or [START_REF] Grojnowski | Affine sl p controls the modular representation theory of the symmetric group and related Hecke algebras[END_REF]Lemma 4.7] we know that any eigenvalue of X a+1 differs from an eigenvalues of X a by a power of q.

Hence, as the elements X 1 , . . . , X n pairwise commute, we can write M as a direct sum of generalised simultaneous eigenspaces:

M = k∈K n M (k), where M (k) = M (i, j) is defined by, for k = (i, j) ∈ K n ≃ I n × J ′ n : M (i, j) := m ∈ M : (X a -v ja q ia ) N m = 0 for all 1 ≤ a ≤ n ,
where N ≫ 0. Note that all but finitely many M (k) are reduced to {0}. We now consider the family {e(k)} k∈K n of projections associated with the decomposition above. In particular:

• we have e(k)e(k ′ ) = δ k,k ′ e(k);
• we have k∈K n e(k) = id (this is a finite sum since all but finitely many e(k) are zero);

• we have e(k)M = M (k).

Remark 3.6. We already used the notation e(k) for some generators of R Λ n (Γ). This abuse of notation will be justified by the proof of Theorem 3.4, where we prove that these elements can be identified.

Since

e(k) is a polynomial in X 1 , . . . , X n we have e(k) ∈ H Λ n (q, v). If now α |= K n is a K-composition of n, the following element: e(α) := k∈K α e(k) ∈ H Λ n (q, v),
is a central idempotent (the reader should compare this definition to (2.3)). We thus get a subalgebra:

H Λ α (q, v) := e(α)H Λ n (q, v).
(3.7)

Remark 3.8. The subalgebra H Λ α (q, v) is either {0} or a block of H Λ n (q, v) (see [LyMa]). This block has unit e(α).

Recall that the elements y a ∈ R Λ α (Γ) for 1 ≤ a ≤ n are nilpotent (Lemma 2.8). Hence, each power series f (y 1 , . . . , y n ) ∈ F [[y 1 , . . . , y n ]] in these elements is a well-defined element of R Λ α (Γ). In particular, for a ∈ {1, . . . , n -1} and k ∈ K α the following power series is well-defined in R Λ α (Γ):

P a (k) := 1 if k a = k a+1 , (1 -q)(1 -y a (k)y a+1 (k) -1 ) -1 if k a = k a+1 , ∈ F [[y a , y a+1 ]], (3.9)
where:

y a (k) := v ja q ia (1 -y a ) if k = (i, j).
(3.10)

For k = (i, j) ∈ K = I × J ′ , we define: (3.11) in particular we obtain y a (k) = v ka q ka (1 -y a ) for any k ∈ K n . Note that:

q k := q i , q -k := q -i , v k := v j , v -k := v -1 j ,
1 -y a (k)y a+1 (k) -1 = y a+1 (k) -y a (k) y a+1 (k) = (v ka+1 q ka+1 -v ka q ka ) + v ka q ka y a -v ka+1 q ka+1 y a+1 y a+1 (k) ;
thus, by (3.1) we know that this expression is indeed invertible when k a = k a+1 . For (w, f ) ∈ S n ×F [[y 1 , . . . , y n ]], we denote by f w ∈ F [[y 1 , . . . , y n ]] the usual right action of w on f . For instance, if w = τ is a transposition then f τ (y 1 , . . . , y n ) = f (y τ (1) , . . . , y τ (n) ). Let us give a lemma involving this action (see, for instance, [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF](2.6)]). Lemma 3.12. For any f ∈ F [[y 1 , . . . , y n ]], a ∈ {1, . . . , n -1} and k ∈ K α we have:

f ψ a e(k) = ψ a f sa e(k) + ∂ a (f )e(k) if k a = k a+1 , ψ a f sa e(k) if k a = k a+1 , where ∂ a (f ) := f sa -f ya-ya+1 ∈ F [[y 1 , . . . , y n ]].
Proof. This is a consequence of (2.1f), (2.1h) and (2.1i).

We say that a family {Q a (k)} a∈{1,...,n-1},k∈K α of elements of F [[y 1 , . . . , y n ]] satisfies the property (BK) if:

Q a (k) is an invertible element of F [[y a , y a+1 ]], (3.13) Q a (k) = 1 -q + qy a+1 -y a if k a = k a+1 , (3.14) Q a (k)Q a (s a • k) sa =                      (1 -P a (k))(q + P a (k)) if k a -k a+1 , (3.15a) (1 -P a (k))(q + P a (k)) y a+1 -y a if k a → k a+1 , (3.15b) (1 -P a (k))(q + P a (k)) y a -y a+1 if k a ← k a+1 , (3.15c)
(1

-P a (k))(q + P a (k)) (y a+1 -y a )(y a -y a+1 ) if k a ⇆ k a+1 , (3.15d)
Q a+1 (s a+1 s a • k) sa = Q a (s a s a+1 • k) sa+1 . ( 3.16)
We can now give the key of Theorem 3.4.

Theorem 3.17. Let {Q a (k)} a∈{1,...,n-1},k∈K α be a family of elements of F [[y 1 , . . . , y n ]] which satisfies (BK). There exist unique F -algebra homomorphisms f : and, recalling (3.11):

H Λ α (q, v) → R Λ α (Γ) and g : R Λ α (Γ) → H Λ α (q, v) such that: f (X a ) := k∈K α y a (k)e(k), f (T a ) := k∈K α (ψ a Q a (k) -P a (k))e(k),
g(e(k)) := e(k), g(y a ) := k∈K α (1 -v -ka q -ka X a )e(k), g(ψ a ) := k∈K α (T a + P a (k))Q a (k) -1 e(k).
Moreover, these homomorphisms are inverse to each other, hence H Λ α (q, v) ≃ R Λ α (Γ). We will explain at the beginning of §3.3.2 how the elements P a (k) and Q a (k) are considered as elements of H Λ α (q, v). We note that there exist such families {Q a (k)} a,k , see §4.1 for further details.

Proof of Theorems 3.4 and 3.17

In this subsection, we first check that the maps of Theorem 3.17 indeed define algebras homomorphisms: we check that the different defining relations (1.2b)-(1.2f), (1.6) (for f ) and (2.1), (2.5) (for g) are satisfied. The proof is exactly as in [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Section 4]: we will only give some details when some v j are involved. The remaining parts of the argument require only notational changes from [BrKl].

The map f is a homomorphism

We prove that the images of the generators of H Λ n (q, v) by f satisfy the defining relations. The proof of the quadratic relation (1.2b) is exactly the same as the one for [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Theorem 4.3]. Namely, it suffices to check that for any k ∈ K α we have f (T a ) 2 e(k) = (q -1)f (T a )e(k) + qe(k), and the result follows since k∈K α e(k) = 1 in R Λ α (Γ). The equality f (X 1 )f (X 2 ) = f (X 2 )f (X 1 ) is clear. Hence, to check the length 4-braid relation (1.2c) it suffices to prove that qf (X 2 ) = f (T 1 )f (X 1 )f (T 1 ). We will in fact prove that for any a ∈ {1, . . . , n -1}:

qf (X a+1 ) = f (T a )f (X a )f (T a ).
(3.18)

Since we have just checked the relation (1.2b) for f (T a ), it suffices to prove that for any

k ∈ K α : f (X a )f (T a )e(k) = (f (T a ) + 1 -q)f (X a+1 )e(k).
Once again, the rest of the proof is exactly the same as in the corresponding part of the proof of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Theorem 4.3]. We write down here some of the details since we have to add some v j in the calculations. We have:

X a T a e(k) = (y a (s a • k)ψ a Q a (k) -y a (k)P a (k)) e(k) = ψ a y a+1 (k)Q a (k) + δ ka,ka+1 v ka q ka Q a (k) -y a (k)P a (k) e(k),
and:

(T a + 1 -q)X a+1 e(k) = (ψ a Q a (k) -P a (k) + 1 -q) y a+1 (k)e(k).

Considering the two cases k a = k a+1 and k a = k a+1 separately and using (3.9), (3.10) and (3.14), we can easily prove that the two above quantities are equal. The commutation relations (1.2d) and (1.2e) are straightforward from the defining relations in R Λ α (Γ), and for (1.2f) we can reproduce the corresponding part of the proof of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Theorem 4.3] Finally, let us prove that the cyclotomic relation (1.6) is satisfied, that is:

k∈K f (X 1 ) -v k q k Λ k = 0.
We have, using (2.1a) and (2.1b):

k∈K f (X 1 ) -v k q k Λ k = k∈K k∈K α v k1 q k1 (1 -y 1 ) -v k q k e(k) Λ k = k∈K k∈K α v k1 q k1 (1 -y 1 ) -v k q k Λ k e(k) = k∈K α k∈K v k1 q k1 (1 -y 1 ) -v k q k Λ k e(k) .
By (2.5), for k ∈ K α the term for k = k 1 vanishes, hence we get the result.

To conclude, the map f : H Λ n (q, v) → R Λ α (Γ) defined on the generators X 1 , T 1 , . . . , T n-1 yields a homomorphism of algebra. By restriction, we get an algebra homomorphism f :

H Λ α (q, v) → R Λ α (Γ).
In particular, the image of X a for a > 1 is the one given in Theorem 3.17, thanks to (1.3) and (3.18).

The map g is a homomorphism

In this paragraph, for any m ∈ R Λ α (Γ) we also write m := g(m) ∈ H Λ α (q, v). In particular, we have:

y a = k∈K α (1 -v -ka q -ka X a )e(k) ∈ H Λ α (q, v),
thus we can consider the power series P a (k) and Q a (k) as elements of H Λ α (q, v), namely:

ψ a = k∈K α (T a + P a (k))Q a (k) -1 e(k) ∈ H Λ α (q, v).
Following Lusztig, define the following "intertwining element" in H Λ α (q, v) for a ∈ {1, . . . , n-1} by: Φ a := T a + (1 -q)

k∈K α ka =ka+1 (1 -X a X -1 a+1 ) -1 e(k) + k∈K α ka=ka+1 e(k),
where (1 -X a X -1 a+1 )

-1 e(k) denotes the inverse of (1 -X a X -1 a+1 )e(k) in e(k)H Λ α (q, v)e(k). Noticing that y a (k)e(k) = X a e(k), we can check the following equality:

Φ a = k∈K α (T a + P a (k))e(k).
We can give an analogue of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Lemma 4.1]. Once again, we just have to write a (respectively k, k) instead of their r (resp. i, i), both in the statements and the proofs. Among all the relations in the lemma, we will make here an explicit use of the following one:

X a+1 Φ a e(k) = Φ a X a e(k) if k a = k a+1 , Φ a X a e(k) + (qX a+1 -X a )e(k) if k a = k a+1 . (3.19)
We now check the different relations of R Λ α (Γ). Relations (2.1a)-(2.1g), (2.1j)-(2.1k) and (2.5) follow as in the corresponding part of the proof of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Theorem 4.2]. To check (2.1i), again we just follow the corresponding part of the proof of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Theorem 4.2], but we need to add some v j 's. We have:

y a+1 ψ a e(k) = (1 -v -ka q -ka X a+1 )Φ a Q a (k) -1 e(k).
If k a = k a+1 , using (3.19) we get:

y a+1 ψ a e(k) = Φ a Q a (k) -1 (1 -v -ka q -ka X a )e(k) = ψ a y a e(k),
whereas if k a = k a+1 we obtain:

y a+1 ψ a e(k) = (1 -v -ka q -ka X a+1 )(T a + 1)Q a (k) -1 e(k) = (T a + 1)(1 -v -ka q -ka X a ) + v -ka q -ka X a -v -ka q 1-ka X a+1 Q a (k) -1 e(k) = (ψ a y a + 1)e(k), since (v -ka q -ka X a -v -ka q 1-ka X a+1 )e(k) = Q a (k)e(k)
. The proof of (2.1h) is similar.

Conclusion

As in [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Lemma 3.4], we have:

f (e(k)) = e(k) ∈ R Λ α (Γ),
for all k ∈ K α . It is now an easy exercise to show that f • g is the identity of R Λ α (Γ), and then that g • f is the identity of H Λ α (q, v). Hence, the homomorphisms f and g are inverse isomorphisms and Theorem 3.17 is proved. Summing the F -isomorphism

H Λ α (q, v) ≃ R Λ α (Γ)
over all α |= K n, we thus get the statement of Theorem 3.4. Note that since H Λ α (q, v) is zero for all but finitely many α, the same thing happens for R Λ α (Γ). In particular, the direct sum:

R Λ n (Γ) = α|=K n R Λ α (Γ),
has a finite number of non-vanishing terms.

An unexpected corollary

For j ∈ J ′ , let us write Λ j for the restriction of Λ to I × {j} ≃ I. Since Γ is given by p ′ disjoint copies of the quiver Γ e , we know from [START_REF] Rostam | Cyclotomic Yokonuma-Hecke algebras are cyclotomic quiver Hecke algebras[END_REF]Theorem 6.30] that there is an algebra isomorphism:

R Λ n (Γ) ≃ λ|= J ′ n Mat m λ R Λ 1 λ1 (Γ e ) ⊗ • • • ⊗ R Λ p ′ λ p ′ (Γ e ) ,
where

m λ := n! λ1!•••λ p ′ ! .
For any j ∈ J ′ , we set:

H Λ j λj (q) := H Λ j λj (q, v triv ), where v triv has only one coordinate, equal to 1. In particular, we saw from Theorem 3.4 or [BrKl] that we have the F -isomorphism H Λ j λj (q) ≃ R Λ j λj (Γ e ). We deduce the following result.

Theorem 3.20. Let v ∈ (F × ) p ′ as in §3.1. We have an (explicit) F -algebra isomorphism:

H Λ n (q, v) ≃ λ|= J ′ n Mat m λ H Λ 1 λ1 (q) ⊗ • • • ⊗ H Λ p ′ λ p ′ (q) .
In particular, the algebras

H Λ n (q, v) and ⊕ λ|= J ′ n H Λ 1 λ1 (q) ⊗ • • • ⊗ H Λ p ′ λ p ′ (
q) are Morita equivalent. Note that since the following condition is satisfied (recall (3.1)):

1≤j<j ′ ≤p ′ i,i ′ ∈I -n<a<n q a (v j q i ) -v j ′ q i ′ ∈ F × ,
the Morita equivalence is known by [START_REF] Dipper | Morita equivalences of Ariki-Koike algebras[END_REF]Theorem 1.1]. Therefore, Theorem 3.20 provides an explicit isomorphism for which the Morita equivalence of [DiMa] follows. [START_REF] Poulain D'andecy | Invariants for links from classical and affine Yokonuma-Hecke algebras[END_REF]Corollary 3.2] or [START_REF] Rostam | Cyclotomic Yokonuma-Hecke algebras are cyclotomic quiver Hecke algebras[END_REF] we know that the algebra of Theorem 3.20 is a cyclotomic Yokonuma-Hecke algebra of type A, as introduced in [ChPA].

Remark 3.21. If Λ 1 = • • • = Λ p ′ , by

A presentation for H Λ p,n (q)

In this section, we prove our second main result, given in Corollary 4.16: we give a cyclotomic quiver Hecke-like presentation for H Λ p,n (q). The key is to make a careful choice for the family {Q a (k)} a,k .

A nice family

We consider the quiver Γ with vertex set V = {v j q i } i∈I,j∈J ′ ≃ K = I × J ′ of §3.1, where v 1 , . . . , v p ′ ∈ F × satisfy (3.1). We give here a particular choice for the family {Q a (k)} a,k . We recall the definition of the family {P a (k)} a,k of (3.9). Lemma 4.1 [START_REF] Stroppel | Quiver Schur algebras and q-Fock space[END_REF](5.4)]). The family {Q a (k)} 1≤a<n,k∈K n given by:

Q a (k) :=      1 -q + qy a+1 -y a if k a = k a+1 , 1-Pa(k) ya+1-ya if k a ← k a+1 or k a ⇆ k a+1 , 1 -P a (k)
otherwise, satisfies the property (BK). and:

σ -1 • f (T a ) = k∈K n σ -1 [ψ a Q a (k) -P a (k)]e(k) = k∈K n [ψ a Q a (k) -P a (k)]e(σ -1 (k)) = k∈K n [ψ a Q a (σ(k)) -P a (σ(k))]e(k) = k∈K n [ψ a Q a (k) -P a (k)]e(k) = f (T a ) = f • σ(T a ).
Note that the above sums over K n are in fact finite, since all but finitely many e(k) ∈ R Λ n (Γ) are zero (recall, for instance, §3.3.3).

Remark 4.15. Theorem 4.14 fails if we consider the homomorphism f built from the family {Q BK a (k)} a,k . For instance, Lemma 4.13 is no longer valid with

Q BK a (k), since if k a ← k a+1 : Q BK a (σ(k)) = ζζ ka q ka = ζQ BK a (k)
, and the same result holds if k a → k a+1 .

We now recall some notation and facts from §2.3. If α |= K n, we denote by [α] its orbit under the action of σ (this action is defined in Lemma 2.13). We have an associated subset

K [α] = ⊔ β∈[α] K β of K n (see (2.16)). The quotient set of K [α] by the equivalence relation ∼ generated by k ∼ σ(k) for all k ∈ K [α] is K [α] σ (Definition 2.21). Each equivalence class γ ∈ K [α]
σ has cardinality p, and is given by γ = {k, σ(k), . . . , σ p-1 (k)} for some k ∈ K [α] (see (2.22)). Finally, thanks to the canonical map K n /∼ → (K/∼) n and Lemma 4.9, for any γ ∈ K

[α]
σ and a ∈ {1, . . . , n} we have well-defined statements γ a = γ a+1 , γ a → γ a+1 , etc. (see Lemma 2.12 and before Remark 2.26). Moreover, since Λ is σ-stable (Proposition 4.12) the integer Λ γa is well-defined.

Corollary 4.16. The F -algebra isomorphism f : H Λ n (q, ζ) → R Λ n (Γ) induces an isomorphism between H Λ p,n (q) and R Λ n (Γ) σ . Hence, we have the following F -algebra isomorphism:

H Λ p,n (q) ≃ [α] R Λ [α] (Γ) σ ,
where [α] runs over the orbits of the K-compositions of n under the action of σ , and the subalgebra H Λ p, [α] (q) has a presentation given by the generators

{e(γ)} γ∈K [α] σ ∪ {y 1 , . . . , y n } ∪ {ψ 1 , . . . , ψ n-1 },
and the relations (2.29) and (2.36).

Proof. Using Theorem 4.14, for h ∈ H Λ n (q, ζ) we have:

σ(h) = h ⇐⇒ f • σ(h) = f (h) ⇐⇒ σ -1 • f (h) = f (h) ⇐⇒ f (h) = σ • f (h), hence: h is fixed under σ ⇐⇒ f (h) is fixed under σ.
Using Corollary 1.18, we get:

H Λ p,n (q) ≃ H Λ n (q, ζ) σ ≃ R Λ n (Γ) σ ,
as desired. We deduce the second statement from the equality R Λ n (Γ) σ = ⊕ [α] R Λ [α] (Γ) σ (note that this direct sum is finite by Theorem 3.17) and Theorem 2.43, where we gave a presentation for R Λ

[α] (Γ) σ .

Theorem A.2. The algebra homomorphisms φ : H Λ 1,n (q) → H Λ n (q) and ψ : H Λ n (q) → H Λ 1,n (q) given by: Proof. We first check that ψ is an algebra homomorphism: all relations are straightforward except (1.2c), but it follows from (1.14h) and (A.1). Concerning φ, again all relations are straightforward, except (1.14d) (if n ≥ 3). Note the following consequence of (1.2c):

S -1 T 1 ST 1 = T 1 ST 1 S -1 . (A.3)
In the following calculation, we adopt the following conventions:

• we use color when a quantity simplifies;

• we use underbrace when we will use a relation;

• we use parenthesis when we did use a relation.

We have:

φ(t ′ 1 )φ(t 1 )φ(t 2 )φ(t ′ 1 )φ(t 1 )φ(t 2 ) = φ(t 2 )φ(t ′ 1 )φ(t 1 )φ(t 2 )φ(t ′ 1 )φ(t 1 ) ⇐⇒ [S -1 T 1 S]T 1 T 2 [S -1 T 1 S]T 1 (A.3) T 2 = T 2 [S -1 (1.2d) T 1 S]T 1 T 2 [S -1 T 1 S]T 1 (A.3) . ⇐⇒ S -1 T 1 S T 1 T 2 (T 1 (1.2f)
ST 1 S -1 )T 2 = (S -1 T 2 )T 1 S T 1 T 2 (T 1 (1.2f) ST 1 S -1 ) ⇐⇒ T 1 S(T 2 T 1 T 2 )S T 1 (T 2 S -1 ) = T 2 T 1 S(T 2 T 1 T 2 )S T 1 S -1 ⇐⇒ T 1 (T 2 S)T 1 (S T 2 )T 1 T 2 = T 2 T 1 (T 2 S)T 1 (ST 2 )T 1 ⇐⇒ T 1 T 2 ST 1 S(T 1 T 2 T 1 ) = (T 1 T 2 T 1 )ST 1 ST 2 T 1 ⇐⇒ ST 1 ST 1 = T 1 ST 1 S, which allows us to conclude. Finally, the composition φ • ψ is the identity on the set of generators {S, T 1 , . . . , T n-1 }, and using (A.1) we find that ψ • φ is the identity on the set of generators {s, t ′ 1 , t 1 , . . . , t n-1 }. Hence, the algebras homomorphisms φ and ψ are inverse isomorphisms and this concludes the proof.

A.2 Two equivalent relations

We suppose that p ≥ 2. We prove here the statement of Remark 1.16: in the algebra H Λ p,n (q), the relations (1.14i) and (1.14i') are equivalent. We will even prove a slightly more general statement (cf. Proposition A.6). Let A be a unitary ring and q ∈ A × an invertible element. Let s, t ′ 1 , t 1 some symbols which satisfy:

(t ′ 1 + 1)(t ′ 1 -q) = (t 1 + 1)(t 1 -q) = 0. (A.4) Lemma A.5. We have:

(q -1 t ′ 1 t 1 ) 2-p t 1 st ′ 1 + (q -1)

p-2 k=1 (q -1 t ′ 1 t 1 ) 1-k st ′ 1 = (t -1 1 t ′ 1 -1 t -1 1 t ′ 1 -1 . . . p-2 factors )(. . . t 1 t ′ 1 t 1 t ′ 1 p-2 factors )t 1 st ′ 1 .
Proof. For p = 2 we get:

t 1 st ′ 1 = t 1 st ′
1 , which is obviously true. If the equality is satisfied for p -1 ≥ 2, we get: (q -1 t ′ 1 t 1 ) 2-p t 1 st ′ 1 + (q -1)

p-2 k=1 (q -1 t ′ 1 t 1 ) 1-k st ′ 1 = (q -1 t ′ 1 t 1 ) 2-p t 1 st ′ 1 + (q -1)

p-2 k=2 (q -1 t ′ 1 t 1 ) 1-k st ′ 1 + (q -1)st ′ 1 = (q -1 t ′ 1 t 1 ) -1 (q -1 t ′ 1 t 1 ) 3-p t 1 st ′ 1 + (q -1)(q -1 t ′ 1 t 1 ) -1 p-3 k=1 (q -1 t ′ 1 t 1 ) 1-k st ′ 1 + (q -1)st ′ 1 = (q -1 t ′ 1 t 1 ) -1   (q -1 t ′ 1 t 1 ) 2-(p-1) t 1 st ′ 1 + (q -1)

(p-1)-2 k=1 (q -1 t ′ 1 t 1 ) 1-k st ′ 1   + (q -1)st ′ 1 = (q -1 t ′ 1 t 1 ) -1 (t -1 1 t ′ 1 -1 . . . p-3 )(. . . t 1 t ′ 1 p-3 )t 1 st ′ 1 + (q -1)st ′ 1 = q(t -1 1 t ′ 1 -1 . . . p-1 )(. . . t 1 t ′ 1 p-3
)t 1 st ′ 1 + (q -1)st ′ 1 .

Let us now distinguish between two cases. If p is even, using qt -1 1 = t 1 -(q -1) we get:

(q -1 t ′ 1 t 1 ) 2-p t 1 st ′ 1 + (q -1)

p-2 k=1 (q -1 t ′ 1 t 1 ) 1-k st ′ 1 = q(t -1 1 t ′ 1 -1 . . . t ′ 1 -1 t -1 1 p-1 )(t ′ 1 t 1 . . . t 1 t ′ 1 p-3 )t 1 st ′ 1 + (q -1)st ′ 1 = (t -1 1 t ′ 1 -1 . . . t ′ 1 -1 p-2
)[t 1 -(q -1)](t ′ 1 t 1 . . . t 1 t ′ 1 p-3

)t 1 st ′ 1 + (q -1)st ′ 1

= (t -1 1 t ′ 1 -1 . . . t ′ 1 -1 p-2 )(t 1 t ′ 1 t 1 . . . t 1 t ′ 1 p-2 )t 1 st ′ 1 -(q -1)(t -1 1 t ′ 1 -1 . . . t ′ 1 -1 p-2 )(t ′ 1 t 1 . . . t 1 t ′ 1 p-3 )t 1 st ′ 1 + (q -1)st ′ 1 = (t -1 1 t ′ 1 -1 . . . t ′ 1 -1 p-2 )(t 1 t ′ 1 t 1 . . . t 1 t ′ 1 p-2 )t 1 st ′ 1 -(q -1)t -1 1 t 1 st ′ 1 + (q -1)st ′ 1 = (t -1 1 t ′ 1 -1 . . . t ′ 1 -1 p-2 )(t 1 t ′ 1 t 1 . . . t 1 t ′ 1 p-2 )t 1 st ′ 1 ,
thus we are done. If p is odd, similarly we obtain, now using qt ′ 1 -1 = t ′ 1 -(q -1):

(q -1 t ′ 1 t 1 ) 2-p t 1 st ′ 1 + (q -1)

p-2 k=1 (q -1 t ′ 1 t 1 ) 1-k st ′ 1 = q(t -1 1 t ′ 1 -1 . . . t -1 1 t ′ 1 -1 p-1 )(t 1 t ′ 1 . . . t 1 t ′ 1 p-3 )t 1 st ′ 1 + (q -1)st ′ 1 = (t -1 1 t ′ 1 -1 . . . t -1 1 p-2 )[t ′ 1 -(q -1)](t 1 t ′ 1 . . . t 1 t ′ 1 p-3 )t 1 st ′ 1 + (q -1)st ′ 1 = (t -1 1 t ′ 1 -1 . . . t -1 1 p-2 )(t ′ 1 t 1 t ′ 1 . . . t 1 t ′ 1 p-2 )t 1 st ′ 1 -(q -1)(t -1 1 t ′ 1 -1 . . . t -1 1 p-2 )(t 1 t ′ 1 . . . t 1 t ′ 1 p-3 )t 1 st ′ 1 + (q -1)st ′ 1 = (t -1 1 t ′ 1 -1 . . . t -1 1 p-2 )(t ′ 1 t 1 t ′ 1 . . . t 1 t ′ 1 p-2
)t 1 st ′ 1 -(q -1)t -1 1 t 1 st ′ 1 + (q -1)st ′ 1

  Lemma 2.8 ([START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF] Lemma 2.1]). The elements y a ∈ R Λ α (Γ) are nilpotent for any a ∈ {1, . . . , n}.

  φ(s) := S, φ(t ′ 1 ) := S -1 T 1 S, φ(t a ) := T a , for all a ∈ {1, . . . , n -1}, and: ψ(S) := s, ψ(T a ) := t a , for all a ∈ {1, . . . , n -1}, are well-defined and inverse to each other.

Acknowledgements I am grateful to Maria Chlouveraki, Nicolas Jacon, Noah White

Remark 2.39. The linear map µ is a linear projection onto the subspace R [α] (Γ) σ . Lemma 2.40. We have the following equality:

In particular, µ(I Λ

Proof. Since for γ ∈ K

[α]

σ we have y

σ . We now consider an element h of I Λ [α] . Because of (2.1a), (2.1b) and Theorem 2.10, we know that we have: h = w1,w2∈Sn m 1 ,m 2 ∈N n k∈K [α] h k ψ w1 y

where the h k ∈ F are some scalars which depend on k and on the other various indices of the sums. For m ∈ {0, . . . , p -1} we have:

w1,w2∈Sn m 1 ,m 2 ∈N n k∈K [α] h k ψ w1 y

Summing all these equalities from m = 0 to p -1 and using (2.24) we get, where

w1,w2∈Sn m 1 ,m 2 ∈N n k∈K [α] h k ψ w1 y

(2.41) Since:

• we have p = 0;

• the elements ψ w1 y

,σ , and we deduce the last statement from Lemma 2.38.

Lemma 2.42. For each

. Hence, by Lemma 2.33 we obtain:

for all m ′ ∈ {0, . . . , p -1}, thus, by summing:

We are now ready to state the main theorem of this section. We recall that the idempotents {e(γ)} γ of (2.28) are indexed by the set K

σ , which is defined in Definition 2.21. Remark 4.2. The condition "k a ← k a+1 or k a ⇆ k a+1 " is equivalent to "v ka q ka = qv ka+1 q ka+1 ".

With k = (i, j), that means "i a = i a+1 + 1 and j a = j a+1 ". The family given in [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF](4.36)] would be the following one:

(4.3)

We will see in Remark 4.15 why the choice of Lemma 4.1 is more adapted to our problem.

For the convenience of the reader, we will now give a proof of Lemma 4.1.

In particular, if k a ← k a+1 or k a ⇆ k a+1 we get (recall Remark 4.2):

which is well-defined. As suggested in [StWe], we now notice that, if k a = k a+1 :

(1

This is a straightforward consequence of the equality P a (k) + P a (s a • k) sa = 1 -q (see [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF](4.28)]). Let us now check that (BK) is satisfied. First, the element Q a (k) is of course invertible when k a = k a+1 (since 1 -q = 0), and the invertibility in the remaining cases follows from the above calculations so (3.13) holds. Moreover, equation (3.14) is true by definition.

We now check the different relations (3.15

Finally, to prove equation (3.16) it suffices to see that P a+1 (s a+1 s a • k) sa = P a (s a s a+1 • k) sa+1 . This equality follows from [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF](4.29)] and the braid relation s a s a+1 s a = s a+1 s a s a+1 .

Remark 4.5. We deduce from the calculations made at the beginning of the proof of Lemma 4.

Intertwining

In this subsection, we show how our previous works allow us to prove our main result (Corollary 4.16). For j ∈ J ′ , let us set v j := ζ j : it follows from the definition of p ′ that v 1 , . . . , v p ′ satisfy the distinct orbit condition (3.1). In particular, the vertex set of Γ is V = {ζ j q i } i∈I,j∈J ′ . Let us consider a weight Λ = (Λ k ) k∈K of level r, such that:

We suppose that the associated tuple Λ = (Λ i ) i∈I , of level ωd, satisfies the condition of Proposition 1.33, that is (recall the notation η of (1.23)):

so that the algebras H Λ n (q, ζ), H Λ p,n (q) (recall Definition 1.35) and the shift automorphism of H Λ n (q, ζ) (recall Proposition 1.9) are well-defined. We will use the above condition (4.7) and the results of §2.3 to define a particular automorphism σ of R Λ n (Γ). Let us define σ : V → V by: σ(v) := ζv, (4.8) for all v ∈ V . Note that σ is well-defined since V is also given by {ζ j q i } i∈I,j∈J . Moreover, the reader may have noticed the similarity with the map of Proposition 1.9.

Lemma 4.9. The map σ : V → V defined on the vertices of Γ satisfies the assumptions of §2.3, that is:

is also an edge of Γ;

• for any p 1 ∈ {1, . . . , p -1} and any vertex v ∈ V we have

Proof. Since ζ is a primitive pth root of unity, we deduce that the first and third points are satisfied. It remains to prove the second one. Let (v, v ′ ) be an edge of Γ. By definition, we have v ′ = qv, thus ζv ′ = ζ(qv) = q(ζv). Hence, we have σ(v ′ ) = qσ(v): we have proved that (σ(v), σ(v ′ )) is an edge of Γ.

The action of σ on V is algebraically easy. Let us now describe how σ acts "graphically" on the set V of the vertices of Γ, that is, on K = I × J ′ . Let i ∈ I, j ∈ J ′ and set v := ζ j q i . We have:

Hence, if j < p ′ then σ just translates the vertex v to the copy of Γ e directly on its right. If j = p ′ , we have j + 1 = p ′ + 1 / ∈ J ′ thus we write:

It means that v is translated to the first copy of Γ e and rotated by η. Note that depending on Γ, there may not be any translation or rotation. With the examples of Figure 2, that gives:

case (e, p) = (2, 3) we have p ′ = 3, η = 0 and the map σ is given by the product of 3-cycles (0 1 , 0 2 , 0 3 )(1 1 , 1 2 , 1 3 );

case (e, p) = (2, 6) we have p ′ = 3, η = 1 and the map σ is given by the 6-cycle (0 1 , 0 2 , 0 3 , 1 1 , 1 2 , 1 3 );

case (e, p) = (∞, 2) we have p ′ = 2, η = 0 and the map σ is given by the product of transpositions i∈I (i 1 , i 2 ).

In particular, note that σ has indeed order p.

By Theorem 2.14 and Lemma 2.31, the permutation σ of the vertices of

Let us now check that the weight Λ satisfies the σ-stability condition (2.32). Proposition 4.12. For any k ∈ K = I × J ′ we have Λ k = Λ σ(k) .

Proof. We have seen above that for (i, j) ∈ I × J ′ :

• if j < p ′ then σ(i, j) = (i, j + 1);

• if j = p ′ then σ(i, j) = (i + η, 1). Thus, we deduce the result from (4.6) and (4.7).

By Lemma 2.33, we know that the map σ induces an automorphism of the cyclotomic quiver Hecke algebra R Λ n (Γ). We will refer to it as the shift automorphism of R Λ n (Γ). Lemma 4.13. The power series y a (k), P a (k) and Q a (k) of R Λ n (Γ) are shift-invariant. Moreover:

Proof. The first statement is clear since y a and y a+1 are shift-invariant (by definition, just recall Theorem 2.14). Recall that

In particular, the image of σ(k) in V n is (ζζ k1 q k1 , . . . , ζζ kn q kn ). Thus, we have:

Hence, if k a = k a+1 then by Lemma 2.12 we have P a (k) = P a (σ(k)), and if k a = k a+1 we have:

Let us now denote by σ : H Λ n (q, ζ) → H Λ n (q, ζ) the shift automorphism of H Λ n (q, ζ) (defined in Proposition 1.9). Recalling the choice for v that we made at the beginning of §4.2, we consider the F -algebra isomorphism f : H Λ n (q, ζ) → R Λ n (Γ) from Theorems 3.4 and 3.17, defined with the family {Q a (k)} a,k of Lemma 4.1.

Theorem 4.14 (Main theorem). We have σ

Proof. Since we deal with algebra homomorphisms, it suffices to check the equality on the generators S, T 1 , . . . , T n-1 of H Λ n (q, ζ). We successively have, using Lemma 4.13 (recall that, by definition, S = X 1 ):

Recall from Remark 2.47 that R Λ n (Γ) is naturally Z-graded. From this grading, Theorem 3.4 and the isomorphism f , we can endow H Λ n (q, ζ) with a (non-trivial) Z-grading. Corollary 4.17. The shift automorphism σ :

) is homogeneous with respect to the previous grading. Moreover, the subalgebra H Λ p,n (q) is a graded subalgebra of H Λ n (q, ζ).

Proof. Recall from Remark 2.47 that σ : R Λ n (Γ) → R Λ n (Γ) is homegeneous and that R Λ n (Γ) σ is a graded subalgebra. We thus deduce the first assertion from Theorem 4.14 and the second one from Corollary 4.16.

We now give an analogue of a classical corollary of [START_REF] Brundan | Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras[END_REF]Theorem 1.1].

Corollary 4.18. If q ∈ F \ {0, 1} has the same order e ∈ N ≥2 ∪ {∞} as q then:

as F -algebras.

Proof. We know from Lemma 1.21 and Theorem 3.4 that the algebras H Λ n (q) and H Λ n (q) are isomorphic to the same cyclotomic quiver Hecke algebra R Λ n (Γ). Moreover, we have the following isomorphism:

where σ is uniquely determined by the quiver Γ and the element η ∈ I such that q η = ζ p ′ .

To prove that H Λ p,n (q) ≃ H Λ p,n (q), it thus suffices to prove that there is a primitive pth root of unity ζ ∈ F × such that: qη = ζp ′ (recall from Lemma 1.21 that p ′ does not depend on the chosen primitive pth root of unity).

To deal with the case e = ∞, it suffices to set ζ := ζ. Recall that, in that case, we have η = 0 and p ′ = p. Hence, we now assume that e < ∞. Since q and q are both primitive eth roots of unity, there is some a ∈ Z, invertible modulo e, such that q = q a . In particular, for any k ∈ Z we have q = q a+ke . Since q η = ζ p ′ , we get:

Therefore, it suffices to prove that there is some k ∈ Z such that a+ke is invertible modulo p, that is, such that ζ := ζ a+ke is a primitive pth root of unity. A quick (but very powerful) argument is to use Dirichlet's theorem about arithmetic progression (see also [START_REF] Hu | Crystal bases and simple modules for Hecke algebras of type G(p, p, n)[END_REF]Lemma 3.5]): since a and e are coprime, the set {a + ke} k∈N contains infinitely many prime numbers.

In particular, it contains a prime ℘ which does not divide p, hence which is coprime to p. It now suffices to choose k ∈ N such that ℘ = a + ke.

A About H Λ p,n (q)

The aim of this appendix is to give details for the statements of Remarks 1.15 and 1.16.

A.1 Case p = 1

We prove here the statement of Remark 1.15: the presentation of H Λ p,n (q) given at §1.2 gives the Ariki-Koike algebra H Λ n (q) = H Λ n (q, 1) of §1.1. Note that since p = 1, the relation (1.14i) becomes st ′ 1 = t 1 s, thus we have:

The result follows from the theorem below.

thus we are done.

Proposition A.6. We assume that s, t ′ 1 , t 1 satisfy, in addition to (A.4), the following relation:

The relations:

and: ) are equivalent.

Proof. By Lemma A.5, relation (Ar) is equivalent to:

If p is even, this reads:

whence we obtain:

Thus, using (A.7) to bring s to the left on both sides, we get:

, which is the desired result: the relations (Ar) and (BMR) are equivalent. If now p is odd, relation (A.8) reads:

whence we obtain:

Thus, using (A.7) to bring s to the left on both sides, we get:

, which is the desired result: the relations (Ar) and (BMR) are thus equivalent.

Using Proposition A.6, it is now clear that the algebra H Λ p,n (q) is isomorphic to the one defined by Ariki in [START_REF] Ariki | Representation theory of a Hecke algebra of G(r, p, n)[END_REF] as stated in Remark 1.16.


Paula Barros

Anne Dunan-Page

W R Owens

come

Introduction

The members of the Quaker congregation portrayed in Egbert van Heemskerck's famous painting, A Puritan Meeting, dated 1678, are engaged in a somewhat disorderly performance of religious worship. 1 In the gallery, one man seems to be testifying while another one is wiping the sweat from his brow. In spite of the open window, the room seems stifling hot. In the centre foreground below, a woman collapses into a fit. One of her fellow worshippers is holding her hand, his solicitude in stark contrast with the sardonic smile worn by the painter in the left-hand corner -presumably a portrait of the artist himself. On the bench to the right, a man with upturned eyes experiences a form of religious ecstasy. Some of the members of the congregation are praying or singing, at least one of them appears to be sleeping, others are talking. The painting captures the meeting in full sensory detail. Dominated by a multiplicity of voices, the assembly's soundscape seems to dissolve into an indecorous cacophony. Robert Raines has noted that the painting calls to mind Celia Fiennes's comment from Scarborough in 1697: 'I was at a Quakers Meeting in the town where 4 men and 2 women spoke […] but it seem'd such a confusion and so incoherent that it very much moved my compassion and pitty to see their delusion and ignorance.' [START_REF] Raines | quoting The Journeys of Celia Fiennes[END_REF] The confusion of voices in van Heemskerck's satirical representation of a Quaker meeting reflects both the soundscape of religious meetings and the polemical use of noise in early modern religious disputes. [START_REF] Craig | Psalms, groans and dogwhippers: the soundscape of worship in the English parish church, 1547-1642[END_REF] As Laura Feitzinger Brown has shown, 'many early modern English religious writers, whether separatist, Puritan, conformist or recusant […] rhetorically invoke [d] noise to condemn worship practices with which [they] disagree[d]'. [START_REF] Feitzinger Brown | Brawling in Church: Noise and the Rhetoric of Lay Behavior in Early Modern England[END_REF] In a Protestant context such as van Heemskerck's, the central role of hearing in the process of salvation gave added justification to the dismissal of potentially subversive expressions of Dissent as noise. [START_REF] On | The Art of Hearing. English Preachers and Their Audiences[END_REF] Recent scholarship has highlighted the sensory dimension of religious expression in the Long Reformation, with a special focus on speaking and hearing. [START_REF]For a discussion of the full range of the senses in early modern religious life, see Matthew Milner, The Senses and the English Reformation[END_REF] Diarmaid MacCulloch has described the Reformation as a period of noise, not only because of its repudiation of monastic silence, but also because of its emphasis on preaching and psalm singing, not to mention its taste for controversy on theological and ecclesiological issues. [START_REF] Macculloch | Silence: a Christian History[END_REF] Additionally, publications on music and singing, public and private devotion, spaces of worship, emotions, concepts of community and identity, and book history have complicated our understanding of religious experience in early modern Europe, and in 2009 the French social historian Arlette Farge published her Essai pour une histoire des voix. [START_REF] Bayard | See also Dissenting Praise: Religious Dissent and the Hymn in England[END_REF] To probe further into these issues and explore their implications for the history of Protestant Dissent, the Eighth Triennial Conference of the International John Bunyan Society, which convened in Aix-en-Provence, 6-9 July 2016, concentrated on the representation and expression of Dissent, Nonconformity and Puritanism in the Long Reformation, with an emphasis on the relationship between written and oral cultures.

Among the themes that were discussed during the conference was the variety of exchanges and interactions not only within Dissenting communities, but also between Dissenting and non-Dissenting traditions. [START_REF] Jenna | Voicing Dissent in the Long Reformation[END_REF] Taken together, these essays testify to the enduring legacy of Dissenting voices and the persistence of Dissenting art and culture from the eighteenth century to contemporary societies, from Britain to the United States and China. Following such seminal works as Isabel Hofmeyr's The Portable Bunyan (2004), they remind us that much still needs to be done to gauge the heritage of Dissent. And that heritage can be seen (or indeed heard) not only through the political, philosophical, religious texts that have shaped the religious history of Protestantism but also through the works of artists and cultural intermediaries -whether they be musicians, painters, printers, craftsmen, translators, missionaries, or mothers who gave good books at Christmas.

Paula Barros

Université Paul-Valéry, IRCL-UMR 5186, France

Anne Dunan-Page

Aix Marseille Univ, LERMA, Aix-en-Provence, France

W. R. Owens

The Open University and University of Bedfordshire, UK Notes between written and oral cultures. Topics might include:

  Papers and plenary lectures dealing with the circulation and adaptation of ideas, texts and practices across confessional, geographical or chronological boundaries brought to the fore the diversity of expressions of Dissent in the Long Reformation and identified multiple echoes of early modern Dissenting voices in more recent periods. The essays collected in this special issue explore these interactions from a literary and artistic point of view, examining expressions of Dissent in practices such as poetry, hymn-writing, 'gifting' of books, stained-glass work, and translation. 10 Focusing on George Herbert and John Bunyan, they highlight conversations between different religious denominations in the seventeenth century as well as in the broader context of British religious history from the Long Reformation to the present day. They reveal patterns of reception and strategies of adaptation which indicate that, in certain circumstances, the noise of religious controversy is muted or silenced as common denominators are emphasized; in other cases, however, processes of circulation and appropriation result in intergenerational transmission of family piety and reaffirmation of denominational identities. In the first essay, Helen Wilcox explores the manifold uses of poetry by post-Restoration seventeenth-century Dissenting writers. She argues that Dissenters as a whole acknowledged poetry as a rich medium of spiritual expression and showed little reluctance to take their inspiration from earlier Conformist poets. Their special affinity with George Herbert's lyrics, whose emotional and devotional qualities chimed in with Dissenting spirituality, is also the topic of Jenna Townend's essay, which focuses on seventeenthcentury appropriations of poems from The Temple in the Dissenting tradition of psalm translation, hymn-writing, and devotional singing. Townend identifies strategies of adaptation that aligned Herbert's poetic voice with expressions of Dissenting experience and identity and contributed to create 'communities of godly devotion'. Turning from George Herbert to John Bunyan, Sylvia Brown traces the practice of 'gifting' copies of The Pilgrim's Progress within social and familial networks from the seventeenth to the nineteenth centuries, as well as the special role of women in transmitting Bunyan's most famous work on both sides of the Atlantic. Material traces left by several generations of readers enable her to underline the role played by the circulation of 'good books' such as The Pilgrim's Progress in the transmission of Dissenting identity. Nathalie Collé, on the other hand, addresses the legacy of Dissent beyond the boundaries of Dissenting communities. Examining representations of Bunyan and The Pilgrim's Progress in stainedglass windows set up in British churches from the end of the nineteenth century, Collé (who specialises in art history and the relationships between texts and images) shows how a key figure of English Dissent is being reclaimed by the Anglican tradition. As she argues, this is a striking enlargement of reputation for a figure who symbolized a British identity defined by resistance to oppression. Stained-glass windows have barely been touched upon in Bunyan scholarship. This is a topic that would certainly deserve further treatment, taking into account the spectacular variety of Victorian art and the Victorians' special relationship with Bunyan. Finally, Xie Jiapeng and Su Yuxiao take us to China with an essay that reviews Chinese translations of the The Pilgrim's Progress from the nineteenth to the twenty-first centuries. The authors argue that the variations between the different translations they examine must be read as so many attempts to adapt Bunyan's work to changing priorities: the evangelisation of the Chinese population by British missionaries, who authored the first translations, the appropriation of Bunyan by Chinese Christians, who translated The Pilgrim's Progress to give voice to their individual spiritual experiences, and Bunyan's place in the canon of classic world literature, which has fostered translations aiming to make The Pilgrim's Progress accessible to Chinese university students. 11

On Egbert Van Heemskerck, see Robert Raines, 'Notes on Egbert van Heemskerck and the English Taste