Owen Garnier
email: o.garnier@u-picardie.fr.

SPRINGER CATEGORIES FOR REGULAR CENTRALIZERS IN WELL-GENERATED COMPLEX BRAID GROUPS

Keywords: Primary 20F36, 51F15 ; Secondary 20F05 Garside Category, Complex braid groups

In his proof of the K(π, 1) conjecture for complex reflection arrangements, Bessis defined Garside categories suitable for studying braid groups of centralizers of Springer regular elements in well-generated complex reflection groups. We provide a detailed study of these categories, which we call Springer categories.

We describe in particular the conjugacy of braided reflections of regular centralizer in terms of the Garside structure of the associated Springer category. In so doing we obtain a pure Garside theoretic proof of a theorem of Digne, Marin and Michel on the center of finite index subgroups in complex braid groups in the case of a regular centralizer in a well-generated group. We also provide a "Hurwitz-like" presentation of Springer categories. To this aim we provide additional insights on noncrossing partitions in the infinite series. Lastly, we use this "Hurwitz-like" presentation, along with a generalized Reidemeister-Schreier method we introduce for groupoids, to deduce nice presentations of the complex braid group B(G31).

Introduction 1. Preliminaries on Garside categories 1.1. Definitions 1.2. Normal forms, groupoid and conjugacy 1.3. Periodic elements and divided categories 2. Complex reflection groups, braid groups and braid categories 2.1. Reminders on complex reflection groups and their braid groups 2.2. Dual braid monoid for well-generated groups 2.3. Lyashko-Looijenga map, cyclic labels 2.4. Regular numbers and topological groupoids 3. Properties of Springer categories 3.1. Elementary properties 3.2. The case p = 1 3.3. Presentation by Hurwitz relations 3.4. Braided reflections and atomic loops 3.5. Conjugacy of atomic loops and centers of finite index subgroups 4. Presentations of the braid group B(G 31 ) 4.1. The method 4.2. Computational data 4.3. Presentation associated to representatives of the φ 2 -orbits Appendix A. The Reidemeister-Schreier method for groupoids Introduction Let W be an irreducible complex reflection group. In the case where W is well-generated (i.e it can be generated by a number of reflections equal to its rank), Bessis introduced in [Bes15, Section 8] a Garside monoid, whose group of fractions is naturally isomorphic to the complex braid group B(W ) of W . It is the the so-called dual braid monoid.

This Garside monoid has then been used to study group theoretic questions on complex braid groups. For instance the center of any finite index subgroup is determined in [START_REF] Digne | The center of pure complex braid groups[END_REF] through the use of various Garside monoids, including the dual braid monoid. More recently, González-Meneses and Marin defined in [START_REF] Gonzàlez-Meneses | Parabolic subgroups of complex braid groups I[END_REF] a general notion of parabolic subgroups of a complex braid group. They also studied these subgroups using the dual braid monoid notably.

Unfortunately, this approach is not available when W is not well-generated. This proves especially problematic for the exceptional group G 31 , which is both badly-generated and of high rank. This lack of a well-studied Garside structure was a hindrance in completing the study program of complex braid groups. For instance the proof of [DMM11, Theorem 1.4] for the complex braid group B(G 31 ) requires external arguments involving Krammer representations.

However, Bessis also defined in [START_REF] Bessis | Finite complex reflection arrangements are K(π, 1)[END_REF] Section 11] a Garside category suitable for studying the braid group of the centralizer of a Springer regular element in a well-generated group. This approach applies in particular to G 31 , which appears as the centralizer of a Springer regular element inside of the well-generated group G 37 . The problem is now to understand these Garside categories. Indeed, the theory of Garside categories, which was in its infancy when Bessis wrote his article, is now much more developed, and allows for the generalization of many (if not all) properties of the dual braid monoid to these categories. This article aims to be a first step in this program.

Let W be a well-generated irreducible complex reflection group, and let g ∈ W be a regular element for the eigenvalue ζ d := exp( 2iπ d ), where d is a positive integer. The group W g := C W (g) is again a complex reflection group, and the braid group B(W g ) is isomorphic to the centralizer in B(W ) of a so-called regular braid [START_REF] Bessis | Finite complex reflection arrangements are K(π, 1)[END_REF]Theorem 12.4]). The proof of this statement by Bessis relies on heavy topological arguments, which boils down to the construction of a Garside category C, whose enveloping groupoid (obtained by formally inverting all morphisms) is equivalent to B(W g ). The dual braid monoid can be seen as a particular case of this topological construction (associated to the regular element Id). We refer to the category C as the Springer category (associated to W g and W ) from now on.

On the other hand, the construction of the category C can be formulated in a pure Garside theoretic manner as a category of periodic elements. We recall this construction in Section 1, before we restate the topological construction in Section 2.

The interaction between these two construction then allows us to describe the braided reflections of the group B(W g ) in terms of the Garside structure of C. In the classical case, the atoms of the dual braid monoid are known to be a set of braided reflections generating the braid group B(W ). The same result cannot hold in the case of a category, as an atom of the category C may have a different source and target, and thus it cannot be identified with an element of B(W g ). However we have a similar result. For every atom a in C, we define two morphisms a ♭ and a # which have same target and same source as a, respectively. We then get an element λ(a) := aa # a ## • • • a ♭♭ a ♭ ∈ C(u, u), which we call an atomic loop (associated to a).

Theorem. (Theorem 3.31) Let u be an object of C. Any atomic loop inside C(u, u) is a braided reflection of the group B(W g ) ≃ B(u, u). Conversely, any braided reflection σ ∈ B(W g ) ≃ B(u, u) is conjugate in B to some atomic loop.

In the case where C is a monoid, the atomic loop associated to an atom is simply the atom itself, and we recover the situation of the classical case. This explicit description of the braided reflections in terms of the Garside structure of B and C enables us to study the conjugacy of braided reflections and their powers using Garside theory. The following theorem ensures that the conjugacy of powers of braided reflections is "the same as" that of braided reflections.

Theorem. (Theorem 3.37) Let λ(s) ∈ C(u, u) be an atomic loop of some object u, and let f ∈ G. If there is some endomorphism z ∈ C such that λ(s) n f = f z for some n 1, then z = λ(s ′ ) j for some atomic loop λ(s ′ ) such that λ(s)f = f λ(s ′ ).

This result was already proven by Digne, Marin and Michel in the case of the dual braid monoid ([DMM11, Proposition 2.2]). As a corollary, we obtain the analogue of [DMM11, Theorem 1.4] in our case:

Corollary. (Corollary 3.39) Let W be a well-generated irreducible complex reflection group, and let g ∈ W be a ζ d -regular element for some integer d (where ζ d := exp( 2iπ d )). If U ⊂ B(W g ) is a finite index subgroup, then we have Z(U ) ⊂ Z(B(W g )).

This was already proven in [START_REF] Digne | The center of pure complex braid groups[END_REF] for all complex braid groups, although the proof in the case of B(G 31 ) relied on computations and the use of the generalized Krammer representation.

Our other point of focus in this article concerns presentations of Springer categories, and a way to deduce from such a presentation a presentation of the associated group B(W g ). First, we have a presentation of the Springer category C associated to B(W g ) Theorem. (Theorem 3.27) A Springer category is presented by its atoms, endowed with the Hurwitz relations, that is, all the relations induced by commutative squares of atoms.

Again, this is known to hold for dual braid monoids ([Bes15, Lemma 8.8]). The proof of this theorem relies on the particular case where the Springer category happens to be a monoid. In this case we show (Theorem 3.10 and Corollary 3.11)) that the Springer category is naturally isomorphic to the dual braid monoid associated to W g (which is wellgenerated in this case). As a byproduct we obtain (Corollary 3.22) an isomorphism between the lattice of simples of the dual braid monoid associated to G(d, 1, n) and the lattice of noncrossing partitions of type (d, 1, n) defined in [START_REF] Bessis | Non-crossing partitions of type (e, e, r)[END_REF]Definition 1.11].

In the case of well-generated groups, following the indication of [START_REF] Bessis | Finite complex reflection arrangements are K(π, 1)[END_REF]Remark 8.9], one can use the Hurwitz presentation of the dual braid monoid to prove other presentations of the associated braid group, like the ones of [BMR98, Table 3] or [BM04, Section 2].

Following the idea of [START_REF] Bessis | Finite complex reflection arrangements are K(π, 1)[END_REF]Remark 11.29], we propose in Section 4 a generalization of this work to the case of categories, which we apply to the complex braid group B(G 31 ). The underlying method is an analogue of the Reidemeister-Schreier method adapted for groupoids. We define a Schreier transversal for a connected free groupoid F(S) as a set of paths starting from a fixed object u 0 (the root of the transversal) and arriving at every object of F(S).

Proposition. (Reidemeister-Schreier method for groupoids, Proposition A.13) Let G = S | R be a connected presented groupoid. A Schreier transversal T for F(S) rooted in the object u 0 induces an explicit presentation of the group G(u 0 , u 0 ).

We consider the Springer groupoid B 31 associated to the embedding of G 31 inside G 37 as the centralizer of a i-regular element. By choosing a particular Schreier transversal rooted in an object u, we are able to obtain a presentation of B 31 (u, u) ≃ B(G 31 ), where the generator are atomic loops. By considering a particular object of B 31 , we obtain the following theorem.

Theorem. (Theorem 4.2 and Section 4.3.1) The complex braid group B(G 31 ) is given by the following presentation B(G 31 ) ≃ s, t, u, v, w ts = st, vt = tv, wv = vw, suw = uws = wsu, svs = vsv, vuv = uvu, utu = tut, twt = wtw.

where s, t, u, v, w are braided reflections. This is the presentation of B(G 31 ) that was conjectured in [BMR98, Table 3] and [START_REF] Bessis | Explicit Presentations for Exceptional Braid Groups[END_REF]Conjecture 2.4]. Another approach to prove this presentation was proposed in [Bes15, Section 4], using the data of [BM04, Figure 3]. Unfortunately, this method cannot readily be carried out with these data as the formula for the discriminant of G 31 given by [BM04, Figure 3] does not appear to satisfy the conditions of [START_REF] Bessis | Finite complex reflection arrangements are K(π, 1)[END_REF]Corollary 4.5] (the discriminant considered in [Bes15, Corollary 4.5.(iii)] is zero everywhere).

By considering other objects of B 31 , we obtain other new presentations of B(G 31 ). They are all positive homogeneous with braided reflections (atomic loops) as generators. They give explicit examples of presentations whose existence was proven in [Bes01, Theorem 0.1] in the case of G 31 . For each presentation, we give in particular an explicit isomorphism to the presentation given above in terms of images of the generators.

This work is part of my PhD thesis, with some results originating from my Master's thesis. Both are done under the supervision of Pr. Ivan Marin. I thank him for his precious advice during the preparation and redaction of this article.

Preliminaries on Garside categories

Garside categories were originally introduced near the end of the 2000s as a natural generalization of Garside monoids (see for instance [START_REF] Krammer | A class of Garside groupoid structures on the pure braid group[END_REF] or [START_REF] Bessis | Garside categories, periodic loops and cyclic sets[END_REF]). A comprehensive survey of the general theory of Garside categories is made in [DDGKM]. Throughout this paper, all categories are assumed to be small categories.

1.1. Definitions. We start with a category C. The set of objects of C will be denoted by Ob(C). For u, v ∈ Ob(C), the set of morphisms from u to v in C will be denoted by C(u, v). We follow the usual convention for composition of arrows in Garside categories: the composition of the diagram x f / / y g / / z will be denoted by f g. For u ∈ Ob(C), we consider C(u, -) (resp. C(-, u)) the set of morphisms in C with source u (resp. with target u). We define a relation on C(u, -) by ∀f, g ∈ C(u, -), f g ⇔ ∃h | f h = g.

In particular, the source of h is the target of f , and the target of h is the target of g. We say that g is a right-multiple of f and that f left-divides g. Likewise, we define a relation on C(-, u).

We want the relations and to be preorders. A classical condition to study this problem is that of (left-and right-)Noetherianity (cf [DDGKM, Definition II.2.26]). However, for the purpose of this article, we can use the stronger condition of homogeneity.

Definition 1.1. ([Bes07, Definition 2.2]) Let C be a category. A length functor is a functor ℓ : C → (Z 0 , +), such that C is generated by morphisms of positive length. A category C endowed with a length functor ℓ is called a homogeneous category.

A homogeneous category admits no nontrivial invertible morphism. Indeed if f is invertible, then we have ℓ(f ) + ℓ(f -1 ) = ℓ(1) = 0, so ℓ(f ) = ℓ(f -1 ) = 0. Since C is generated by elements of positive lengths this implies that f is trivial.

A convenient way to prove that a category (or a monoid) is homogeneous is to define it by a homogeneous presentation (cf Lemma A.7).

Lemma 1.2. Let C = (C, ℓ) be a homogeneous category, and let u ∈ Ob(C). The relations and are preorders on C(u, -) and C(-, u), respectively.

Proof. We prove the result for , the result for is obtained by working in C op . The relation is obviously reflexive and transitive, it only remains to show that it is antisymmetric. Let f, g ∈ C(u, -) be such that f g and g f . We have ℓ(f ) ℓ(g) and ℓ(g) ℓ(f ), so ℓ(g) = ℓ(f ). Let h be such that f h = g. We have ℓ(h) = ℓ(g) -ℓ(f ) = 0, thus h is trivial and f = g.

Let C be category. A nontrivial element in C which admits no left-divisor (other than itself and the identity) is called an atom . In a homogeneous category (C, ℓ), a morphism of length f can always be written as a composition of at most ℓ(f ) atoms.

Definition 1.3. ([DDGKM, Definition 2.9]) Let u be an object of a homogeneous category C. • The left-gcd of f, g ∈ C(u, -) is the meet of f and g in (C(u, -), ) (should it exist), we denote it by f ∧ g. • The right-lcm of f, g ∈ C(-, u) is the join of f and g in (C(u, -), ) (should it exist),
we denote it by f ∨ g. • Likewise, we define f ∧ R g and f ∨ L g the right-gcd and the left-lcm of f and g, respectively.

Of course, gcds and lcms need not exist in C. We need two more general definitions before we move on to the definition of a Garside category.

Definition 1.4. ([DDGKM, Definition II.2.52]) Let C be a category. We say that C is cancellative if every equality of the form f gh = f g ′ h in C implies g = g ′ . This is equivalent to the statement that every morphism in C is both a monomorphism and an epimorphism. (2) The map ∆ is target injective: if u = v, then φ(u) = φ(v).

(3) The families

Div(∆) := u∈Ob(C) ([1 u , ∆(u)], ) and Div R (∆) := u∈Ob(C) ([1 φ(u) , ∆(u)], )
are equal. We say that ∆ is a balanced map. (4) The family S := Div(∆) is finite and generates C. We call its elements the simple morphisms.

(5) For every g ∈ C(u, -), the elements g and ∆(u) admit a left-gcd. A homogeneous cancellative category C endowed with a Garside map ∆ will be called a homogeneous Garside category.

Remark 1.6. This definition is actually stronger than [DDGKM, Definition V.2.19], we notably assume that S = Div(∆) is finite. This implies in particular that C must have a finite number of objects.

Let from now on (C, ∆) be a homogeneous Garside category. To avoid heavy expressions, we will often replace the assertion "f = ∆(u) where u is the source of f " with "f ∈ ∆" or even "f = ∆". This causes no confusion because f = ∆(u) implies that u is the source of f , thus ∆(u) is the only morphism of the form ∆(x) to which f may be equal.

For s ∈ S, we denote by s (resp. s * ) the morphism in S such that ss ∈ ∆ (resp. s * s ∈ ∆). Both s and s * are unique by cancellativity.

Proposition 1.7. ([DDGKM, Proposition V.1.28 and Proposition V.2.32]) Let (C, ∆) be a homogeneous Garside category. We define an automorphism φ of C by setting -for u ∈ Ob(C), φ(u) is the target of ∆(u).

-for f ∈ C(u, v), φ(f ) is the unique morphism in C such that f ∆(v) = ∆(u)φ(f ).
Furthermore, φ has finite order and ∆ is a natural transformation from the identity functor

1 C of C to φ.
Let s be a simple morphism. We have φ(s) = s because s∆ = (ss)s = ∆s. In particular we see that φ maps S into itself. Since S is finite and generates C, φ has finite order. We also get that φ is uniquely determined by the property that, for all s ∈ S, we have φ(s) = s. Remark 1.9. If we consider the particular case of monoids, that is categories with only one object, we recover the classical definition of a homogeneous Garside monoid. The relations and are the classical left-and right-divisibility relations. The Garside map ∆ corresponds to an element of the monoid, which is the Garside element.

Let (C, ∆) be a homogeneous Garside category with set of simples S, and let φ be the automorphism of C introduced in Proposition 1.7. In the sequel, we will often be interested in subcategories of fixed points under some power of φ. Let q 0 be an integer, and let C φ q be the subcategory of C consisting of φ q -invariant morphisms. We also consider the set S φ q of simple morphisms which are φ q -invariant.

Lemma 1.10. Let a, b ∈ C. If two of a, b and ab lie in C φ q , then so does the third.

Proof. If both a and b are φ q -invariant, then φ q (ab) = φ q (a)φ q (b) = ab. If both a and ab are φ q -invariant, then φ q (a)φ q (b) = φ q (ab) = ab = φ q (a)b, and φ q (b) = b by cancellativity. The third case is dual to the second. Lemma 1.11. Let s ∈ S and suppose that the source of s is φ q -invariant. We denote by ψ(s) the right-lcm in S of all the φ iq (s) for i ∈ Z 0 . We have ψ(s) ∈ S φ q , and for every y ∈ S φ q , we have s y in S if and only if ψ(s) y in S φ q .

Proof. Let u be the source of s, and let n denote the order of φ q . We have ψ(s) = s ∨ φ q (s) ∨ • • • ∨ φ (n-1)q (s). As φ q induces an automorphism of the lattice S(u, -), we have

φ q (ψ(s)) = φ q (s) ∨ φ 2q (s) ∨ • • • ∨ φ nq (s) = ψ(s) and ψ(s) ∈ S φ q . Let now y ∈ S φ q , if ψ(s) y, then s ψ(s) y. Conversely, if x y, then for all i ∈ [[1, n -1]],
we have φ iq (s) φ iq (y) = y, thus ψ(s) y by definition of the right-lcm.

By [START_REF] Dehornoy | Foundations of Garside Theory[END_REF]Proposition VII.4.2], the category C φ q , endowed with the restriction of the map ∆, is again a Garside category. Its simple morphisms are the elements of S φ q . In particular we see that (S φ q (u, -), ) is always a lattice. 1.2. Normal forms, groupoid and conjugacy. From now on, we fix a homogeneous Garside category (C, ∆), and S its set of simple morphisms. Recall from [DDGKM, Definition II.1.28] that a S-path is a sequence of composable elements of S in C. By definition of a homogeneous Garside category, every morphism in C can be expressed by a S-path.

Definition 1.12. ([DDGKM, Corollary V.1.54]) A S-path st of length 2 in C is called greedy if s is the left-gcd of st with ∆, or equivalently if s and t are left-coprime. In general, a S-path

s 1 • • • s r is called greedy if each subpath s i s i+1 is greedy for i ∈ [[1, r -1]].
Remark 1.13. Let s 1 • • • s r be a S-path in C. This path is greedy if and only if the path φ(s 1 ) • • • φ(s r ) is greedy.

Proposition 1.14. ([DDGKM, Proposition V.3.4]) Every morphism f in C is expressed by a unique greedy S-path. This path is called the greedy normal form of f . Lemma 1.15. Let s and t be two composable simple morphisms in C. We set d := s ∧ t, s ′ := sd ∈ S and t ′ := d -1 t ∈ S. We have st = s ′ t ′ in C, and the path s ′ t ′ is greedy.

Proof. First, we have st = sdt ′ = s ′ t ′ . Then, let p be such that dp = s. We have sdp = ∆, so p = s ′ and d(s

′ ∧ t ′ ) = ds ′ ∧ dt ′ = s ∧ t = d. Thus s ′ ∧ t ′ is
trivial by cancellativity, and s ′ t ′ is a greedy path. This lemma gives an algorithmic way to compute the greedy normal form of a morphism in C, provided that we know how to compute the left-gcd of two simple morphisms.

For every category C, one can consider the enveloping groupoid G(C) of C, defined by formally inverting all morphisms in C (see Lemma A.8). In the case where (C, ∆) is a homogeneous Garside category, the natural functor C → G(C) is an embedding, and morphisms in G(C) can be conveniently described.

Notation 1.16. For a positive integer m, ∆ m (u) denotes the path ∆(u) • • • ∆(φ m-1 (u)). If m is a negative integer, ∆ m (u) denotes the inverse in G(C) of ∆ -m (u).
Occasionally, we will write ∆ m for ∆ m (u) when there is no need to specify the source u explicitly (cf. [DDGKM, Convention 3.7]).

Proposition-Definition 1.17. ([DDGKM, Definition V.3.17 and Proposition V.3.18]) Let G(C) be the enveloping groupoid of C, and let f be a morphism in G(C). There is unique way to express f as a path of the form

f = ∆ p s 1 • • • s r such that p ∈ Z, s 1 = ∆ and s 1 • • • s r is a greedy path. The path ∆ p s 1 • • • s r in G(C) is called the left-weighted factorization of f .
In particular we see that, for every morphism f in G(C). There is a positive integer m such that ∆ m f lies in C. In particular we see that an morphism f in G(C) lies in C is and only if inf(f ) 0.

Definition 1.19. ([DDGKM, Definition VIII.1.1]) Let x and x ′ be two endomorphisms in G(C). A morphism f ∈ G(C) conjugates x to x ′ if xf = f x ′ in G(C).
The endomorphism x ′ will be denoted by x f . As usual, conjugacy in G(C) induces an equivalence relation, and we can consider the conjugacy class in G(C) of the endomorphism x.

Definition 1.20. ([DDGKM, Definition VIII.2.3 and Definition VIII.2.8]) Let x be an endomorphism in G(C), with left-weighted factorization ∆ p s 1 • • • s r . The initial factor (resp. final factor) of x is defined as φ -p (s 1 ) (resp. s r ). The cycling of x is defined as cyc(x

) := x φ -p (s 1 ) = ∆ p s 2 • • • s r φ -p (s 1 ). The decycling of x is defined as dec(x) = x s -1 r = s r ∆ p s 1 . . . s r-1 = ∆ p φ p (s r )s 1 . . . s r-1 .
Note that the expressions given for cyc(x) and dec(x) are not left-weighted factorizations a priori. Definition 1.21. ([BGG07, Definition 3.1]) Let x be an endomorphism in G(C), with left-weighted factorization ∆ p s 1 . . . s r . The morphism x is called rigid if the path s r φ -p (s 1 ) is greedy, or if r = 0.

Lemma 1.22. Let x ∈ G(C) be a rigid endomorphism with left-weighted factorization ∆ p s 1 • • • s r . The left-weighted factorizations of cyc(x) and dec(x) are given by

cyc(x) = ∆ p s 2 • • • s r φ -p (s 1 ) and dec(x) = ∆ p φ p (s r )s 1 . . . s r-1 .
Furthermore, the morphisms cyc(x) and dec(x) are also rigid.

Proof. By definition, the path s 2 • • • s r φ -p (s 1 ) is greedy, and s 2 = ∆ (otherwise the path s 1 s 2 would not be greedy). Thus the path given for cyc(x) is its left-weighted factorization. The same reasoning holds for dec(x). Lastly, the fact that s 1 • • • s r is a greedy path gives directly that cyc(x) and dec(x) are both rigid.

Proposition-Definition 1.23. ([DDGKM, Definition VIII.2.12] Let x be an endomorphism in G(C). The conjugacy class of x in G(C) admits a well-defined subset SSS(x) on which each one of inf and sup takes a constant value. Furthermore, for every conjugate x ′ of x in G(C), we have inf(x ′ ) inf(SSS(x)) and sup(x ′ ) sup(SSS(x))

The set SSS(x) is called the supper-summit set of x.

Note that SSS(x) must be finite, because there is only a finite number of morphisms in G(C) with given inf and sup. Also, if x ∈ C, then inf(x) 0, and SSS(x) is included in C. Proposition 1.24. ([DDGKM, Proposition VII.2.16]) Let x be an endomorphism in G(C). One can go from x to an element of SSS(x) by a finite sequence of cycling, followed by a finite sequence of decycling.

This proposition, combined with Lemma 1.22, shows that a rigid element always lies inside its own supper-summit set.

Proposition 1.25. ([DDGKM, Lemma VIII.2.19 and Proposition VIII.2.20]) Let x be an endomorphism in G(C), and let x ′ , x ′′ be in SSS(x). Let also f be a morphism in C with x ′f = x ′′ . If f = s 1 . . . s r is the greedy normal form of f , then for all i ∈ [[1, r]] the morphism x ′s 1 •••s i lies in SSS(x).

Notice that, as φ preserves left-weighted factorizations, it stabilizes supper-summit sets. In particular, last proposition also applies to the case where the conjugating element lies in G(C).

1.3. Periodic elements and divided categories. We fix (C, ∆) a homogeneous Garside category, and G(C) its enveloping groupoid. The study of supper-summit sets provides a solution to the conjugacy problem in G(C) for any two pair of endomorphisms. If we restrict our attention to the so-called periodic elements of G(C), then we obtain a more convenient solution to the conjugacy problem through the construction of a particular Garside category.

Definition 1.26. ([DDGKM, Definition V.3.2]) Let u be an object of G(C), and let p, q 1 be integers. An endomorphism ρ of G(C)(u, u) is called (p, q)-periodic if ρ p = ∆ q (u).

In particular, we need ∆ q to be an endomorphism of u, that is φ q (u) = u. Note that a (p, q)-periodic element in G is (np, nq)-periodic for every positive integer n.

Proposition 1.27. ([DDGKM, Corollary VIII.3.31 and Proposition VII.3.34]) Let p, q 1 be positive integers. Let d be the gcd of p and q, with dp ′ = p and dq ′ = q. Any (p, q)-periodic element in G(C) is conjugate to a (p ′ , q ′ )-periodic element lying in C.

This proposition allows us to restrict our attention to (p, q)-periodic elements with p and q coprime. The study of periodic elements in a general Garside context was introduced in [START_REF] Bessis | Garside categories, periodic loops and cyclic sets[END_REF], in which the following notion of divided category is introduced. We follow the exposition given in [DDGKM, Section XIV.1.1].

For m ∈ Z 1 , define

D m (∆) := {(u 0 , . . . , u m-1 ) ∈ S m | u 0 • • • u m-1 = ∆}.
In particular, we require the u i to be composable: the target of u i must by the source of

u i+1 for i ∈ [[0, m -2]].
For every positive integer m, we define an action of the automorphism φ on m-tuples by setting

φ.(u 0 , . . . , u m-1 ) := (u 1 , . . . , u m-1 , φ(u 0 )) ∈ D m (∆).
This not formally an action of the group φ , as φ k = 1 C doesn't necessarily mean that φ k .u = u for all p-tuple u. This is rather an action of the free group Z, which we denote by φ for convenience. Note that this action preserves D m (∆). Let now m, n be two positive coprime integers. We define

D n m (∆) := {u ∈ D m (∆) | φ n .u = u} .
In order to study (p, q)-periodic elements for p and q coprime, we are going to define a categorical presentation using the sets D kq kp (∆) with k ∈ {1, 2, 3} (see Appendix A.1 for reminders on categorical presentations).

The subcategory C φ q introduced in the end of Section 1.1 is useful for giving a more efficient description of the sets D kq kp (∆). Those sets are a priori described by kp parameters lying in C, but since p and q are coprime, we show that they depend only on k parameters lying in C φ q .

Lemma 1.28. Let p, q be coprime positive integers with p > 1. There is a well-defined integer η such that, for all f := (f 0 , . . . , f p-1 ) ∈ C p , we have

φ q .f = f ⇔ f 0 ∈ C φ q and ∀i ∈ [[1, p -1]], f i = φ iη (f 0 ).
In particular, f depends only on f 0 .

Proof. We begin with some arithmetic results. Let q = kp + r be the Euclidean division of q by p (we have r > 0 since p = 1). For n ∈ Z 1 , let nr = a n p + b n be the Euclidean division of nr by p. We have

∀n 1, b n+1 = b n + r if b n < p -r b n + r -p if b n p -r and a n+1 = a n if b n < p -r a n + 1 if b n p -r and b n ∈ [[0, p -1]
] is equal to nr modulo p. Consider now the sequence k nr , defined by

k nr = k if b n < p -r k + 1 if b n p -r = k + a n+1 -a n , ∀n 1.
We set

h nr = n-1 i=0 k ir = nk + a n -a 0 = nk + a n .
We have h nr p = -b n modulo q. Indeed, as pk + r = q, we have

ph nr = p(nk + a n ) = -nr + pa n = -b n modulo q
As p and q are coprime, there is a smallest m ∈ [[1, p -1]] such that b m = 1 (that is, mr = 1 modulo [p]). We have, for all n 1 b n h mr = -ph nr h mr = h nr b m = h nr modulo q

We define η = -h mr , and we have ∀n 0, b n η = -h nr . Let now f := (f 0 , . . . , f p-1 ) be a p-tuple of morphisms in C. For the sake of readability, the index i of f i is seen as an element of Z/pZ, so that f i = f i+p , f r = f q and f nr = f bn by definition. We have φ q .f = (φ k (f r ), . . . , φ k (f p-1 ), φ k+1 (f 0 ), . . . , φ k+1 (f r-1 )).

As p and q are coprime, for every i ∈ [[0, p -1]], there is a unique

n i ∈ [[0, p -1]] such that b n i = i. We obtain φ q .f = f ⇔ ∀i ∈ [[0, p -1]], f i = f n i r = φ kn i r (f (n i +1)r ) = φ kn i r (f i+r ).
By an immediate induction, we get φ q .f = f ⇔ ∀n > 0, f 0 = φ hnr (f nr ).

If φ q .f = f , then we have in particular

f 0 = φ hpr (f 0 ) = φ pk+ap (f 0 ) = φ pk+r (f 0 ) = φ q (f 0 ). Also, since -h nr ≡ b n η modulo q, we deduce that ∀i ∈ [[0, p -1]], f i = f n i r = φ -hn i r (f 0 ) = φ bn i η (f 0 ) = φ iη (f 0 ).
Conversely, if f is such that φ q (f 0 ) and

f i = φ iη (f 0 ) for all i ∈ [[1, p -1]]
, then for n > 0, we have f 0 = φ -bnη (f bn ) = φ hnr (f nr ), which shows that φ q .f = f . Remark 1.29. In the above proof, we saw that η = -mk -a m , where q = pk + r is the Euclidean division of q by p, and m ∈ [[0, p -1]] is such that mr = a m p + 1 is the Euclidean division of mr by p. In particular we see that η depends only on p and q.

The above result is also vacuously true for p = 1. From now on, we fix the integer η associated to p and q by Lemma 1.28 (we fix η = 1 when p = 1).

Proposition 1.30. Let p, q be coprime positive integers. (a) The map (u 0 , . . . , u p-1 ) → u 0 induces a bijection between D q p (∆) and the set u 0 ∈ S φ q | u 0 φ η (u 0 ) . . . φ (p-1)η (u 0 ) = ∆ .

(b) The map (a 0 , b 0 , . . . , a p-1 , b p-1 ) → (a 0 , b 0 ) induces a bijection between D 2q 2p (∆) and the set {(a, b) ∈ (S φ q ) 2 | ab ∈ D q p (∆)}. (c) The map (x 0 , y 0 , z 0 , . . . , x p-1 , y p-1 , z p-1 ) → (x 0 , y 0 , z 0 ) induces a bijection between D 3q 3p (∆) and the set {(x, y, z) ∈ (S φ q ) 3 | xyz ∈ D q p (∆)}. Proof. We first consider the case where p = 1. We have

D q 1 (∆) = {u 0 ∈ S | u 0 = ∆ and φ q (u 0 ) = u 0 }. D 2q 2 (∆) = {(a, b) ∈ S 2 | ab = ∆ and (a, b) = (φ q (a), φ q (b))} = {(a, b) ∈ (S φ q ) 2 | ab = ∆}.
By Lemma 1.10, this set is equal to {(a, b) ∈ (S φ q ) 2 | ab ∈ D q 1 (∆)}. The same reasoning applies to D 3q 3 (∆). We now assume that p > 1. The first claim is a direct consequence of Lemma 1.28 and the fact that u 0 • • • u p-1 = ∆. For the second claim, let (a 0 , . . . , b p-1 ) be a 2p-tuple. This 2p-tuple is φ 2q -invariant if and only if the two p-tuples (a 0 , . . . , a p-1 ) and (b 0 , . . . , b p-1 ) are both φ q -invariant. By Lemma 1.28, this is equivalent to φ q (a 0 ) = a 0 , φ q (b 0 ) = b 0 and (a 0 , b 0 , . . . , a p-1 , b p-1 ) = (a 0 , b 0 , φ η (a 0 ), . . . , φ (p-1)η (b 0 )) which shows that the map (a 0 , b 0 , . . . , a p-1 , b p-1 ) → (a 0 , b 0 ) is injective and maps D 2q 2p (∆) into the considered set. Conversely, for (a, b) in the considered set, the 2p-tuple (a 0 , b 0 , φ η (a 0 ), . . . , φ (p-1)η (b 0 )) lies inside D 2q 2p (∆) and gives the preimage of (a, b). Again, the same reasoning applies to the third claim.

From now on, we will use the alternative descriptions of D q p (∆), D 2q 2p (∆) and D 3q 3p (∆) given by Proposition 1.30.

D q p (∆) ≃ u ∈ S φ q | uφ η (u) • • • φ (p-1)η (u) = ∆ . D 2q 2p (∆) ≃ (a, b) ∈ (S φ q ) 2 | ab ∈ D q p (∆) . D 3q 3p (∆) ≃ (x, y, z) ∈ (S φ q ) 3 | xyz ∈ D q p (∆)
. Under these descriptions, the action of φ is given by

∀u ∈ D q p (∆), φ.u = φ η (u). ∀a, b ∈ D 2q 2p (∆), φ.(a, b) = (b, φ η (a)). ∀x, y, z ∈ D 3q 3p (∆), φ.(x, y, z) = (y, z, φ η (x)
). This description of the action of φ also holds for p = 1 with η = 1.

We are now going to construct an oriented graph S q p , which will serve as generators for our categorical presentation (see Appendix A.1 for definitions regarding oriented graphs).

Definition 1.31. Let p, q be coprime positive integers. We denote by S q p the following oriented graph -The objects are the elements of D q p (∆). -The arrows are the elements of D 2q 2p (∆). -For (a, b) ∈ D 2q 2p (∆), the source (resp. the target) of (a, b) is given by ab (resp. bφ η (a)). We call S q p the graph of simples (for the couple (p, q)). Note that, for (a, b) ∈ S q p , the source and target of (a, b) are objects of S q p . Indeed we have (a, b) ∈ D 2q 2p (∆) and φ.(a, b) = (b, φ η (a)) ∈ D 2q 2p (∆), so both ab and bφ η (a) lie in D q p (∆) by Proposition 1.30.

Lemma 1.32. The action of φ 2 on D 2q 2p (∆) induces an automorphism φ p of the graph S q p , with (φ p ) q = 1 S q p . For an object u of S q p , we have φ p (u) = φ.u.

Proof. By definition, we have φ 2 .(a, b) = (φ η (a), φ η (b)) for (a, b) ∈ D 2q 2p (∆). The source and target of φ 2 .(a, b) are φ η (ab) = φ.(ab) and φ η .(bφ η (a)), respectively. As both φ and φ 2 are automorphisms of C, they induce bijections on the sets D q p (∆) and D 2q 2p (∆), respectively. The fact that (φ p ) q = 1 S q p comes from the definition of the sets D q p (∆) and D 2q 2p (∆). We now endow the graph S q p with a set of relations R q p . The set R q p is in bijection with D 3q 3p (∆). A 3p-tuple (x, y, z) in D 3q 3p (∆) induces the relation (x, yz)(y, zφ η (x)) = (xy, z) between elements of D 2q 2p (∆). Definition 1.33. The presented category C q p := S q p | R q p + is called the category of (p, q)-periodic elements of C. The enveloping groupoid of C q p will be denoted by G q p . Lemma 1.34. The function ℓ defined on arrows of S q p by ℓ(a, b) := ℓ(a) extends to a length functor on C q p , making C q p into a homogeneous category.

Proof. First, the function ℓ extends to C q p as it is compatible with R q p . Indeed, for (x, y, z) ∈ D 3q 3p (∆), we see that ℓ(xy, z) = ℓ(xy) = ℓ(x)+ℓ(y) = ℓ(x, yz)+ℓ(y, zφ η (x)). We then have to show that C q p is generated by elements of positive length. As C q p is generated (by definition) by S q p , we have to show that the elements of length 0 in S q p are trivial in C q p . By definition of ℓ, the only elements of S q p of length 0 are elements of the form (1, u) for u ∈ D q p (∆). We claim that the element (1, u) ∈ S q p is the identity morphism of the object u in C q p . The source and target of (1, u) are both u. Furthermore, for (a, b) ∈ S q p with source u, the element The category C q p is cancellative. Furthermore, defining ∀u ∈ Ob(C q p ), ∆ p (u) := (u, 1) ∈ C q p (u, φ p (u)) induces a Garside map ∆ p on C q p . The simple morphisms associated to ∆ p are exactly the image in C q p of the graph S q p . Lemma 1.36. The automorphism φ p of S q p extends to an automorphism of C q p . Moreover, φ p is the automorphism induced by the Garside map ∆ p .

Proof. As we already know that φ p is an automorphism of S q p , we only have to show that it is compatible with the set of relations. Let (x, y, z)

∈ D 3q 3p (∆), the element (φ η (x), φ η (y), φ η (z)) of D 3q 3p (∆) induces the relation (φ η (x), φ η (y)φ η (z))(φ η (y), φ η (x)φ 2η (z)) = (φ η (x)φ η (y), φ η (z)) ⇔(φ η (x), φ η (yz))(φ η (y), φ η (xφ η (z))) = (φ η (xy), φ η (z)) ⇔φ p (x, yz)φ p (y, zφ η (x)) = φ p (xy, z)
which proves that φ p is compatible with R q p . Now, in order to prove that φ p is indeed the automorphism induced by the Garside map ∆ p , we only need to show that φ p (s) = s for any simple morphism s. Let s = (a, b) ∈ S q p , the element (a, b, 1) in D The map defined on S q p by (a, b) → a induces a functor π p : C q p → C, which is called the collapse functor.

Lemma 1.38. Let u be an object of C q p , and let x := π p (u) be the source of u in C. The collapse functor restricts to an isomorphism of posets

(S q p (u, -), ) ≃ ({s ∈ S φ q | s u}, ) ⊂ S φ q (x, -) ⊂ S(x, -). In particular, if s := (a, b) and s ′ := (a ′ , b ′ ) are two simples in C q p with source u, then s s ′ in C q p if and only if a a ′ in C. Proof.
Let π denote the restriction of the collapse functor π p to the set S q p (u, -). For s := (a, b) ∈ S q p (u, -), we have ab = u by definition, so a u and φ q (a) = a. Conversely, if a u is such that φ q (a) = a, then there is a unique b ∈ S such that ab = u. We have φ q (b) = b by Lemma 1.10, and the simple morphism (a, b) is then the unique preimage by π of a in S q p (u, -). Let now s := (a, b) and s ′ := (a ′ , b ′ ) be two elements of S q p (u, -). If s s ′ , then there is some (x, y, z) ∈ D 3q 3p (∆) such that (x, yz) = (a, b) and (xy, z) = (a ′ , b ′ ). We have in particular ay = xy = a ′ and a a ′ . Conversely, if a a ′ , there is some y ∈ S such that ay = a ′ . Again we have φ q (y) = y by Lemma 1.10, and the relation induced by (a, y, b) then gives s s ′ .

Corollary 1.39. Let s := (a, b) and t := (α, β) be two composable simple morphisms in C q p . The greedy normal form of the path st in C q p is given by st

= (ad, d -1 b)(d -1 α, βφ η (d)))
where d = α ∧ b. In particular the path st is greedy if and only if b and α are coprime.

Proof. The composability of s and t is equivalent to αβ = bφ η (a). By Lemma 1.38, we have t ∧ s = (α, β) ∧ (b, φ η (a)) = (d, e), where e is defined by de = αβ. We have

st = (a, b)(α, β) = (a, b)(d, e)(d -1 α, β) = (ad, d -1 b)(d -1 α, βφ η (d)).
This last term is the greedy normal form of st because of Lemma 1.15.

Theorem 1.40. ([DDGKM, Proposition XIV.1.8]) (1) Let u be an object of G q p . The element π p (∆ q p (u)) is a (p, q)-periodic element in G(C).
(2) Every (p, q)-regular element of G(C) is conjugate to an element of the form π p (∆ q p (u)).

(3) The collapse functor induces a functor G q p → G(C), which in turn induces a group isomorphism between G q p (u, u) and the centralizer of π p (∆ q p (u)) in G(π p (u), π p (u)).

1.3.1. The case p = 1. If p = 1, then the graph S q p is simply the subgraph S φ q of S. Indeed we have

D q 1 (∆) = {u ∈ S φ q | u = ∆} = {∆(u) ∈ S | φ q (∆(u)) = ∆(φ q (u)) = ∆(u)} ≃ {u ∈ Ob(C) | φ q (u) = u}. D q 2 (∆) = {(a, b) ∈ (S φ q ) 2 | ab = ∆} = {(a, a) ∈ (S φ q ) 2 } ≃ S φ q .
Let (x, y, z) be in D 3q 3 (∆). We have z = xy, thus (x, y, z) expresses the relation x.y = (xy) between elements of S φ q .

By [DDGKM, Proposition VI.1.11], the category C q 1 is simply the subcategory C φ q . The Garside map of Theorem 1.35 is then the restriction to C φ q of the Garside map of C. The same goes for the automorphism φ 1 .

The collapse functor π 1 sends a ∈ S φ q to a ∈ S: it is the inclusion C φ q ֒→ C. Lemma 1.38 expresses that, for an object u of C φ q , the inclusion S φ q (u, -) ⊂ S(u, -) is a morphism of lattices, which is also a consequence of Lemma 1.10. Corollary 1.39 ensures that the greedy normal form of a path in C φ q is the same as its greedy normal form in C.

Lastly, Theorem 1.40 expresses that every (1, q)-periodic element of C is conjugate to some ∆ q (u), and that G(C) φ q (u, u) is the centralizer of ∆ q (u) inside of G(C)(u, u).

Complex reflection groups, braid groups and braid categories

2.1. Reminders on complex reflection groups and their braid groups.

2.1.1. Definitions, basic invariants. We follow the exposition of [START_REF] Lehrer | Unitary reflection groups[END_REF]. Let V be a finite dimensional complex vector space. An nontrivial element s ∈ GL(V ) is called a reflection if it pointwise fixes some hyperplane of V and has finite order. The hyperplane Ker(s-Id) is then called the reflecting hyperplane of s. A subgroup W < GL(V ) is called a complex reflection group if it is finite and generated by reflections of V . The integer n = dim V is the rank of W .

A complex reflection group W < GL(V ) is irreducible if there are no nontrivial Wstable subspaces in V . It is well-known that every complex reflection group decomposes as a product of irreducible complex reflection groups. This allows us to restrict our attention to irreducible groups. These groups were classified by Shephard and Todd in [START_REF] Shephard | Finite unitary reflection groups[END_REF], with on the one hand an infinite series of monomial matrices (groups denoted by G(de, e, n) for integers d, e, n), and on the other hand a list of 34 exceptional cases, labeled G 4 to G 37 .

An irreducible complex reflection group of rank n is called well-generated if it can be generated by a set of n reflections. An irreducible group of rank n which is not wellgenerated is badly-generated . Such a group can always be generated by n + 1 reflections.

From now on we fix W < GL(V ) an irreducible complex reflection group of rank n. The action of W on V extends to an action of W on the algebra S of polynomial functions on V . By the Chevalley-Shephard-Todd theorem ([LT09, Theorem 3.20]), the algebra S W of Winvariant elements of S is a polynomial algebra in n variables. A n-tuple f = (f 1 , . . . , f n ) of homogeneous algebraically independent generators of S W is called a system of basic invariants. The degrees d i of the f i doesn't depend on the choice of f . They are the degrees of the reflection group W .

Likewise, one can define the codegrees d * 1 , . . . , d * n of W by considering the module of invariant derivations on the algebra S ([Bes15, Definition 1.2]).

2.1.2. Braid groups, braided reflections. We consider the subset X ⊂ V consisting of points not belonging to any of the reflecting hyperplanes associated to reflections of W . By Steinberg's Theorem ([LT09, Theorem 9.44]), the action of W on X is free and induces a covering map from X to X/W . We fix a basepoint x 0 ∈ X and we set Let f := (f 1 , . . . , f n ) be a system of basic invariants. The isomorphism between S W and C[f 1 , . . . , f n ] induces in turn an algebraic isomorphism V /W ≃ C n . Let H denote the image in V /W of the reunion of the reflecting hyperplanes of W , it is an algebraic hypersurface of V /W ≃ C n . Since the braid group B(W ) is defined as the fundamental group of the complement of H in V /W , it is generated by particular elements called braided reflections (around the irreducible divisors of H).

We quickly recall from [BMR98, Appendix 1] the definition of braided reflections, as it will be useful in Section 3.4. Let H be a reflecting hyperplane of W , and let D ⊂ H be the image of

H in V /W . Definition 2.1. A path from x 0 to D in X/W is a path γ : [0, 1] → V /W such that γ(0) = x 0 , γ(1) ∈ D and γ(t) ∈ X/W for t < 1. Two paths γ, γ ′ from x 0 to D in X/W are D-homotopic if there exist a homotopy T : [0, 1] × [0, 1] → V /W from γ to γ ′ such that • For all t ∈ [0, 1[ and u ∈ [0, 1] T (t, u) ∈ X/W . • For all u ∈ [0, 1], T (0, u) = x 0 and T (1, u) ∈ D. The D-homotopy class of γ is denoted by [γ].
Let γ be a path from x 0 to D in X/W . Let also U be a connected open neighborhood of γ(1) in X/W ∪ D such that U ∩ X/W has a fundamental group free of rank 1. Let u ∈ [0, 1[ be such that γ(t) ∈ U for t u. The orientation of U ∩ X allows us to choose a "positive" generator λ of π 1 (U ∩ X/W, γ(u)).

We set γ u (t) := γ(ut) for t ∈ [0, 1] and ρ γ,λ := γ u * λ * γ -1 u . The homotopy class of ρ γ,λ depends only on [γ] and is denoted by ρ

[γ] . It is by definition a braided reflection of B(W ) (around the divisor D). The image of ρ [γ] inside W is a reflection with hyperplane H. Proposition 2.2. ([BMR98, Appendix 1])
All meridians around D form a conjugacy class of B(W ). In particular the set of all braided reflections is stable under conjugacy in B(W ).

Remark 2.3. Let x 0 → x 1 be a change of basepoint. The induced isomorphism between B(W ) and π 1 (X/W, W.x 1 ) maps braided reflections to braided reflections.

Lastly, we give the definition of a particular element in B(W ), which plays a particular role when considering centers and centralizers.

Definition 2.4. ([BMR98, Notation 2.3])
The full-twist is the homotopy class in B(W ) of the loop

γ π : [0, 1] -→ X/W t -→ exp(2iπt)x 0 . It lies inside P (W ) ∩ Z(B(W )).
2.1.3. Regular elements, regular braids. Let d be a positive integer. We denote by µ d the group of d-th roots of unity in C. We also consider µ * d the subgroup of µ d consisting of primitive d-th roots of unity, and Let g ∈ W be a ζ d -regular element, and let W g := C W (g). It is possible to study the braid group B(W g ) by embedding it inside B(W ). Let V g denote the eigenspace Ker(g -ζ d Id) on which W g acts as a reflection group. Let also X g denote the space of regular vectors inside V g . By [LT09, Theorem 11.33], we have

ζ d := exp( 2iπ d ) ∈ µ * d . Definition 2.5. ([LT09, Definition 11.21]) Let ζ be a root of unity in C. An element g ∈ W is called ζ-regular if it admits a ζ- eigenvector lying in X. An integer d is regular for W if there is a ζ d -regular element g in W . If g ∈ W is a ζ-regular element, then for every integer k, g k is ζ k d -regular. Thus, if g is ζ-regular for some ζ ∈ µ * d ,
X g = X ∩ V g .
The scalar action of C * on V induces in particular an action of µ d on V /W . By [Bes15, Theorem 1.9], the embedding V g → V induces two homeomorphisms V g /W g ≃ (V /W ) µ d and X g /W g ≃ (X/W ) µ d . In particular B(W g ) identifies with the fundamental group of (X/W ) µ d . It was shown in [START_REF] Bessis | Finite complex reflection arrangements are K(π, 1)[END_REF]Theorem 12.4] and [Gar23, Theorem 1.2] that the fundamental group (X/W ) µ d can itself be identified with the centralizer in B(W ) of any d-th root of the full-twist. Such d-th roots of the full-twist will be called d-regular braids.

Example 2.7. A guiding example is the case of the groups G 31 and G 37 . The degrees (resp. codegrees) of G 37 are 2 8 12 14 18 20 24 30 0 6 10 12 16 18 22 28 The integer 4 is regular for G 37 . The centralizer of a i = ζ 4 -regular element in G 37 is isomorphic to G 31 . Its degrees (resp. codegrees) are 8, 12, 20, 24 (resp. 0, 12, 16, 28). Note that 4 is the gcd of the degrees of G 37 which it divides.

Lemma 2.8. Let W < GL n (R) ⊂ GL n (C) be a real reflection group which contains -Id and for which 4 is regular. If g ∈ W is a i-regular element and r is a reflection of W , then r and r g commute.

Proof. First, the element -Id is a -1-regular element in W . Since -Id is central in W , and since all -1-regular elements in W are conjugate to -Id, we get that -Id is in fact the only -1-regular element in W . Now, since g is i-regular, g 2 is i 2 = -1-regular, hence equal to -Id.

We assume that r = r g (otherwise our claim is immediate). We fix -, -a W -invariant scalar product on R n . Let α be a root for r (that is, a generator of Ker(r -Id) ⊥ ). We have

g 2 rg -2 = r ⇒ grg -1 = g -1 rg.
Thus g -1 (α) and g(α) are two roots of the same reflection r g . Let α ∈ R such that g -1 (α) = λg(α). We have α = λg 2 (α) = -λα and λ = -1,thus

g(α), α = α, g -1 (α) = α, -g(α) = -g(α), α .
The two roots α and g(α) are then orthogonal for -, -and r and r g commute. This lemma applies to the case of G 31 seen inside G 37 as the centralizer of a i-regular element. Indeed, G 37 is the complexified version of the exceptional Coxeter group E 8 , which contains -Id.

2.2. Dual braid monoid for well-generated groups. Let W < GL(V ) be a wellgenerated complex reflection group of rank n. In [START_REF] Bessis | The dual braid monoid[END_REF] and [START_REF] Bessis | Finite complex reflection arrangements are K(π, 1)[END_REF], Bessis defines a particular Garside monoid, the dual braid monoid , which admits the braid group B(W ) as its group of fraction. This monoid is the first step from which it is possible to define a Garside category suitable for studying centralizers of regular braids in B(W ).

Here we consider the combinatorial definition of the dual braid monoid as an interval monoid. We will be discussing the topological definition of the dual braid monoid in the next section. We follow [Bes15, Section 8].

Let R be the set of reflections of W . We have R = R -1 , and R generates W by definition. For w ∈ W , one can define ℓ R (w) to be the minimal length of a decomposition of w as a product of reflections. Because the set R is globally invariant under conjugation, the value of ℓ R (w) depends only on the conjugacy class of w in W . The function ℓ R induces relations and on W defined by

∀w, v ∈ W, v w ⇔ ℓ R (v) + ℓ R (v -1 w) = ℓ R (w), ∀w, v ∈ W, w v ⇔ ℓ R (wv -1 ) + ℓ R (v) = ℓ R (w).
Let v, w be in W . Since ℓ R takes constant values on conjugacy classes in W , we get that v w ⇔ w v. Thus we will always work with and only consider for readability purposes.

As W is well-generated, we know that the highest degree h of W is regular for W (see the proof of [Bes15, Theorem 2.4]). A Coxeter element of W is then an element c ∈ W which is regular for the eigenvalue

ζ h := exp( 2iπ h ). Let c be a Coxeter element in W . We define R c := {r ∈ R | r c} and I c := [1, c] = {w ∈ W | w c}
We also consider formal copies R ⊂ I of R c ⊂ I c (their elements will be denoted in bold font). The interval monoid associated to R and c is then defined by the presentation:

M (c) := I | st = u ⇔ (st = u and ℓ R (s) + ℓ R (t) = ℓ R (u))
We know that (I c , ) is a lattice ([Bes15, Lemma 8.6]). We will denote s ∧ t (resp. s ∨ t) the gcd (resp. lcm) of s and t in I c . We do not have to distinguish between left-and rightgcd and lcm since all simple elements are balanced (their left-and right-divisor are the same).

Let c and c ′ be two Coxeter elements of W . Since they are regular elements of W for the same eigenvalue ζ h , there is some w ∈ W such that wcw -1 = c ′ . Since ℓ R is invariant under conjugacy, one readily sees that conjugation by w induces an isomorphism between the two intervals I c and I c ′ . In particular we have M (c) ≃ M (c ′ ). The Hurwitz relations with respect to W and c are the formal relations of the form

rr ′ = r ′ r ′′
where r, r ′ , r ′′ ∈ R c are such that r = r ′ , rr ′ ∈ I c and rr ′ = r ′ r ′′ holds in W . We know from [Bes15, Lemma 8.8] that the monoid M (c) admits a particular presentation with R as a set of generators, endowed with the Hurwitz relations.

Remark 2.10. We can be a little less specific and say that M (c) (and G(c)) is presented by relations of the form ss ′ = rr ′ with s, s ′ , r, r ′ ∈ R, and ss ′ ∈ I c . That is M (c) is presented by its atoms and the equality between decompositions of simple elements of length 2. It is this precise rephrasing which we will show holds in Springer categories (see Theorem 3.27).

Lemma 2.11. For a ∈ I, we have a 2 ∈ I if and only if a = 1.

Proof. If a 2 ∈ I, the relation aa = a 2 must be a defining relation of M , so we must have 2ℓ R (a) = ℓ R (a 2 ). If a = 1, then let s ∈ R c be a divisor of a with sa ′ = a and a ′′ s = a. We have

a 2 = a ′′ ssa = a ′′ s 2 a, so ℓ R (a 2 ) ℓ R (a ′′ ) + ℓ R (a ′ ) + 1 = 2ℓ R (a) -1, which contradicts 2ℓ R (a) = ℓ R (a 2 ).
Proposition 2.12. Let a, a ′ be in I. If the product aa ′ is also in

I, then aa ′ = a ∨ a ′ and a ∧ a ′ is trivial. Proof. First, note that in I c , we have aa ′ = a ′ a a ′ = a a ′ a. Now, because ℓ R is invariant under conjugation, we have ℓ R (aa ′ ) = ℓ R (a) + ℓ R (a ′ ) = ℓ R (a ′ ) + ℓ R a a ′ = ℓ R a a ′ + ℓ R (a)
This means that aa ′ a a ′ and a a ′ aa ′ : we have a a ′ , a a ′ ∈ I c , and aa ′ is a common left and right multiple of a and a ′ . Now let ab = a ′ b ′ be the right-lcm of a and a ′ . By definition, there is some x ∈ I such that bx = a ′ and b ′ x = a a ′ . In particular, we have a ′ x, a a ′ x and x a a ′ since every element is balanced. So aa ′ admits x 2 as a factor and x 2 is a simple. By Lemma 2.11,

x = 1, a ′ = b, b ′ = a a ′ and aa ′ = a ∨ a ′ .
If x is a common divisor of a and a ′ , then we have a x and x a ′ . We get that x 2 is a simple element in M (c), it must then be trivial.

Remark 2.13. By definition of an interval monoid, there is a natural bijection between the simples of M (c) and the interval I c = ([1, c], R ) in W . This bijection induces in turn a bijection between the sets D m (∆) and the sets D m (c), defined by 2.3. Lyashko-Looijenga map, cyclic labels. Here we recall the preliminaries which are necessary to construct both the dual braid monoid from a topological point of view, and the Springer category associated to a regular number. Most of this section comes from [Bes15, Section 5, 6]. Let again W < GL(V ) be an irreducible well-generated complex reflection group with highest degree h, together with a system of basic invariants f such that the discriminant has the following form:

D m (c) := (u 0 , . . . , u p-1 ) ∈ W p u 0 • • • u p-1 = c ℓ R (u 0 ) + • • • + ℓ R (u p-1 ) = ℓ R (c) .
∆ f = X n n + α 2 X n-2 n + • • • + α n where α i ∈ C[X 1 , • • • , X n-1
] (such a system always exists for well-generated groups by [Bes15, Theorem 2.4 and Section 2.5]).

The system of basic invariants f provides an isomorphism

V /W ≃ C n , which sends the orbit W.v of v ∈ V to (f 1 (v), • • • , f n (v)) ∈ C n . We denote v i := f i (v). Definition 2.14. ([Bes15, Definition 7.23]) Let x = W.v ∈ V /W . The multiset LL(x) is defined as the solutions in T of the polynomial ∆ f (v 1 , • • • , v n-1 , T + v n ) ∈ C[T ] It is by definition an element of E n := C n /S n . The map LL : V /W → E n is called the (extended) Lyashko-Looijenga morphism.
Remark 2.15. The original approach in [START_REF] Bessis | Finite complex reflection arrangements are K(π, 1)[END_REF] was to study a slightly different application, denoted by LL (cf [Bes15, Definition 5.1]). Bessis then suggested that the application LL was in fact a far better choice.

For x ∈ V /W , we have x ∈ X/W ⇔ 0 / ∈ LL(x). By definition this is equivalent to LL(x) ∈ E • n := (C * ) n /S n .
We endow V /W with the quotient of the scalar action of C on V . We also endow the space of configurations E n with the scalar action of C. The fat basepoint for X/W is then defined as the following set

U := {x ∈ V /W | LL(x) ∩ iR 0 = ∅}.
That is, x ∈ U if no point of LL(x) is directly above 0. This subset of X/W is open and contractible ([Bes15, Lemma 6.3]). It can be used as a fat basepoint for defining the braid group of W ([Bes15, Definition 6.4]). From now on we consider B(W ) := π 1 (X/W, U ).

Let x ∈ X/W , the points of LL(x) are ordered clockwise starting from "right after noon" (that is, points lying on the half-line iR 0 will be at the end). Points with the same argument are ordered by increasing module. If x ∈ V /W , the sequence (x 1 , . . . , x k ) of points of LL(x) ordered in this way is called the cyclic support of x ([Bes15, Definition 11.8]).

Remark 2.17. The set of points x ∈ U such that the points in the cyclic support of x all have distinct arguments is dense in X/W . The ordering we choose allows us to define univalent desingularization, which preserves the ordering of the points of the cyclic support.

= (x, L) ∈ U × [0, 2π h ].
The path γ T associated with a semitunnel T is the path

γ T : [0, 1] -→ X/W t -→ e itL x
We say that T is a circular tunnel if it satisfies the additional condition γ T (1) ∈ U .

Let T := (x, L) be a circular (semi)tunnel. Since LL is homogeneous of degree h, the path LL • γ T corresponds to a continuous rotation of angle hL. Since the path γ T starts and ends in U , it induces a well defined element in B(W ) = π 1 (X/W, U ). From now on we amalgamate a circular tunnel T with the path γ T it induces in X/W . From now on we fix the Coxeter element c so that we have the isomorphism G(c) ≃ B(W ) induced by the above theorem. By [Bes15, Lemma 6.13], the element ∆ h in B(W ) represents the full-twist.

Remark 2.20. In [START_REF] Bessis | Finite complex reflection arrangements are K(π, 1)[END_REF], the notion of circular tunnel is preceded by the notion of tunnel (cf [Bes15, Definition 6.6]). However, [Bes15, Lemma 11.10] and [Bes15, Corollary 6.18] show that the two notion are actually synonymous, in the sense that an element of B(W ) is represented by a circular tunnel if and only if it is represented by a tunnel.

The notion of circular tunnel allows us to define the notion of cyclic label, which will prove crucial in the study of (X/W ) µ d . Let x ∈ U , and let (x 1 , . . . , x k ) be the cyclic support of x. We assume that all the x i have distinct arguments. There is a minimal θ ∈ R >0 such that e iθ x / ∈ U . The head of x is defined as the element c 1 of B(W ) represented by the circular tunnel T = (x, θ + ε), which doesn't depend on ε > 0 small enough. We can then consider the head of γ T (1) = e i(θ+ε)t x and so on, until all the points of the cyclic support of x have been labeled. Definition 2.21. ([Bes15, Definition 11.9]) Let x ∈ U be such that all the points in LL(x) have distinct arguments. The sequence (s 1 , • • • , s k ) defined above is the cyclic label of x, denoted by clbl(x). If different points in the cyclic support of x have the same argument, we define clbl(x) as the cyclic label of some desingularization of x, as in Remark 2.17.

Since the elements of clbl(x) are defined as circular tunnels, they are simple elements in M (c). Thus we can see them as elements of I c ⊂ W . Theorem 2.19 proves that, for any x ∈ X/W , the product of all the terms of clbl(x) is equal to ∆ in B(W ), as it is represented by the circular tunnel (x, 2π h ). Thus, if the cyclic support of x contains k points, then clbl(x) lies inside D k (c).

Theorem 2.22. ([Bes15, Proposition 11.13]) A pair (x, (a 1 , • • • , a k )) ∈ E • n × D k (c
) is compatible if the cyclic support of x contains k points, and their respective multiplicities coincide with (ℓ R (a 1 ),

• • • , ℓ R (a k )).
The map (LL, clbl) induces a bijection between X/W and the set E • n ⊡ D(c) of compatible pairs. This bijection induces a topology on the latter.

In particular, considering some x ∈ X/W , a path starting from LL(x) in E • n induces a path inside X/W , provided that we know how moving the points of LL(x) impacts clbl(x).

Lemma 2.23. (Hurwitz Rule) Let x ∈ X/W , and let T := (x, L) be a circular tunnel. Let also γ : [0, 1] → X/W be a path starting at x and such that, for all t ∈ [0, 1], (γ(t), L) is a circular tunnel. For all t ∈ [0, 1], the elements in B(W ) represented by (γ(t), L) and T are equal.

Proof. Let t ∈ [0, 1]. Defining H(r, s) := e isL γ(rt) yields a homotopy between the paths associated to T and to the circular tunnel (γ(t), L). This easy result is actually quite useful. For instance it implies directly that altering the module of one (or several) points of the cyclic support of some x ∈ X/W does not affect the cyclic label.

Let now x be in X/W and let (x 1 , • • • , x k ) be its cyclic support. Suppose that we swap two consecutive points x i and x i+1 of LL(x), there are two natural ways to do so, one going "farther" than the other: Let γ + i and γ - i denote the first path and the second path, respectively. Lemma 2.24. Let x ∈ X/W , with cyclic support (x 1 , . . . , x k ) and cyclic label (s 1 , . . . , s k ). The cyclic labels of γ + i (1) and γ - i (1) are given by clbl(γ + i (1)) = (s 1 , . . . , s i-1 , s i+1 , s

s i+1 i , s i+2 , . . . , s k ) clbl(γ - i (1)) = (s 1 , . . . , s i-1 , s i s i+1 , s i , s i+2 , . . . , s k ).
Proof. The concatenation of γ - i and γ + i gives a homotopically trivial path from x to itself. In particular the assertion about clbl(γ - i (1)) follows from that on clbl(γ + i (1)). By construction of the cyclic label, we can assume that all the arguments of the points in LL(x) are distincts. Let then θ be some angle strictly between the arguments of x i-1 and x i . By [START_REF] Bessis | Finite complex reflection arrangements are K(π, 1)[END_REF]Lemma 11.1], one can replace x by e iθ/h x and consider x ′ := γ + 1 (x). Now if θ ′ denotes an angle strictly included between the arguments of x 2 and x 3 , then the circular tunnels (x, θ ′ ) and (x ′ , θ ′ ) represent the same element in B(W ) by the Hurwitz rule. That is the product of the first two terms of clbl(x) and clbl(x ′ ) are equal. The Hurwitz rule also shows that the terms of clbl(x) and clbl(x ′ ) are equal for i > 2.

Lastly, the assertion that the first two terms of clbl(x ′ ) are (s 2 , s s 2 1 ) follows from [Bes15, Lemma 11.11], using the notion of reduced label ([Bes15, Definition 7.14]) and a modified version of the Hurwitz rule ([Bes15, Lemma 6.15]).

By induction, we see that in general, moving a point of the cyclic support of some x ∈ X/W may only affect the terms of the cyclic label corresponding to points of the support with lower module.

2.4. Regular numbers and topological groupoids. We can now move on to the definition of the Springer category associated to a regular number. We keep the definitions and notation of the last section. Let d be a regular number for W , let g ∈ W be a ζ d -regular element. Let also W g := C W (g) be the centralizer in W of g. We set

p := d d ∧ h , q := h d ∧ h
where d ∧ h is the gcd of d and h. The categories we are going to consider are defined as fundamental groupoids with fat basepoints (cf. [Bes15, Appendix A]) having several contractible connected components. Define

D := ζ∈µp ζiR 0 , U p := {x ∈ X/W | LL(x) ∩ D = ∅}, U µ d := {x ∈ (X/W ) µ d | LL(x) ∩ D = ∅}.
The set D is composed of p half lines starting at 0. It cuts the plane in p sectors P 1 , . . . P p , labeled clockwise starting from the vertical halfline iR + ⊂ D. In the following example we have p = 3:

D P 1 P 2 P 3 Lemma 2.25. ([Bes15, Lemma 11.22]) Let x ∈ U p with cyclic label (c 1 , • • • , c k ). For i ∈ [[1, d ′ ]]
, define u i as the product of the c j corresponding to points inside the sector P i . The cyclic content of x is defined as cc p (x) := (u 1 , . . . , u p ). The map cc p induces a bijection between the connected components of U p (resp. U µp ) and D p (c) (resp D q p (c)). Furthermore the connected components of U p (resp. U µ d ) are contractible.

Since the connected components of U p and U

B q p (W ) = π 1 ((X/W ) µ d , U µ d )
The functoriality of π 1 gives a natural functor B q p (W ) → B(W ). Note that B q p (W ) is equivalent to the braid group of W g (in particular, it is a connected groupoid). We now establish an isomorphism between the topological groupoid B q p (W ) and the category M (c) q p . Let s := (a, b) ∈ D 2q 2p (c) be a morphism in M (c) q p . By Theorem 2.22, there is a unique element x s ∈ U µ d such that clbl(x) = s and LL(x s ) consists of the points e iπ( 1 2 -2j+1 2p ) such that the j-th term of s is nontrivial. The circular tunnel (x s , π d ′ h ) then defines an element of B q p (W ) which we denote by b s . Theorem 2.27. ([Bes15, Theorem 11.28]) The map s → b s extends to a groupoid isomorphism G(M (c) q p ) → B q p (W ). We have the following diagram of functors

M (c) q p _ πp / / M (c) _ G(M (c) q p ) ≃ / / G(c) ≃ B q p (W ) / / B(W )
As ∆ h ∈ M (c) represents the full-twist in B(W ), a d-th root of the full-twist is a (d, h)periodic element of M (c). Such an element is conjugate to a (p, q)-periodic element of M (c) by Proposition 1.27. Let u be an object of B q p (W ), that is an object of M (c) q p or an element of D q p (c). By Theorem 1.40, the morphism B q p (W ) → B(W ) sends the automorphism group B q p (W )(u, u) to the centralizer in B(W ) of some (p, q)-periodic element of M (c), in particular a d-th root of.

Lemma 2.28. Let u be an object of B q p (W ). The group isomorphisms

B(W g ) ≃ B q p (W )(u, u) ≃ G(M (c) q p )(u, u) maps the full-twist in B(W g ) to ∆ ph (u).
Proof. The morphism ∆ p (u) in M (c) q p corresponds to a rotation of angle 2π ph , while the full-twist corresponds to a rotation of angle 2π.

Properties of Springer categories

Theorem 2.27 gives a way to study the braid group of the centralizer of a regular element in a well-generated complex reflection group using Garside theory. This section is dedicated to the study of the categories of periodic elements associated to dual braid monoids, and to the results we deduce for the associated braid groups.

Let W < GL(V ) be an irreducible well-generated complex reflection group of rank n, with highest degree h. Let d be a regular number for W . We set again

p := d d ∧ h , q := h d ∧ h
We choose some Coxeter element c in W and we denote by C the category M (c) q p of (p, q)-periodic elements of the dual braid monoid M (c). The enveloping groupoid B of C is equivalent to the braid group of the centralizer of some ζ d -regular element in W by Theorem 2.27.

Note again that the full-twist of B(W ) is equal to ∆ h , where h is the highest degree of W . The category of (1, h)-periodic elements is then simply the monoid M (c) itself. Thus the content of this sections also applies to the dual braid monoid of a well-generated irreducible complex reflection group.

We denote by I q c the set of elements of I c which are invariant under φ q (that is, invariant under conjugation by c q ). 3.1. Elementary properties. Recall from Definitions 1.31 and 1.33 that C depends on the sets D q p (∆), D 2q 2p (∆) and D 3q 3p (∆). Thanks to Remark 2.13 and Proposition 1.30, we can replace these by

Ob(C) := D q p (c) ≃ u ∈ I q c pℓ R (u) = ℓ R (c) = n u(u c η ) • • • (u c (p-1)η ) = c S q p := D 2q 2p (c) = (a, b) ∈ (I q c ) 2 | ab ∈ Ob(C) R q p := D 3q 3p (c) = (x, y, z) ∈ (I q c ) 3 | xyz ∈ Ob(C)
where η is the integer associated to (p, q) introduced in Lemma 1.28 and Remark 1.29.

Under these definitions, a simple morphism (a, b) in the graph of simples S q p goes from ab to ba c η in C, and an element (x, y, z) of D 3q 3p (c) induces the relation (x, yz)(y, zx c η ) = (xy, z) in R q p . The following lemma is an obvious consequence of the definition of Ob(C). Lemma 3.1. Let u ∈ Ob(C), the morphism ∆ p (u) = (u, 1) has length ℓ R (u) = n p . This length doesn't depend on u and every simple morphism in C has length at most n p . Let u ∈ D q p (c) be an object of C. The collapse functor π p : C → M (c) sends ∆ q p (u) to some (p, q)-periodic element in M (c). We know that ∆ p (u) = (u, 1) and ∆ p (φ n p (u)) = (u c ηn , 1). We then have

π p (∆ p p (u)) = π p (u, 1)(u c η , 1) • • • (u c (p-1)η , 1) = uu c η • • • u c (p-1)η = c.
If q := pk + r is the Euclidean division of q by p, then we get

π p (∆ q (u)) = c k u c pkη • • • u c (pk+r-1)η .
We know from Theorem 1.40 that this is a (p, q)-regular element in M (c). This gives an explicit formula for roots of the full-twist in B(W ), provided that one knows how to compute elements of the sets D n m (c).

Lemma 3.2. The atoms of C are exactly the simples s = (a, b) ∈ S q p (c) such that a admits no proper divisors in I q c . In particular if φ q is trivial, a simple morphism of C is an atom if and only if it has length 1.

Proof. The first statement follows directly from Lemma 1.38. If φ q is trivial (that is, if ∆ q is central in M (c)), then Lemma 1.38 gives an isomorphism between S q p (u, -) and [1, u] where u is the source of s. Thus s is an atom if and only if a lies in R c . Since by definition ℓ(s) = ℓ(a), we get the desired result.

Proposition 3.3. If p > 1, then there are no pairs of parallel simples in C that is, a simple morphism is uniquely determined by its source and target. In particular, for u ∈ Ob(C), we have S q p (u, u) = {1 u } and S q p (u, φ p (u)) = {∆ p (u)}. Proof. Let s := (a, b) be a simple morphism in C, and let u, v denote the source and target of s, respectively. We have u = ab and v = ba c η . We show that b is the left-gcd of u and v in I c . The result is obvious when a = 1, so we assume a = 1. We have ab = ba b , so b is an obvious left-divisor of ab and ba c η . If b is not the gcd of ab and ba c η , then there is a nontrivial common divisor d of a and a c η . Since p > 1, we have ∆ = aba c η b c η • • • (ab) c (p-1)η and aba c η is simple. Since all elements of I c are balanced, we obtain that d 2 divides aba c η , thus d 2 ∈ I c , which contradicts Lemma 2.11. Now, let (a ′ , b ′ ) ∈ S q p have the same source and target as (a, b). We have

u = ab = a ′ b ′ and v = ba c η = b ′ a ′c η . We have b b ′ and b ′ b, so b = b ′ and (a, b) = (a ′ , b ′ ).
In the case p = 1. There is only one object in C, which corresponds to ∆ ∈ D 1 (∆). The proposition is then false in this case.

Lemma 3.4. (Lifting words expressing simples) Let s := (a, b) be a simple morphism in C, and let a 1 • • • a r be a word in I q c expressing a in M (c). There is a unique path

s 1 • • • s r in S q p expressing s in C and such that π p (s i ) = a i for all i ∈ [[1, r]]
Proof. Let u := ab be the source of s. We proceed by -induction on s. If s is an atom, then a is an atom of I q c by Lemma 3.2. The only word in I q c expressing a is then a itself and the result is trivial. Now for the general case, we have a 1 x = a with x = a 2 • • • a r . By Lemma 1.38, s 1 := (a 1 , xb) is the only atom in C with source u and such that π p (s 1 ) = a 1 . By induction hypothesis, there is a unique path

s 2 • • • s r expressing (x, ba c η 1 ) in C and such that π p (s i ) = a i for i ∈ [[2, r]]. The path s 1 s 2 • • • s r is then the unique path expressing s and such that π p (s i ) = a i for i ∈ [[1, n]].
Let s := (a, b) ∈ S q p be a simple morphism in C. Its source is ab = bφ η (a bc -η ) and its target is bφ η (a) = a c η b -1 b. By Lemma 1.38 and Lemma 1.10, we deduce the existence of the following simple morphisms in C In particular a and b are coprime by Proposition 2.12. We get that the path s ♭ s is greedy by Corollary 1.39. The path ss # = (s # ) ♭ s # is also greedy by the same argument. (d) We have

(φ p (s)) ♭ = (a c η , b c η ) ♭ = (a c η b c η c -η , b c η ) = (a b , b c η ) = φ p (s ♭ ) and (φ p (s)) # = (φ p (s #♭ )) # = ((φ p (s # )) ♭ ) # = φ p (s # )
The transformation s → s # is a bijection of the finite set D 2q 2p (c): it has finite order. As the source of s # is the target of s, there is then a smallest integer n such that s (n#) = s. Definition 3.6. Let s : u → v be a simple morphism in C. The simple loop (of the object u) associated to s is the morphism

λ(s) := ss # s ## • • • s ♭♭ s ♭ ∈ C(u, u)
If s is an atom of C, then we say that λ(s) is an atomic loop. Lemma 3.7. Let s be a simple morphism in C. The simple loop λ(s) is rigid, in particular it lies in its own supper-summit set.

Proof. By Lemma 3.5 (c), the greedy normal form of λ(s) is given by

λ(s) := ss # s ## • • • s ♭♭ s ♭ .
Since s ♭ s is greedy, we get that λ(s) is rigid. By Lemma 1.22, we have cyc(λ(s)) = λ(s # ) and dec(λ(s)) = λ(s ♭ ). In particular we see that cycling and decycling a simple loop doesn't change its inf or its sup. By Proposition 1.24, we deduce that λ(s) lie in its own supper-summit set.

Remark 3.8. Let u be an object of M (c) p . We have

∆ p (u) # = (u, 1) # = (u c η , 1) = φ p (∆ p (u)) = ∆ p (φ p (u)) So λ(∆ p (u)) = ∆ k p (u)
where k is the smallest integer such that φ k p (u) = u. By definition of C, we have that k divides q: there is some k ′ such that kk ′ = q. The element (∆ p ) q (u) is then equal to λ(∆ p (u)) k ′ . By Theorem 1.40, the group B(u, u) then identifies with the centralizer in G(c) of π p (λ(∆ p (u)) k ′ ).

3.2. The case p = 1. Exceptionally in this Section, we denote by I c (W ) the interval associated to c in W for the reflection length. Let q be a positive integer. A (1, q)-periodic element in the dual braid monoid M (c) is simply given by ∆ q . The associated category of (1, q)-periodic elements is then the submonoid M (c) φ q of M (c). Its set of simple elements corresponds to the set I c (W ) c q of simple elements which commute with c q in W . We have by Theorem 2.27 that M (c) φ q is a Garside monoid for the braid group of the centralizer W c q := C W (c q ). Lemma 3.9. Let W be a well-generated irreducible group, and let q be a positive integer. The group W c q = C W (c q ) (acting on the eigenspace V c q = Ker(c q -ζ q h )) is well-generated. Proof. By [Bes15, Theorem 2.4], we known that an irreducible complex reflection group is well-generated if and only if the sum of its i-th degree (in increasing order) with its i-th codegree (in decreasing order) is constant and equal to the highest degree. Let n be the rank of W , we denote the degrees (resp. codegrees) of W by d 1 , .

i j + d * i j = h = d i k and W c q is well-generated.
We have c ∈ W c q and, as Ker(c -ζ h ) ⊂ Ker(c q -ζ q h ) = V c q , it is a Coxeter element of W c q . Because of this lemma, we can consider the interval I c (W c q ) associated to c in W c q for the reflection length in W c q . This section is devoted to the proof of the following theorem.

Theorem 3.10. Let W be a well-generated irreducible group with a Coxeter element c, and let q be a positive integer. The posets I c (W ) c q = I c (W c q ) are equal. That is, we have I c (W ) c q = I c (W c q ) as subsets of W c q and, for s, t ∈ I c (W c q ), we have s t in W if and only if s t in W c q , for the respective reflection length of W and W c q . Corollary 3.11. Let W be a well-generated irreducible group with a Coxeter element c, and let q be a positive integer. The monoid M (c) φ q is isomorphic to the dual braid monoid M ′ (c) associated to W c q .

The corollary comes from the fact that the defining presentation of an interval monoid only depends on the poset structure of the associated interval (I, ): the defining relations are all relations of the form s(s -1 t) = t for s, t ∈ I with s t.

The proof of Theorem 3.10 is ultimately going to rely on a case by case analysis, but we can do some easy reductions. Lemma 3.12. We keep the notation of Theorem 3.10. If Theorem 3.10 holds when q divides the highest degree of W , then it holds for all values of q.

Proof. Let h be the highest degree of W . Since h is the order of c, we have that the order d of c q divides h. we set q ′ := h/d. The two elements c q and c q ′ both have order d in the cyclic group c of order h. There are two integers a and b with c aq = c q ′ and c q ′ b = c q . We then have

W c q = W c q ′ , I c (W ) c q = I c (W ) c q ′
and M (c) φ q = M (c) φ q ′ . Thus Theorem 3.10 holds for q if and only if it holds for q ′ , which divides h.

From now on we assume that the integer q divides h, and we set d = h/q. The element c q is ζ d = ζ q h -regular. If d divides all the degrees of W , then c q is central in W and we have W c q = W , Theorem 3.10 is obvious in this case. If h is the only degree of W divisible by d, then W c q is a complex reflection group of rank 1: it is cyclic and equal to c . We have I c (W c q ) = {Id, c}. On the other hand, any s ∈ I q c lies in W c q = c . We then have I q c = {Id, c} by [Bes15, Lemma 12.2]. The poset structure is induced by Id c in both cases and Theorem 3.10 holds.

Note that these two extreme cases are sufficient to prove Theorem 3.10 when W has rank 2. We now distinguish whether or not W belongs in the infinite series.

The case where W is a well-generated irreducible exceptional group is handled by computer.

If W is a well-generated irreducible group of rank n 3 belonging to the infinite series then we have either -W ≃ S n+1 acting the the hyperplane H in C n+1 given by the equation

x 1 +• • •+x n+1 = 0.
This is not the group G(1, 1, n + 1), as the latter acts on a space of dimension n + 1 and is not irreducible. However, the sets of reflections of these two groups are equal (they are the transpositions in S n+1 ), and their sets of Coxeter elements are the same (they are the n + 1-cycles in S n+1 ). Thus the interval monoids given by W and G(1, 1, n + 1) are the same and we can restrict our attention to the latter group. -W ≃ G(m, 1, n) for m 2.

-W ≃ G(e, e, n) for e 2 (this group is always irreducible when n 3).

3.2.1. W = G(1, 1, n) for n 4. Let ℓ W denote the reflection length in W . Our approach is largely modeled on that of [BW02, Section 3 and Section 4], which covers the case where n is even and q = n 2 .

Lemma 3.13. Let w = c 1 • • • c k be a product of disjoint cycles in W . We have

ℓ W (w) = k i=1 ℓ W (c i ) = k i=1 (ℓ(c i ) -1)
where ℓ(c) denotes the size of the support of c.

Proof. The group W is a complexified real reflection group by [DDGKM, Lemma IX.2.19].

The reflection length ℓ W (w) is then given by the codimension of the fixed space of w acting on C n . Since the c i all have disjoint support, the first equality is immediate. For the second equality, let c = (i 1 • • • i k ) be a k cycle in W . The fixed space of c acting on C n is given by {(x 1 , . . . ,

x n ) ∈ C n | x i 1 = x i 2 = • • • = x i k } and has codimension k -1.
The highest degree of W is n. Let q be a divisor of n, with dq = n. We label the canonical basis of C n in the following way {e(0, 1), e(0, 2), . . . , e(0, q), e(1, 1), . . . , e(d -1, q)} A guiding idea is to think of e(j, i) as ζ j d e i in the vector space C q (with canonical basis e 1 , . . . , e q ). We consider the following element of W c(1, 1, n) := (e(0, 1) e(0, 2)

• • • e(0, q) e(1, 1) • • • e(d -1, q))
It is a Coxeter element of W as the vector

q i=1 d-1 j=0 ζ n-i-qj n e(j, i) is a ζ n -eigenvector for c(1, 1, n) which is regular. We have c(1, 1, n) q = (e(0, 1) e(1, 1) • • • e(d -1, 1))(e(0, 2) • • • e(d -1, 2)) • • • (e(0, q) • • • e(d -1, q))
Lemma 3.14. The eigenspace V := Ker(c(1, 1, n) q -ζ d Id) has dimension q and admits a basis v := {v 1 , • • • , v q }, where

v i := d-1 j=0 ζ d-j d e(j, i).
The isomorphism of vector spaces V ≃ C q given by the basis v induces an isomorphism between W c(1,1,n) q and G(d, 1, q).

Proof. The degrees of W c(1,1,n) q acting on V are the degrees of G(1, 1, n) which are divided by d, that is d, 2d, . . . , dq. In particular, W c(1,1,n) q has rank q and V has dimension q. As the v i are clearly linearly independent, v is a basis for V . The groups W c(1,1,n) q and G(d, 1, q) share the same degrees (cf. [BMR98, Table 2]), and the same cardinality (since the order of a complex reflection group is the product of its degrees). We then only have to show that the image of W c(1,1,n) q in GL q (C) contains a generating set of G(d, 1, q), like the one given in [LT09, Section 2.7]. This is a direct check: -The cycle (e(0, 1) e(1, 1) • • • e(d -1, 1)) acts trivially on v j for j = 1 and sends

v 1 to ζ d v 1 . -For i ∈ [[1, q -1]], the permutation (e(0, i) e(0, i + 1)) • • • (e(d -1, i) e(d -1, i + 1)) swaps v i and v i+1 .
From now on, we set W ′ = W c(1,1,n) q and ℓ W ′ the reflection length for elements of W ′ regarding the reflections of W ′ acting on V . The action of c(1, 1, n) on V is given (in the basis v) by the matrix

c(d, 1, q) :=      0 ζ d 1 . . . . . . 0 1 0     
which is indeed a Coxeter element for the group G(d, 1, q) (see Section 3.2.2). The group G(d, 1, q) is endowed with a character χ, sending a monomial matrix to the product of its nonzero entries. The isomorphism W ′ ≃ G(d, 1, q) of Lemma 3.14 allows us to define χ on W ′ . Let c := (e(j 1 , i 1 ) • • • e(j k , i k )) ∈ W be a cycle. We set

c (1) := c c(1,1,n) q = (e(j 1 -1, i 1 ) • • • e(j k -1, i k )).
Proposition 3.15. An element σ ∈ W lies in W ′ if and only if can be written as decomposition as a product of disjoint cycles of the form

σ = c 1 c (1) 1 • • • c (d-1) 1 c 2 c (1) 2 • • • c (d-1) 2 • • • c a c (1) a • • • c (d-1) a γ 1 • • • γ b where γ (1) i = γ i for i ∈ [[1, b]].
Proof. Our proof is an adaptation of the proof of [BW02, Proposition 3.1], which deals with the case d = 2. First, it is clear that elements of the given form lie in

W ′ . Conversely, let σ = c 1 • • • c r be a product of disjoint cycles in S n . We have that c(1, 1, n) q centralizes σ if and only if c 1 • • • c r = c (1) 1 • • • c (1)
r . By uniqueness (up to reordering) of cycle decompositions in S n , for each i either c i = c

(1) j for some j = i or else c i = c

(1)

i . An immediate induction then gives that σ has the required decomposition.

We note in particular that the cycles c i • • • c , c (d-1) are all disjoint. By assumption, i 1 , . . . , i k are all distinct. One readily sees that c acts on V by

c.v im = ζ j m+1 -jm d v i m+1 ∀m ∈ [[1, k -1]], c.v i k = ζ j 1 -j k d v i 1 .
In particular, we have

χ( c) = ζ j 2 -j 1 d • • • ζ j k -j k-1 d ζ j 1 -j k d = 1
. The fixed space of c acting on V is generated by all the v i with i / ∈ {i 1 , . . . , i k } and

v i 1 + • • • + v i k . Thus ℓ W ′ ( c) = k -1 = ℓ(c) -1 by Lemma 3.14 and [Shi07, Theorem 2.1].
Let γ be a balanced cycle. It can be written as

γ = (e(0, i 1 ) e(j 2 , i 2 ) • • • e(j k , i k ) e(1, i 1 ) • • • e(j k + d -1, i k )).
The action of γ on V is then given by

     γ.v i 1 = ζ j 2 d v i 2 , γ.v im = ζ j m+1 -jm d v i m+1 ∀m ∈ [[2, k -1]], γ.v i k = ζ 1-j k d v 1 .
In particular we have

χ(γ) = ζ j 2 d • • • ζ j k -j k-1 d ζ 1-k d = ζ d .
The fixed space of γ acting on V is generated by all the v i with i / ∈ {i 1 , . . . , i k }. Thus ℓ W ′ (γ) = k = ℓ(γ)/d again by Lemma 3.14 and [Shi07, Theorem 2.1].

By combining Lemma 3.17 and Lemma 3.13, we obtain the following proposition.

Proposition 3.18. Let σ = c 1 • • • c a γ 1 • • • γ b ∈ W ′ . The reflection length of σ in W and W ′ are given by ℓ W (σ) = a i=1 d(ℓ(c i ) -1) + b j=1 (ℓ(γ i ) -1) and ℓ W ′ (σ) = a i=1 (ℓ(c i ) -1) + b j=1 ℓ(γ i ) d .
In particular, we have ℓ

W (σ) + b = dℓ W ′ (σ).
This relation between reflection lengths in W and in W ′ will be the key element in our proof of Theorem 3.10 for the case W = G(1, 1, n).

Lemma 3.19. Let D be a nontrivial diagonal matrix in G(d, 1, q). We have D c(d, 1, q) in G(d, 1, q) if and only if D is a diagonal reflection s with χ(s) = ζ d .

Proof. By [Shi07, Theorem 2.1], the reflection length ℓ G(d,1,q) (D) of D in G(d, 1, q) is the number of nontrivial diagonal entries of D. The underlying permutation of D -1 c(d, 1, q) is an n-cycle. Thus, again by [Shi07, Theorem 2.1], we have

ℓ G(d,1,q) (D -1 c(d, 1, q)) = n -1 if χ(D) = χ(c(d, 1, q)) = ζ d n otherwise
If D is non trivial (i.e if its reflection length is nonzero), we get D c(d, 1, q) if and only if χ(D) = ζ d and D has exactly one nontrivial diagonal entry.

Proposition 3.20.

Let σ = c 1 • • • c a γ 1 • • • γ b ∈ W ′ . If σ ∈ I c(1,1,n) (W ′ ) or if σ ∈ I c(1,1,n) (W ), then b ∈ {0, 1} and χ(σ) = ζ b d . Proof. Let γ = (e(0, i 1 ) e(j 2 , i 2 ) • • • e(j k , i k ) e(1, i 1 ) • • • e(j k + d -1, i k )) be a balanced cycle. The diagonal reflection of W ′ sending v i to ζ d v i is given by s i := (e(0, i) • • • e(d-1, i)). We have that s -1 i 1 γ = (e(0, i 1 ) • • • e(j k , i k )) • • • (e(d -1, i 1 ) • • • e(j k + d -1, i k )) is a saturated cycle. Proposition 3.18 then gives that ℓ W ′ (s i 1 ) + ℓ W ′ (s -1 i 1 γ) = 1 + k -1 = k = ℓ W ′ (γ) and s i 1 γ in W ′ .
Assume that σ ∈ I c(1,1,n) (W ′ ) and b > 1. Since γ 1 and γ 2 have disjoint support, we deduce that there are two diagonal reflections s, s ′ of W ′ such that s γ 1 , s ′ γ 2 , ss ′ = Id and χ(ss ′ ) = ζ 2 d . We then have ss ′ c(1, 1, n) in W ′ which contradicts Lemma 3.19. Assume now that σ ∈ I c(1,1,n) (W ). By [DDGKM, Proposition IX.2.7], the set theoretic partition of the set µ n induced by σ is noncrossing (in the sense of [BC06, Section 1.2]). If b > 1, then the orbits of γ 1 and γ 2 induce two parts of µ n whose convex hull contain 0. Thus the partition of µ n induced by σ is crossing and σ / ∈ I c(1,1,n) (W ). The assumption on χ(σ) is a direct consequence of Lemma 3.17.

Proposition 3.21. The two posets

I c(1,1,n) (W ) c(1,1,n) q and I c(1,1,n) (W ′ ) are equal.
Proof. First, we show that the two sets

I c(1,1,n) (W ) c(1,1,n) q and I c(1,1,n) (W ′ ) are equal. Let σ ∈ I c(1,1,n) (W ′ ). By Proposition 3.20, we have χ(σ) ∈ {1, ζ d } and χ(σ -1 c(1, 1, n)) = χ(σ) -1 ζ d . Proposition 3.18 then gives ℓ W (σ) + ℓ W (σ -1 c(1, 1, n)) = dℓ W c(1,1,n) q (σ) + dℓ W c(1,1,n) q (σ -1 c(1, 1, n)) -1 = d(ℓ W c(1,1,n) q (σ) + ℓ W c(1,1,n) q (σ -1 c(1, 1, n))) -1 = dℓ W c(1,1,n) q (c(1, 1, n) q ) -1 = ℓ W (c(1, 1, n)) and σ ∈ I c(1,1,n) (W ). The same reasoning gives I c(1,1,n) (W ) c(1,1,n) q ⊂ I c(1,1,n) (W ′ ).
Let now σ, τ ∈ I c(1,1,n) (W ′ ), we have to show that σ τ in W ′ if and only if σ τ in W . We consider four cases -χ(σ) = 1 and χ(τ

) = 1. We have ℓ W (σ) = dℓ W ′ (σ) and ℓ W (τ ) = dℓ W ′ (τ ). If σ τ in W (resp. in W ′ ), then σ -1 τ ∈ I c(1,1,n) (W ′ ) is such that χ(σ -1 τ ) = 1. We then have ℓ W (σ -1 τ ) = dℓ W ′ (σ -1 τ ) and σ τ in W ′ (resp. in W ). -χ(σ) = ζ d and χ(τ ) = 1. We cannot have σ τ in either W or W ′ since this would imply that σ -1 τ is an element of I c(1,1,n) (W ′ ) with χ(σ -1 τ ) = ζ -1 d (we can assume ζ d = -1 since the case d = 2 is known by [BW02, Lemma 4.8]). -χ(σ) = 1 and χ(τ ) = ζ d . We have ℓ W (σ) = dℓ W ′ (σ) and ℓ W (τ ) + 1 = dℓ W ′ (τ ). If σ τ in W (resp. in W ′ ), then σ -1 τ ∈ I c(1,1,n) (W ′ ) is such that χ(σ -1 τ ) = ζ d . We then have ℓ W (σ -1 τ ) + 1 = dℓ W ′ (σ -1 τ ) and σ τ in W ′ (resp. in W ). -χ(σ) = ζ d and χ(τ ) = ζ d . We have ℓ W (σ) + 1 = dℓ W ′ (σ) and ℓ W (τ ) + 1 = dℓ W ′ (τ ). If σ τ in W (resp. in W ′ ), then σ -1 τ ∈ I c(1,1,n) (W ′ ) is such that χ(σ -1 τ ) = 1. We then have ℓ W (σ -1 τ ) = dℓ W ′ (σ -1 τ ) and σ τ in W ′ (resp. in W ).
This closes the proof of Theorem 3.10 in this case. We have the following corollary, which substantiate [BC06, Remark at the end of Section 7], for which we could not find a proof in the literature.

Corollary 3.22. The map from G(d, 1, q) to the set of partitions of the regular n-gon that sends every element of G(d, 1, q) to the partition given by the orbits of its image in S n induces a poset isomorphism from I c(d,1,q) (G(d, 1, q)) to NCP(d, 1, q) := NCP(1, 1, n) µ d , where NCP(1, 1, n) denotes the set of noncrossing partitions of the regular n-gon (in the sense of [BC06, Section 1.2]). q) . On the other hand, the map from G(1, 1, n) to the set of partitions of the regular n-gon that sends every element of G(1, 1, n) to the partition given by its orbits induces a poset isomorphism from

Proof. The embedding G(d, 1, q) → G(1, 1, n) as the centralizer of c(1, 1, n) q identifies the posets I c(d,1,q) (G(d, 1, q)) and I c(1,1,n) (W ′ ) = I c(1,1,n) (W ) c(1,1,
I c(1,1,n) (G(1, 1, n)) to NCP(1, 1, n) ([DDGKM, Proposition IX.2.7]).
Under this isomorphism, the action of c(1, 1, n) by conjugation corresponds to a rotation of angle 2π n . Thus the isomorphism

I c(1,1,n) (W ) ≃ NCP(1, 1, n) induces an isomorphism between I c(1,1,n) (W ) c(1,1,n) q and NCP(1, 1, n) µ d as claimed. 3.2.2. W = G(m, 1, n) for m 2. The highest degree of W is mn. We endow W with the Coxeter element c(m, 1, n) =        0 ζ m 1 0 1 . . . . . . 0 1 0        A ζ mn -eigenvector for c(m, 1, n) which is regular is given by ζ n-1 mn e 1 + ζ n-2 mn e 2 + • • • + e n .
We have seen in Section 3.2.1 that G(m, 1, n) can be realized as the centralizer in W ′ := G(1, 1, mn) of c(1, 1, mn) m , and this identification sends c(m, 1, n) to c(1, 1, mn). Let q be a divisor of mn. We have

W c(m,1,n) q ≃ (W c(1,1,mn) m ) c(1,1,mn) q = W ′ c(1,1,mn) q∨m
This identification induces the following isomorphisms of posets

I c(m,1,n) (W ) c(m,1,n) q ≃ (I c(1,1,mn) (W ′ c(1,1,mn) m )) c(1,1,mn) q I c(m,1,n) (W c(m,1,n) q ) ≃ I c(1,1,mn) (W ′ c(1,1,mn) q∨m )
By Section 3.2.1, we have equalities of posets

(I c(1,1,mn) (W ′ c(1,1,mn) m )) c(1,1,mn) q = (I c(1,1,mn) (W ′ ) c(1,1,mn) m ) c(1,1,mn) q = I c(1,1,mn) (W ′ ) c(1,1,mn) m∨q = I c(1,1,mn) (W ′ c
(1,1,mn) q∨m ) which gives the desired results.

3.2.3. W = G(e, e, n) for e 2, n 3. The highest degree of W is e(n -1). We start by considering the group W ′ := G(e, 1, n -1). The character χ : W ′ → C * giving the product of the nonzero entries allows us to define an embedding

i : W ′ -→ W M -→ χ(M ) -1 0 0 M The element c(e, e, n) = i(c(e, 1, n -1)) is a Coxeter element for W . For instance ζ n-2 e(n-1) e 2 + • • • + ζ e(n-1) e n-1 + e n is a ζ e(n-1)
-eigenvector for c(e, e, n) which is regular. Let q be a divisor of e(n -1). We have c(e, e, n) q = ζ -q e 0 0 c(e, 1, n -1) q

We can compute directly the centralizer of this element in W .

Proposition 3.23. Let q be a positive integer dividing e(n -1).

(a) If e(n-1) e∧n divides q, then W = W c(e,e,n) q . In particular Theorem 3.10 holds. (b) If e(n-1) e∧n does not divide q, then i induces an isomorphism between W ′ c(e,1,n-1) q and W c(e,e,n) q .

Proof. (a) Assume that e(n-1) e∧n j = q for some integer j. Since dq = e(n -1), we obtain that d = e∧n j divides e ∧ n, which is the gcd of the degrees of W (cf. [BMR98, Table 2]). The degrees of W c(e,e,n) are then the same as that of W . (b) We denote d the integer such that dq = e(n -1). Let w ∈ W . We write w as a block matrix.

w = X Y Z T with X ∈ M 1 (C), Y ∈ M 1,n-1 (C), Z ∈ M n-1,1 (C) and T ∈ M n-1,n-1 (C). We find w ∈ W c(e,e,n) q ⇔      c(e, 1, n -1) q Z = ζ -q e Z Y c(e, 1, n -1) q = ζ -q e Y T c(e, 1, n -1) q = c(e, 1, n -1) q T
as w is a monomial matrix, Z and Y have at most one nonzero coefficient. Suppose that Z = 0, then c(e, 1, n -1) q has a diagonal coefficient equal to ζ -q d . The underlying permutation of c(e, 1, n -1) is a (n -1)-cycle. Thus c(e, 1, n -1) q has a nonzero diagonal coefficient if and only if (n -1) divides q.

Let j be an integer such that j(n -1) = q. Since dq = e(n -1), we have dj = e. We also have c(e, 1, n -1) q = (c(e, 1, n -1) n-1 ) j = ζ j e Id. Thus c(e, 1, n -1) q Z = ζ -q e Z if and only if ζ j e = ζ -q e , that is, j = -q mod e. Since j(n -1) = q, we obtain that there is some integer k with jn = kjd. Thus kd = n and d divides n. We have already shown that d divides e, we obtain that d divides e ∧ n and that e(n-1) e∧n divides q. The same reasoning shows that Y = 0 implies that e(n-1) e∧n divides q. We obtain Y = 0, Z = 0, and w ∈ W c(e,e,n) q if and only if w = i(T ) with T ∈ W ′ c(e,1,n-1) q .

From now on, we suppose that e(n-1) e∧n does not divide q. The isomorphism between W c(e,e,n) q and W ′ c(e,1,n-1) q induced by i induces in turn an isomorphism of posets I c(e,1,n-1) (W ′ c(e,1,n-1) q ) ≃ I c(e,e,n) (W c(e,e,n) q ). On the other hand, we know from Section 3.2.2 that the two posets I c(e,1,n-1) (W ) c(e,1,n-1) q and I c(e,1,n-1) (W ′ c(e,1,n-1) q ) are equal. To show that Theorem 3.10 holds in this case, it only remains to show that the morphism i induces an isomorphism of posets between I c(e,1,n-1) (W ′ ) c(e,1,n-1) q and I c(e,e,n) (W ) c(e,e,n) q . By [BC06, Remark after Lemma 1.22], c(e, e, n) induces a free action (of a cyclic group of order e(n-1) e∧n ) on the set I c(e,e,n) (W ) \ i(I c(e,1,n-1) (W ′ )). Since e(n-1) e∧n does not divide q by assumption, we get I c(e,e,n) (W ) c(e,e,n) q = i(I c(e,1,n-1) (W ′ )) c(e,e,n) q = i I c(e,1,n-1) (W ′ ) c(e,1,n-1) q which finishes the proof.

3.3. Presentation by Hurwitz relations. As a category of periodic elements, C has been defined by the categorical presentation S q p | R q p + . In practice, we want a presentation with less generators, for instance with only the atoms of C as generators. Such a result is known in the case of the dual braid monoid M (c), which is presented by its atoms, endowed with the Hurwitz relations (see Remark 2.10).

Let A be the set of atoms of C. Recall from Lemma 3.2 that A is made of the simples (a, b) such that a admits no proper φ q -invariant left-divisor in M (c). Lemma 3.24. Let s := (a, b) and t := (d, e) be two composable atoms in C. The product st is either its own greedy normal form, or it is equal to the simple r = (ad, d -1 b).

Proof. If st is not a greedy path, then the left-gcd of t and s is nontrivial. Since t is an atom, we then get that t s, i.e d b in M (c). In which case st = (ad, d -1 b) is a simple morphism.

Lemma 3.25. Consider a square of atoms in C

u s / / σ v t v ′ τ / / w
The square is commutative if and only if one of the following conditions is met: (1) s = σ and t = τ .

(2) π p (στ ) = π p (st) ∈ I c . In this case, st = στ is a simple morphism.

Proof. Let us denote s = (a, b), t = (d, e), σ = (α, β) and τ = (δ, ε). Suppose that we have st = στ and s = σ. Since both s and σ are atoms, we have s σ and σ s: the paths st, στ are not in greedy normal form. By Lemma 3.24, στ = st is a simple morphism, and we have π p (στ ) = π p (st) ∈ I c .

Conversely, assume that π p (στ ) = αδ = ad = π p (st) lies in I c . If p = 1 then π p is injective by Section 1.3.1 and στ = st, thus we can suppose that p > 1. Let αx = ay be the right-lcm of α and a in I c . Let also ψ(x) and ψ(y) be the elements introduced in Lemma 1.11. We have αx αd and x d. By Lemma 1.11, we have ψ(x) d, thus ψ(x) = d by Lemma 3.2. The same argument gives ψ(y) = δ. Since s and σ share the same source, we have ab = αβ. We deduce x b, y β, and d b, δ β again by Lemma 1.11. This proves that st and στ are simple morphisms with the same source and target: they are equal by Proposition 3.3. Definition 3.26. We call Hurwitz relations on C the relations of the form st = τ σ, where s, t, τ, σ are atoms in C, and π p (στ ) = π p (st) ∈ I c . We call H the set of Hurwitz relations in C.

Theorem 3.27. The category C is presented by the atoms and the Hurwitz relations, that is

C = A | H + .
Proof. If p = 1, this is already known from Corollary 3.11, since dual braid monoids are presented by their atoms, endowed with Hurwitz relations (cf. [Bes15, Lemma 8.8]). Suppose now that p > 1. We know that C is generated by its atoms, and that the defining relations of C imply the Hurwitz relations. It remains to show that the defining relations of C are implied by the Hurwitz relations.

Let st = u be a defining relation of C. We consider three paths of atoms in C

s 1 • • • s r , t 1 • • • t k , u 1 • • • u m
expressing s, t and u, respectively. We set

a i = π p (s i ) for i ∈ [[1, r]], b i := π p (t i ) for i ∈ [[1, k]] and α i := π p (u i ) for i ∈ [[1, m]].
In M (c) φ q , the two words

a 1 . . . a r b 1 • • • b k and α 1 • • • α m
express the same element π p (u). Since M (c) φ q is presented by the Hurwitz relations (this is the case p = 1), there is a sequence of words µ 1 , . . . , µ n in the atoms of M (c) φ q such that -

µ 1 = a 1 • • • a r b 1 • • • b r -µ n = α 1 • • • α m -For i ∈ [[1, n -1]], µ i and µ i+1 are related by a Hurwitz relation in M (c) φ q .
In particular, each of the µ i expresses the element π p (u) in M (c) φ q . By Lemma 3.4, each µ i admits a unique lift p i in C, and each of the p i expresses u in C. By Lemma 3.25, the paths p i and p i+1 are related by a Hurwitz relation in

C for i ∈ [[1, n -1]].
In particular the equality st = u is implied by the Hurwitz relations in C.

This presentation can be seen as an analogue of Remark 2.10 for categories of periodic elements of dual braid monoids. This new presentation will be useful for computational purposes in Section 4.

3.4. Braided reflections and atomic loops. In this section we give a description of braided reflections in B(W g ) in terms of atomic loops in the category C. Namely we prove that a braided reflection is conjugate to an atomic loop. We keep the same notation as in the last sections.

Since the homeomorphism between X g /W g and (X/W ) µ d doesn't depend on a choice, we can identify the two spaces. The choice of a basepoint x ∈ U µ d induces an isomorphism between B(W g ) ≃ π 1 (X g /W g , x) and B(u, u), where u is the connected component of x in U µ d . By Remark 2.3, a change of basepoint preserves the set of braided reflections.

We explicit the definition of braided reflections in the context of the topological tools of Section 2.3 and 2.4. Proposition 3.28. ([Bes15, Definition 11.12 and Proposition 11.13]) The bijection (LL, clbl) of Theorem 2.22 restricts to a bijection between (X/W ) µ d and the set (E • n ) µp ⊡ S q p of compatible pairs where the first term is µ p -invariant, and the second term belongs in S q p . Following this proposition, we can now see the cyclic label of some x ∈ (X/W ) µ d as an element of S q p , with the convention introduced in Section 3.1. Let a ∈ (V /W ) µ d be on the smooth part of some irreducible divisor of the discriminant hypersurface in (V /W ) µ d . For ζ ∈ C * , the point ζa lies on the smooth part of the same irreducible divisor as a. In particular, we can replace a by some ζa for ζ ∈ S 1 so that no point of LL(a) lies on D. From now on we assume that this condition is met.

Since LL is a (ramified) covering ([Bes15, Theorem 5.3]), it is an open map. The choice of a neighborhood of a then induces a neighborhood of LL(a) in E n that is, a neighborhood in C of each point of the cyclic support of LL(a).

-If x ∈ C * is in the support of LL(a), the associated neighborhood is called an outer neighborhood . -If a lies on the discriminant hypersurface, that is if 0 ∈ LL(a), we call central neighborhood the neighborhood of 0 ∈ LL(a).

Definition 3.29. Let a ∈ V /W be a point such that LL(a) ∩ D = ∅. We we say that a neighborhood of a in V /W is confining if the induced neighborhood of LL(a) satisfies the following conditions (1) The outer neighborhoods do not overlap in module the central neighborhood.

(2) The outer neighborhoods never contain points in D.

It is easy to see that any confining neighborhood are stable under intersection and that they make up a basis of neighborhoods of a in V /W . The goal of a confining neighborhood is to "isolate" the central neighborhood, so that the exterior points of LL(a) cannot interfere with the points in the central neighborhood, as seen in the example below Let now a ∈ (V /W ) µ d be such that no point of LL(a) lies on D, and let U be a confining neighborhood of a in V /W . The set U ∩ (V /W ) µ d is a neighborhood of a in (V /W ) µ d . In particular we have that U ∩ U µ d is nonempty since

U µ d is dense in (V /W ) µ d .
Lemma 3.30. Let a ∈ (V /W ) µ d be such that no point of LL(a) lies on D, and let U be a confining neighborhood of a in V /W . Let also x ∈ U ∩ U µ d and let u be the connected component of x in U µ d . There is some s = s(x) ∈ S q p (u, -) such that the image of π 1 (U ∩ (X/W ) µ d , x) in B(u, u) contains all the simple loops λ(t) with t s in C.

Proof. We denote by β = β(x) the product (in clockwise order) of the terms of clbl(x) corresponding to the outer points in the sector P 1 . Let γ be a path in U ∩ (X/W ) µ d starting from x. For all t ∈ [0, 1] such that γ(t) ∈ U µ d , we have β(γ(t)) = β(x) by Lemma 2.24 and the following discussion.

We consider the path γ which consists in sliding the central points in each sector together and then counterclockwise next to the associated half-line:

This standardization motion is homotopically trivial and only impacts the cyclic label. Let α = α(x) be the first term of the cyclic label of γ(1). The cyclic label of γ(1) is given by (α, s 1 , . . . , s k ) with s 1 • • • s k = b. We associate to x the well defined element s = s(x) := (α, β) ∈ S q p (α is also defined by the equality u = αβ). Let t = (d, e) be a left-divisor of s in C. There is some m ∈ S φ q such that dm = α and mβ = e. We can desingularize γ(1) so that we obtain a point x ′ ∈ U ∩ U µ d with cyclic label (d, m, s 1 , . . . , s k ). The path in U ∩ (X/W ) µ d starting from x ′ and consisting in rotating the central points to the left, so that the first point of the cyclic support goes into the sector P p , represents the simple morphism t in B.

Let x 1 denote the endpoint of this motion. The cyclic label of x 1 is (m, s 1 , . . . , s k , d c η ). By applying a standardization motion γ 1 , we get a point γ 1 (1) with cyclic label

(d c η ms 1 •••s k , m, s 1 , . . . , s k ) = (d c η e , m, s 1 , . . . , s k )
We see that the circular tunnel associated to γ(1) represents t # in C. By an immediate induction, we obtain that λ(t) lies in the image of π 1 (U ∩ (X/W ) µ d , x) in B(u, u).

Theorem 3.31. Let u be an object of C, and let s ∈ C(u, -) be an atom. The atomic loop λ(s) is a braided reflection in the group B(u, u) ≃ B(W g ). Conversely, any braided reflection in B(W g ) ≃ B(u, u) is conjugate in B to some atomic loop.

Proof. First, if p = 1, then B = M (c) φ q is isomorphic to the dual braid monoid associated to W g by Corollary 3.11. The result is already known in this case: the braided reflections in B(W g ) are exactly the elements that are conjugated to atoms of the dual braid monoid.

From now on we assume that p > 1. Let s = (α, β) be an atom of C. We consider the element x s of s used to define the circular tunnel b s of Theorem 2.27. Let γ be the path consisting of sliding the first point of LL(x s ) in each sector towards the center The endpoint of γ lies on the discriminant hypersurface. Let U be a confining neighborhood of γ(1) and let r > 0 be such that γ(r) ∈ U . The cyclic label of γ(r) is (α, β). By Lemma 3.30, the image of π 1 (U ∩ (X/W ) µ d , γ(r)) in B(u, u) contains λ(s).

Next, let h ∈ π 1 (U ∩ (X/W ) µ d , γ(r)) be represented by a loop θ. We can choose θ so that, at any given t ∈ [0, 1], at most one point of LL(θ(t)) lies in the half line iR + (as points which do not satisfy this form a a subspace of real codimension 2). This expresses θ as a concatenation of paths in U ∩ (X/W ) µ d homotopic to circular tunnels (and their inverses).

Let t = (d, e) be a simple morphism represented by a path inside U ∩ (X/W ) µ d . By Lemma 2.24 and the following discussion, both the source and target of t are elements of M (c) divisible by β. Since p > 1, the proof of Proposition 3.3 gives that e which is the gcd in M (c) of the source and target of t. We then have that β divides e and t leftdivides s. The same reasoning (by induction) gives that the only morphisms represented by a concatenation of paths in U ∩ (X/W ) µ d (starting from γ(r)) homotopic to circular tunnels are of the form ss # s ## • • • or (ss # • • • ) -1 . We obtain that π 1 (U ∩ (X/W ) µ d , γ(r)) is cyclic and generated by a path representing λ(s) in B, which proves that λ(s) is a braided reflection in B(u, u).

Conversely, let γ be a path from some x ∈ u to the smooth part of some irreducible divisor of the discriminant in X g /W g ≃ (X/W ) µ d . Let also ρ γ denote the braided reflection associated to γ. Up to conjugacy in B(u, u), we can assume that the endpoint a of γ is such that no point of LL(a) lies on D and that, for all t ∈ [0, 1[, there is some r

∈ [t, 1[ such that LL(γ(r)) ∈ U µ d .
Let U be a small enough confining neighborhood of a so that the fundamental group of U ∩ (X/W ) µ d is infinite cyclic. Let also r ∈ [0, 1] be such that y := γ(r) lies in U ∩ U µ d . We denote by v the connected component of y in U µ d . Consider the element s = s(y) introduced in Lemma 3.30. We claim that s is an atom of C. Otherwise, s admits two distinct left-divisors σ and σ ′ (indeed, s(x) is the right-lcm of the atoms which divide it). Lemma 3.30 implies that the image of π 1 (U ∩ (X/W ) µ d , y) in B(v, v) contains both λ(σ) and λ(σ ′ ). Since π 1 (U ∩ (X/W ) µ d , x) is cyclic, there are two integers k and k ′ such that λ(σ) k = λ(σ ′ ) k ′ . Since both λ(σ) and λ(σ ′ ) are rigid, this is impossible.

Again, as every element of π 1 (U ∩ (X/W ) µ d , y) is represented by a path homotopic to a concatenation of circular tunnels, we obtain that the image of π 1 (U ∩ (X/W ) µ d , y) in B(v, v) is generated by λ(s). The path γ r : t → γ(rt) then gives a conjugator in B between the braided reflection ρ γ in B(u, u) and λ(s) ∈ B(v, v).

3.5. Conjugacy of atomic loops and centers of finite index subgroups. In this section we study the super-summit set of (powers of) atomic loops in C. We deduce results on the center of finite index subgroups of B(W g ). More precisely we obtain new Garside theoretic proofs of [START_REF] Bessis | Finite complex reflection arrangements are K(π, 1)[END_REF]Corollary 12.7] and of [START_REF] Digne | The center of pure complex braid groups[END_REF]Theorem 1.4] in the case of the centralizer of a regular element in a well-generated irreducible complex reflection group.

Note again that our approach covers in particular the dual braid monoid associated to a well-generated irreducible complex reflection group. Definition 3.32. Let C be a small category. The center of C is the set Z(C) of natural transformation from the functor 1 C to itself.

The data of an element z of Z(C) is equivalent to the data, for every object u of C, of a morphism z u such that ∀f ∈ C(u, v), z u f = f z v If C = M is a monoid, we recover the classical definition of the center. Note that Z(C) is always a monoid. Moreover, the center Z(G) of a groupoid G is a group. Lemma 3.33. Let G be a connected groupoid, and let u be an object of G. The map r :

Z(G) -→ Z(G(u, u)) z -→ z u
is an isomorphism of groups.

Proof. The map r is clearly a morphism of groups. We choose, for every v ∈ Ob(G), a morphism m v : u → v. Let z 0 ∈ Z(G(u, u)) be a central element. We define an element z of Z(G) by defining, for every v ∈ Ob(G):

z v := m -1 v z 0 m v We have in particular z u = m -1 u z 0 m u = z 0 since z 0 is central. Let f : v → w be a morphism in G, we have z v f = m -1 v z 0 m v f = m -1 v z 0 (m v f m -1 w )m w = m -1 v (m v f m -1 w )z 0 m w = f z w and z is indeed in Z(G). The map z 0 → z is the inverse of r.
It is known from [BMR98, Theorem 2.24] and [Bes15, Theorem 12.3 and Corollary 12.7] that the center of an irreducible complex braid group is cyclic. The center of the Springer groupoid B is also cyclic since B is connected. Under a combinatorial assumption on the integer d, we get that the center of C (and B) is actually generated by some power of the Garside map ∆.

Proposition 3.34. If d is the gcd of the degrees of W which it divides, then both Z(C) and Z(B) are cyclic and generated by ∆ q p . Proof. First, as ∆ p is a natural transformation from 1 C to φ p , and as φ q p = 1 C , we get that ∆ q p lies in the center of C. We claim that Z(B) is generated by Z(C) and ∆ -q p . Let z ∈ Z(B), and let u ∈ Ob(B). We set n u := inf(z u ). Since B admits a finite number of objects (D q p (c) is finite by definition), there is some kq for k ∈ Z 0 such that kq + n u > 0 for all object u. We then have that ∆ kq p z ∈ Z(C) and z = ∆ -kq p (∆ kq p z) as claimed. Now, let ρ be a generator of the center of B(W g ) ≃ B(u, u). As ∆ q p (u) ∈ Z(B(u, u)), there is some integer k such that ρ k = ∆ q p (u). That is ρ is a (pk, q)-regular element in B. By applying π p , we get that π p (ρ) is a (pk, q)-regular element of B(W ). By assumption, we have

π p (B(u, u)) = C B(W ) (π p (∆ q p )) ⊂ C B(W ) (π p (ρ)
). We also have C B(W ) (π p (∆ q )) ⊃ C B(W ) (π p (ρ)) since π p (∆ q p ) is a power of π p (ρ). The element π p (ρ) is then a dk-regular braid in B(W ) with the same centralizer as a d-regular braid. Since d is maximal regarding to divisibility, we get that k = ±1. Thus ρ = ∆ ±q p (u) which shows the proposition.

Remark 3.35. The assumption that d is the gcd of the degrees of W which it divides is important, otherwise Z(C) is generated by some root of ∆ q p . For instance in the group W = G 37 , the integers d 1 = 5 and d 2 = 10 are regulars. We have (p 1 , q 1 ) = (1, 6) and (p 2 , q 2 ) = (1, 3). The associated categories of periodic elements are then monoids, given by C M (c) (∆ 3 ) and C M (c) (∆ 6 ), respectively. Because d 1 and d 2 both divide 2 degrees of W , those centralizers are equal, and their centers are both equal to ∆ 3 + . Lemma 3.36. Let λ(s) be an atomic loop of some object u of C. Let t : u → v be an atom in C. If the path s ♭ t is not greedy, then we have λ(s)t = tλ(s ′ ) for some atomic loop λ(s ′ ) of v.

Proof. Let λ(s) = s n • • • s 1 where s 1 = s ♭ and s i+1 := s ♭ i for i ∈ [[1, n -1]]
(in particular we have s n = s). We denote t := (α, β) and s i := (a i , b) for i ∈ [[1, n]]. By Corollary 1.39, if s ♭ t is not greedy, then we have α b. Let then x be such that αx = b. We have

s ♭ t = (a 1 , b)(α, β) = (a 1 α, x) = (α, a α 1 x)(a α 1 , xα c η
). We set t 1 := (α, a α 1 x) and σ 1 = (a α 1 , xα c η ). Again, as α b, the path s 2 t is not greedy, and we have

s 2 t 1 = (a 2 , b)(α, a α 1 x) = (a 2 α, x) = (α, a α 2 x)(a α 2 , xα c η )
, and we set again t 2 := (α, a α 2 x), σ 2 := (a α 2 , xα c η ). By an immediate induction, we get λ(s)t = t n σ n • • • σ 1 , with t n := (α, a α n x) and

σ i := (a α i , xα c η ) for i ∈ [[1, n]].
The two simple morphisms t and t n share the same source, and we have π p (t) = π p (t n ). By Lemma 1.38, we have t = t n . For i ∈ [[1, n -1]], the target of σ i+1 is the source of σ i , and the second terms of σ i and σ i+1 are both equal to xα c η . Thus σ i is equal to σ # i+1 , again by Lemma 1.38.

We claim that

σ n • • • σ 1 = λ(σ n ). Since σ i = σ # i+1 for i ∈ [[1, n -1]],
we only have to show that σ 1 is the first σ i with target v. The target of σ i is x(αa α i ) c η ) = x(a i α) c η , and the source of σ n is the target of σ 1 , that is v = x(a 1 α) c η . By cancellativity of M (c), we get that the target of

σ i is v only if i = 1, for i ∈ [[1, n]].
Theorem 3.37. Let λ(s) ∈ C(u, u) be an atomic loop of some object u, and let f ∈ G. If there is some endomorphism z ∈ C such that λ(s) n f = f z for some n 1, then z = λ(s ′ ) n for some atomic loop λ(s ′ ) such that λ(s)f = f λ(s ′ ).

Proof. We do the proof in several steps. 1. f is an atom. By assumption, we have f λ(s) n f . If λ(s) n f is in greedy normal form, we get f s. Thus f is trivial or f = s. We have λ(s) f = λ(s) in the first case, and λ(s) f = λ(s # ) in the second case. If λ(s) n f is not in greedy normal form, then s ♭ f is not in greedy normal form. By Lemma 3.36, we get that λ(s) f is an atomic loop, hence the desired result. 2. f is a simple morphism. Since C is homogeneous, we can proceed by -induction on f . The case where f is an atom has already been dealt with. If f is not an atom, then λ(s) n f cannot be in greedy normal form as f s is impossible. Thus s ♭ f is not greedy, and there is some decomposition f = tf ′ with t an atom such that s ♭ t is not greedy. By the first point we have that λ(s) t is of the form λ(s ′ ). Since z = λ(s) f = λ(s) tf ′ = λ(s ′ ) f ′ , the induction hypothesis gives the desired result. 3. f is an arbitrary morphism in B. We have that ∆

q p (u) is central in B(u, u), thus (λ(s) n ) ∆ kq p (u)f = (λ(s) n ) f = z for all k ∈ Z.
By choosing some k big enough so that kq + inf(f ) 0, we can replace f with ∆ kq p (u)f and assume that f ∈ C. We claim that both λ(s) n and z lie in SSS(λ(s) n ). First, as atomic loops are rigid, we have

cyc(λ(s)) = (s # • • • s ♭♭ s ♭ )s = λ(s # ) = dec(λ(s)).
Powers of atomic loops are also rigid, and we have cyc(λ(s) n ) = dec(λ(s) n ) = λ(s # ) n . Because of Proposition 1.24, we get that λ(s) n lies in SSS(λ(s) n ). Then, we have inf(z) 0 as z lies in C, so inf(z) = 0 as inf(z) inf(λ(s) n ). Lastly, we have sup(z) sup(λ(s) n ) = nm (where m is the length of λ(s)). Since every simple morphism has length at least 1, and since inf(z) = 0, we get mn = ℓ(z) sup(z) nm and sup(z) = nm, so z ∈ SSS(λ(s) n ). Let s 1 • • • s r be the greedy normal form of f ∈ C. By Proposition 1.25, each (λ(s) n ) s 1 •••s i lies in SSS(λ(s) n ), in particular is positive. By an immediate induction using the second case, we get that z is of the form λ(t) n , with λ(s) f = λ(t).

This theorem is an analogue of [DMM11, Proposition 2.2] in the context of Springer categories. It gives in particular a complete description of supper-summit sets of atomic loops and their powers.

Corollary 3.38. Let u ∈ Ob(B), and let σ ∈ B(W g ) ≃ B(u, u) be a braided reflection. The supper-summit set of σ in B consists of all the atomic loops to which σ is conjugate in B. Furthermore, for n 1, we have

SSS(σ n ) = {λ(s) n | λ(s) ∈ SSS(σ)}
Proof. We already showed that σ is conjugate to some atomic loop λ(s) ∈ B, and that λ(s) ∈ SSS(λ(s)) for all atomic loops λ(s). If g ∈ SSS(λ(s)), then g is a positive conjugate of λ(s) and Theorem 3.37 gives that g is an atomic loop.

We also have λ(s) n ∈ SSS(σ n ). If g ∈ SSS(σ n ), then we also have that g is a positive conjugate of λ(s) n . We get that g is of the form λ(s ′ ) n for some conjugate λ(s ′ ) of λ(s) by Theorem 3.37.

Like in [START_REF] Digne | The center of pure complex braid groups[END_REF], our theorem on conjugacy of atomic loops has the following corollary on the center of finite index subgroups.

Corollary 3.39. Let U be a finite index subgroup of B(W g ) ≃ B(u, u). We have Z(U ) ⊂ Z(B(W g )).

Proof. Let σ ∈ B(W g ) be a braided reflection, and let x ∈ Z(U ). Since U is of finite index, there is some n 1 with xσ n = σ n x. We claim that σx = xσ.

By Theorem 3.31, there is some morphism f : u → v in B such that σ f = λ(s) is an atomic loop. We define x ′ := x f and U ′ := U f ⊂ B(v, v). By Theorem 3.37, since λ(s) nx = λ(s) n , we have λ(s)x = xλ(s ′ ) for some atomic loop λ(s ′ ) such that λ(s ′ ) n = λ(s) n . Since atomic loops are rigid, the equality λ(s) n = λ(s ′ ) n implies λ(s) = λ(s ′ ) and x ′ commutes with λ(s). We obtain that x commutes with every braided reflection in B(u, u), and thus lies in Z(B(u, u)).

3.5.1. The case p = 1. In this case, for any simple s, we have s # = s ♭ = s and λ(s) = s. Lemma 3.36 is then a mere consequence of the fact that every simple element is balanced. Theorem 3.37 is just [DMM11, Proposition 2.2] in the case of the dual braid monoid.

Presentations of the braid group B(G 31 )

The goal of this section is to apply the previous results to the complex braid group B(G 31 ). In particular we prove several positive homogeneous presentations of this group, where the generators are braided reflections.

We first consider the complex reflection group G 37 , which is isomorphic to the real reflection group of type E 8 . Let M (c) be the dual braid monoid associated to some Coxeter element c of G 37 . We denote again its set of simple elements by I c . We have seen in Example 2.7 that the integer 4 is regular for G 37 , and that the associated centralizer is isomorphic to G 31 . Since the Coxeter number of G 37 is h = 30, we have (p, q) = (2, 15) in this case. By Theorem 2.27, the groupoid G(M (c) 15 2 ) is equivalent to the complex braid group B(G 31 ). We denote C 31 := M (c) 15 2 and B 31 := G(M (c) 15 2 ). The integer η associated to p, q by Lemma 1.28 is η = -7. As c 15 = -Id is central in G 37 , we have Proof. The only nontrivial part is that the full-twist is a generator of Z(P (G 31 )). Let B(G 31 ) ≃ B 31 (u, u), the center of P (G 31 ) is cyclic and generated by the smallest power of ∆ 15 2 (u) which lies in P (G 31 ). Since the collapse functor C 31 → M (c) sends ∆ 15 2 (u 0 ) to some element g with g 2 = ∆ 15 2 , the smallest power of ∆ 15 2 lying in P (G 31 ) is the full-twist ∆ 60 . Algorithm 4.4 doesn't always terminate: it may loop indefinitely. If it terminates, its output is a fraction, and we should check that it is trivial. Of course this may be quite long as algorithm 4.2 is far from optimal. This process can be sped up by "partially simplifying" at each step. The solution to the word problem given by Algorithm 4.2 allows for the computation of longest common divisors of elements of H + u . A general word in X u ∪ X -1 u can be written as a product of (short) fractions. We can simplify these fractions at each step of Algorithm 4.4.

We checked by computer that Algorithm 4.4 terminates on every relator in R ′ u , that is every relator can be expressed as a right-fraction of elements of H + u . Finally, we use the following algorithm to prove that every relator, written as a fraction, is trivial in H u .

Algorithm 4.5 Partial solution to the word problem in H u Input: A fraction f g -1 with f, g ∈ H + u . Output: true If there is some n ∈ H + u such that f n = gn. No output otherwise. put i := 1 Compute the set S i of words of length i in X u while f n = gn for all n ∈ S i do put i := i + 1 Compute the set S i of words of length i in X u end while return true This algorithm is useful since the monoid H + u may not be cancellative: we can have f n = gn (and thus f g -1 = 1 in H u ) without having f = g in H + u . The fact that Algorithm 4.5 returns true for all relators of the presentation of B 31 (u, u) finally proves that Theorem 4.2 holds for u.

Computational data. We consider the following elements in

C 8                                  α 1 = 1
2 (1, -1, -1, -1, -1, -1, -1, 1) α 2 = (1, 1, 0, 0, 0, 0, 0, 0) α 3 = (-1, 1, 0, 0, 0, 0, 0, 0) α 4 = (0, -1, 1, 0, 0, 0, 0, 0) α 5 = (0, 0, -1, 1, 0, 0, 0, 0) α 6 = (0, 0, 0, -1, 1, 0, 0, 0) α 7 = (0, 0, 0, 0, -1, 1, 0, 0) α 8 = (0, 0, 0, 0, 0, -1, 1, 0) α 9 = (1, 0, 1, 0, 0, 0, 0, 0)

                                
α 10 = (0, 0, -1, 0, 1, 0, 0, 0) α 11 = (1, 0, 0, 0, 0, 1, 0, 0) α 12 = 1 2 (-1, -1, -1, -1, -1, 1, -1, 1) α 13 = (0, 1, 0, 0, 0, 1, 0, 0) α 14 = 1 2 (-1, -1, -1, -1, -1, -1, 1, 1) α 15 = (0, 1, 0, 0, 0, 0, 1, 0)

α 16 = 1 2 (1, -1, 1, -1, -1, -1, 1, 1) α 17 = 1 2 (1, -1, -1, 1, -1, -1, 1, 1) α 18 = 1 2 (-1, 1, -1, -1, 1, -1, 1, 1) And for each i ∈ [[1, 18
]], we consider the reflection s i of C 8 given by s i (x) = x-2 x,α i α i ,α i α i (where ., . is the usual hermitian product on C 8 ). The set s 1 , . . . , s 8 generates a subgroup W of GL n (C) isomorphic to the complex reflection group G 37 and which contains all the s i for i ∈ [ [1,18]]. A Coxeter element of W is given by c = s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 .

A system of representatives of φ 2 -orbits of objects of the category C 31 = M (c) where each arrow is a simple morphism in C 31 (because of Proposition 3.3, a simple morphism is uniquely determined by its source and target). For each i, j ∈ [[1, 8]], this graph induces a well defined isomorphism ϕ i,j : B 31 (u i , u i ) → B 31 (u j , u j ) which preserves braided reflections.

For i, j ∈ [[1, 8]], we have by definition ϕ i,j = ϕ i,1 ϕ 1,j , so we only need to describe morphisms of the form ϕ i,1 and ϕ

1,i for i ∈ [[1, 8]].
Futhermore, we know that the full-twist in B 31 (u, u) is given by ∆ 60 (u) by Lemma 2.28. By [Gar23, Theorem 1.2 and Proposition 8.1], every root of the full-twist in B(G 31 ) is conjugate to a power of either a 20-th root or a 24-th root of the full-twist. Furthermore, the full-twist admits 20-th roots and 24-th roots. For each presentation we obtain, we give an explicit 20-th root (resp. 24-th root) of the full-twist as a word in the generators. If ρ ∈ B 31 (u, u) is a 24-th root of the full-twist ∆ 60 (u), then ρ 6 is a 4-th root of ∆ 60 (u). By Proposition 1.27 and Theorem 4.1, we get that ρ 6 = ∆ 15 2 (u) is a generator of Z(B 31 (u, u)). 4.3. Presentation associated to representatives of the φ 2 -orbits. The monoid H + u 1 given by this presentation is not cancellative: we have tuwtuw = uwtuwt and stuwtuw = suwtuwt in H + u 1 . Thus H + u 1 cannot be a Garside monoid. The submonoid L + u 1 of C 31 (u 1 , u 1 ) generated by s, t, u, v, w is cancellative, but it doesn't admit right-lcms: we know that utu = tut is the shortest common multiple of t and u. If right-lcms exists in L + u 1 , then tut must be the right-lcm of t and u. We would then have tut tuwtuw ⇒ t wtuw in L + u 1 by cancellativity. We can check that t wtuw doesn't hold in L + u 1 : the element t -1 wtuw does lie in C 31 (u 1 , u 1 ), but not in the submonoid L + u 1 .

left-divisor is sv, which is not a common right-multiple of s and v. This also proves that the submonoid of C 31 (u 6 , u 6 ) generated by g, n, q, s, v does not admit right-lcms. The morphisms ϕ 6,1 and ϕ 1,6 are given by ϕ 6,1 :

               b → s v p → u t f → w t q → t uwtv g → u vt r → u stuwtv n → u v s → s o → u vs v → v and ϕ 1,6 :                s → s t → p vn u → v n v → v w → p vnf
The maximal roots of the full-twist are given by Maximal regular number d 20 24 d-th root of ∆ 60 svgqns svgqn

In particular, we get that (svgqn) 6 = ∆ 15 2 (u 6 ) generates Z(B 31 (u 6 , u 6 )). We also have svgqns = ∆ 3 (u 6 ) in this case. The last line of relations can be omitted, as it is implied by the others. The monoid H + u 7

given by this presentation is not cancellative: we have ompo = mpom and sompo = smpom in H + u 7 . Thus H + u 7 cannot be a Garside monoid. The submonoid L + u 7 of C 31 (u 7 , u 7 ) generated by the atomic loops is cancellative, but it doesn't admit right-lcms: we know that omo = mom is the shortest common multiple of m and o. If right-lcms exists in L + u 7 , then omo must be the right-lcm of m and o. We would then have omo ompo ⇒ o po Which does not hold in L + u 7 (it doesn't even hold in C 31 (u 7 , u 7 )). The relations defining H + u 7 give in particular n = s o and k = n m in H u 7 . By deleting these generators, we get that B 31 (u 7 , u 7 ) is generated by g, m, o, p, s, with relations as follows:

         gm = mg, go = og, smp = psm = mps, sos = oso, pgp = gpg, gsg = sgs, mom = omo, somp = psom. The monoid given by this presentation does not admit right-lcms. We have that somp = psom and smp = psm are common right multiples of s and p, but their longest common left-divisor is ps, which is not a common right-multiple of s and p. This also proves that the submonoid of C 31 (u 7 , u 7 ) generated by g, m, o, p, s does not admit right-lcms. The morphisms ϕ 7,1 and ϕ 1,7 are given by ϕ 7,1 : The monoid H + u 8 given by this presentation is not cancellative: we have psgpsg = gpsgps and mpsgpsg = mgpsgps in H + u 8 . Thus H + u 8 cannot be a Garside monoid. The submonoid L + u 8 of C 31 (u 8 , u 8 ) generated by the atomic loops is cancellative, but it doesn't admit right-lcms: we know that pgp = gpg is the shortest common multiple of p and g. If right-lcms exists in L + u 8 , then omo must be the right-lcm of m and o. We would then have gpg gpsgps ⇒ g sgps Which does not hold in L + u 8 (it does even hold in C 31 (u 8 , u 8 ), but g -1 sgps is not generated positively by atomic loops).

               g → u vt p → u t k → u wtuwvsuvt s → s m → w st n → u v o → u vs
The relations defining H + u 8 give in particular g = n t in H u 8 . By deleting this generator, we get that B 31 (u 8 , u 8 ) is generated by m, n, p, s, t, with relations as follows: The monoid given by this presentation does not admit right-lcms. We have that ntmn = tnmt and tnt = ntn are common right multiples of n and t, but their longest common left-divisor is tn, which is not a common right-multiple of t and n. This also proves that the submonoid of C 31 (u 8 , u 8 ) generated by m, n, p, s, t does not admit right-lcms. The morphisms ϕ 8,1 and ϕ 1,8 are given by ϕ 8,1 :

                   g → u vt m → w st n → u v p → u t s → s t → t
and ϕ 1,8 :

               s → s t → t u → t p v → n tp w → m tmp
The maximal roots of the full-twist are given by Maximal regular number d 20 24 d-th root of ∆ 60 ptsmnp ptsmn

In particular, we get that (ptsmn) 6 = ∆ 15 2 (u 8 ) generates Z(B 31 (u 8 , u 8 )). We also have (ptsmn) 2 = ∆ 5 (u 8 ) in this case. A morphism between two oriented graphs (O, S) and (O ′ , S ′ ) is given by two maps φ 0 : O → O ′ , φ 1 : S → S ′ which preserve the source and target: ∀f ∈ S, s(φ 1 (f )) = φ 0 (s(f )) and t(φ 1 (f )) = φ 0 (t(f )).

Note that we make no assumption regarding the existence of loops or multiple arrows with the same source and target. In practice, we often amalgamate an oriented graph (O, S) with its set of arrows S. The set of objects will then be denoted by Ob(S) like for categories. We will sometimes denote by S(u, -) (resp. S(-, v), S(u, v)) the set of arrows in S with source u (resp. with target v, with source u and target v).

Definition A.2. ([DDGKM, Definition II.1.28]) Let S be an oriented graph. For u, v ∈ Ob(S), a path of length p 1 in S from u to v is a finite sequence (g 1 , . . . , g p ) of elements of S such that s(g 1 ) = u, t(g p ) = v and t(g i ) = s(g i+1 ), ∀i ∈ [[1, p -1]] For u ∈ Ob(C), we also define an empty path from u to itself, denoted by 1 u , and of length 0 by definition. -Composition is given by concatenation of paths.

-The identity of some object u is the empty path 1 u . This category is free in the following sense: Let S be an oriented graph, and let C be a category. Any morphism of oriented graphs φ : S → C induces a unique functor S * → C, sending a path (g 1 , . . . , g p ) to the composition φ 1 (g 1 ) • • • φ 1 (g p ) in C. In practice, a path (f 1 , . . . , f p ) will often be denoted as a formal composition f 1 • • • f p so that we have

φ(f 1 • • • f p ) = φ 1 (f 1 ) • • • φ 1 (f p ).
This convenient definition of free category allows for defining relations and presentations of categories. Recall that a congruence on a category C is an equivalence relation ≡ on C which is compatible with composition, that is, the conjunction of f ≡ f ′ and g ≡ g ′ implies f g ≡ f ′ g ′ (if f g and f ′ g ′ are defined of course). If ≡ is a congruence on a category C, one can form the quotient category C/ ≡. It has the same objects as C, and its morphisms are ≡-equivalence classes of morphisms in C. For readability purposes, it is convenient to write a relation (f, g) as an equality f = g instead of a couple of paths. We use this convention from now on.

Let Hg be the common source of m and m ′ in F(X ). We consider the paths m i [Hg] in F(X ). We have m Thus m and m ′ induce the same element in G, and the natural functor G → G H is then injective.

Let now H = φ -1 (H) ⊂ F (X) be the preimage of H under φ. Recall from [LS01, Proposition I.3.8] that a Schreier transversal (in the classical sense) for H in F (X) is a set of words T such that -The maps t → Ht is a bijection between T and H \ G.

-The set T is stable under prefix (in particular, the empty word lies in T ).

Let T be a Schreier transversal in the classical sense for H and F (X). The set of paths {t[H]} t∈T is a Schreier transversal rooted in H in F(X ) in the sense of Definition A.10. Conversely, if {m Hg } Hg∈Ob(F (X )) is a Schreier transversal rooted in H in the sense of Definition A.10, then the set of words { π(m Hg )} Hg∈H\G is a Schreier transversal for H and F (X) in the classical sense.

Let T be a Schreier transversal for H in F (X), and let {m Hg } Hg∈Ob(F (X )) be the associated Schreier transversal of F(X ) rooted in H. For g ∈ G, let g denote the element of T such that Hg = Hg.

The elements of X are given by

X = {x[Hg] | x ∈ X, Hg ∈ Ob(G H )}.
The set of generators of G H (H, H) = H we obtain by our method is Remark A.17. Let X be a transitive G-set, and let x ∈ X. There is a natural isomorphism between the category of cosets for the stabilizer of x in G and the graph of the action of G on X. In particular, our method gives a way to compute a presentation of the stabilizer starting from the graph of the action.

  Definition 1.5. ([DDGKM, Definition V.2.19]) Let C be a homogeneous cancellative category. A Garside map in C is a map ∆ : Ob(C) → C satisfying the following assumptions (1) For u ∈ Ob(C), the source of ∆(u) is u. The target of ∆(u) is denoted by φ(u).

  Proposition 1.8. ([DDGKM, Proposition V.2.35]) Let (C, ∆) be a homogeneous Garside category, and let u ∈ Ob(C). The posets (C(u, -), ) and (C(-, u), ) are lattices. The posets (S(u, -), ) and (S(-, u), ) are sublattices of these lattices.

  Definition 1.18. ([DDGKM, Definition V.3.23]) Let f be a morphism in G(C), with left-weighted factorization ∆ p s 1 • • • s r . The infimum and supremum of f are defined by inf(f ) := p and sup(f ) := p + r, respectively.

  (1, a, b) ∈ D 3q 3p (∆) expresses the relation (1, u)(a, b) = (a, b). Similarly, for (d, e) ∈ S q p with target u, the element (d, 1, e) in D 3q 3p (∆) expresses the relation (d, e)(1, u) = (d, e). Thus (1, u) represents the morphism 1 u in C q p . Theorem 1.35. ([DDGKM, Proposition XIV.1.5] and [DDGKM, Proposition VII.4.2])

  3q 3p (∆) expresses the relation (a, b)(b, φ η (a)) = (ab, 1) = ∆ p (ab), which proves that (a, b) = (b, φ η (a)). From this we deduce s = (φ η (a), φ η (b)) = φ p (a, b) as claimed. Definition 1.37. ([Bes15, Definition B.23])

  then d is a regular integer for W . It is known that ζ-regular elements (should they exist) form a conjugacy class in W . Theorem 2.6. ([LT09, Theorem 11.24]) Let g ∈ W be a ζ-regular element for some ζ ∈ µ * d . The centralizer C W (g) of g in W acts on the eigenspace Ker(g -ζ Id) as a complex reflection group. Its degrees (resp. codegrees) coincide with those degrees (resp. codegrees) of W which are divisible by d.

  Theorem 2.9. ([Bes15, Theorem 8.2]) The monoid M (c) is a homogeneous Garside monoid, with Garside element ∆ := c and set of simples I. The atoms of M (c) are the elements of R.

  The automorphism φ of M (c) acts on D m (c) by φ.(u 0 , . . . , u p-1 ) := (u 1 , . . . , u p-1 , u c 0 ). The bijection between D m (∆) and D m (c) restricts to a bijection between D n m (∆) and D n m (c) := {(u 0 , . . . , u p-1 ) ∈ D m (c) | φ q .(u 0 , . . . , u p-1 ) = (u 0 , . . . , u p-1 )} The results of Proposition 1.30 also allow for an alternative description of the sets D m n (c).

  Lemma 2.16. ([Bes15, Lemma 11.1]) Let x ∈ V /W and let λ ∈ C * . We have LL(λx) = λ h LL(x).

  Definition 2.18. ([Bes15, Definition 11.24]) A circular semitunnel is a triple T

  Theorem 2.19. ([Bes15, Proposition 8.5 and Lemma 11.10]) (i) The homotopy class ∆ of the circular tunnel T = (x, 2π h ) does not depend on x ∈ U . (ii) The projection c of ∆ in W is a Coxeter element of W . (iii) Consider S the set of homotopy classes of circular tunnels in B(W ). It is endowed with the relation defined by s s ′ if s -1 s ′ is homotopic to a circular tunnel. The projection map B(W ) ։ W induces an isomorphism of posets (S, ) ≃ (I c , ). This isomorphism induces in turn an isomorphism B(W ) ≃ G(c).

  µ d are contractible, we can use them as fat basepoint for groupoids in the sense of [Bes15, Definition A.4]. Definition 2.26. ([Bes15, Definition 11.23]) The Springer category associated to W and d is the groupoid

s

  ♭ := (a bc -η , b) and s # := (a c η b -1 , b). Lemma 3.5. Let s := (a, b) be a simple morphism in C. (a) The target of s ♭ is the source of s, and the target of s is the source of s # . (b) We have (s ♭ ) # = (s # ) ♭ = s. (c) The paths s ♭ s and ss # are both in greedy normal form in C. (d) We have φ p (s # ) = (φ p (s)) # and φ p (s ♭ ) = (φ p (s)) ♭ . Proof. (a) The target of s ♭ is ba b = ab and the source of s # is a c η b -1 b = ba c η . (b) We have s ♭# = (a bc -η , b) # = (a bc -η c η b , b) = (a, b) = s. The same reasoning proves that (s # ) ♭ = s. (c) Since s = (a, b) is a simple morphism, ab = u is a simple element in M (c).

  (d-1) i in the decomposition of Proposition 3.15 are disjoint. Definition 3.16. Let c ∈ W be a cycle such that c, c (1) , . . . , c (d-1) are disjoint. The product cc (1) • • • c (d-1) will be denoted by c and called a saturated cycle. If γ ∈ W is a cycle lying in W ′ , we say that γ is a balanced cycle.Proposition 3.15 states that elements of W ′ are the products of disjoint saturated cycles and disjoint balanced cycles.Lemma 3.17. Let c be a saturated cycle in W ′ . We have χ( c) = 1 and ℓ W ′ (c) = ℓ(c) -1. Let γ be a balanced cycle in W ′ . We have χ(γ) = ζ d and ℓ W ′ (γ) = ℓ(γ)/d.Proof. Let c = (e(j 1 , i 1 ) • • • e(j k , i k )) be a cycle such that c, . . .

  Ob(C 31 ) := D 15 2 (c) = {u ∈ I c | uu c 8 = c and ℓ R (u) = 4}, S 15 2 := D 30 4(c) = {(a, b) ∈ (I c ) 2 | ab ∈ Ob(C)}, R 15 2 := D 45 6 (c) = {(x, y, z) ∈ (I c ) 3 | xyz ∈ Ob(C)}. By Lemma 3.2, the atoms of C 31 are exactly the elements of length 1. Since 4 is the gcd of degrees of G 37 which it divides, we can apply Proposition 3.34 and Corollary 3.39. We get Theorem 4.1. The centers of C 31 and B 31 are cyclic and generated by ∆ 15 2 . If U is a finite index subgroup of B(G 31 ), then Z(U ) ⊂ Z(B(G 31 )). In particular, the center of the pure braid group P (G 31 ) is cyclic and generated by the full-twist.

4.3. 1 .

 1 Orbit of u 1 . For the first orbit, we recover the presentation of B(G 31 ) conjectured in [BMR98, Table 3] and [BM04, Conjecture 2.4]. The object u 1 has 5 atomic loops s, t, u, v, w. The relations we obtain are as follows      ts = st, vt = tv, wv = vw, suw = uws = wsu, svs = vsv, vuv = uvu, utu = tut, twt = wtw. This presentation is usually represented by the following diagram (corresponding to the Broué-Malle-Rouquier diagram for the reflection group G 31 ).

4.3. 7 .

 7 Orbit of u 7 . The object u 7 admits 7 atomic loops g, k, m, n, o, p, s, with relations as follows:               on = so =ns, kn = nm = mk, go = og, gm = mg, pn = np, pok = okp = kpo, psm = smp = mps, omo = mom, gpg = pgp, gkg = kgk, gsg = sgs, gng = ngn, kpon = somp.

  and ϕ 1,7 :               s → s t → g n u → p gn v → p g w → p gkpnThe maximal roots of the full-twist are given by Maximal regular number d 20 24 d-th root of ∆ 60 pgsomp pgsomIn particular, we get that (pgsom) 6 = ∆ 15 2 (u 7 ) generates Z(B 31 (u 7 , u 7 )). 4.3.8. Orbit of u 8 . The object u 8 admits 6 atomic loops g, m, n, p, s, t, with relations as follows:         tg = nt =gn, ts = st, pn = np, gm = mg, psm = smp = mps, tpt = ptp, tmt = mtm, pgp = gpg, sns = nsn, sgs = gsg, nmn = mnm.

  ts = st, pn = np, mps = psm = smp, ntn = tnt, ptp = tpt, tmt = mtm, sns = nsn, nmn = mnm, ntmn = tmnt

  Appendix A. The Reidemeister-Schreier method for groupoids A.1. Presentations of categories and groupoids. Definition A.1. ([DDGKM, Definition II.1.4 and Definition II.1.32]) An oriented graph (or precategory) is a pair of sets (O, S) endowed with two maps s, t : S → O. The elements of O are called objects and those of S are called elements (or morphisms, or arrows). The maps s and t are called source and target, respectively.

  Definition A.3. ([DDGKM, Definition II.1.28 and Proposition 1.33]) Let S be an oriented graph. The free category on S, denoted by S * , is defined as follows -The objects are the objects of S. -The morphisms S * (u, v) are the paths from u to v in S.

  Definition A.4. ([DDGKM, Definition II.1.36 and Lemma II.1.37]) If C is a category, a relation on C is a pair (g, h) of morphisms in C sharing the same source and the same target. If C is a category, and R is a family of relations on C, there exists a smallest congruence ≡ + R of C which includes R. The congruence ≡ + R is the reflexive-transitive closure of {(f gh, f g ′ h) | (g, g ′ ) ∈ R or (g ′ , g) ∈ R}.

  1 [Hg] = m, m p [Hg] = m ′ . For i ∈ [[1, p -1]], the path m i [Hg] is equivalent to m i+1 [Hg] by the use of one relation in R by definition.

  {γ(x[Hg]) = m Hg xm -1 Hgx | x ∈ X, Hg ∈ Ob(G H )} ={γ(x[Ht]) = m Ht xm -1 Htx | x ∈ X, t ∈ T } ={tx(tx) -1 | x ∈ X, t ∈ T }which is the same set as given in [LS01, Proposition 4.1]. Following [LS01, Proposition 4.1], we denote γ(t, x) := tx(tx) -1 . This element is equal to what we denoted by γ(x[Ht]). Now for the relators. Let r = 1 be a relation of G. It induces the following family of relation on G H : {r[Hg] = 1 Hg | Hg ∈ Ob(G H )} = {r[Ht] = 1 Ht | t ∈ T } Each relation r[Ht] = 1 Ht induces a relation on G H (H, H), given bym Ht r[Ht]m -1 Ht = 1 If r = y 1 y 2 y 3 • • • y k is expressed as a word in X ∪ X -1 , we have m Ht r[Ht]m -1 Ht = γ(y 1 [Ht])γ(y 2 [Hty 1 ]) • • • γ(y k [Hty 1 • • • y k-1 ]) = γ(t, y 1 )γ(ty 1 , y 2 ) • • • γ(ty 1 • • • y k-1 , y k ) = γ(1, t)γ(t, y 1 )γ(ty 1 , y 2 ) • • • γ(ty 1 • • • y k-1 , y k )γ(ty 1 • • • y k , t-1 ) since both γ(1, t) and γ(ty 1 • • • y k , t -1 ) are trivial. Thus the relations we obtain with our method are the same as those given in [LS01, Proposition 4.1], and Proposition A.13 applied to a groupoid of cosets gives a new proof of [LS01, Proposition 4.1].

  . . , d n in increasing order (resp. d

* 1 , . . . , d * n in decreasing order). Let d be the order of ζ q h . The integer d is regular for W and we know by Theorem 2.6 that the degrees (resp. codegrees) of W c q are precisely the d i (resp. the d * i ) that are divisible by d. Since d divides h = d i + d * i for all i ∈ [[1, n]], we have that d divides d i if and only if it divides d * i . Thus, if the degrees of W c q are d i 1 , . . . , d i k , then the codegrees of W c q are d * i 1 , . . . , d * i k . In particular for j ∈ [[1, k]], we have d

  = s 10 s 12 s 13 s 18 u 2 = s 9 s 10 s 12 s 16 u 3 = s 3 s 11 s 12 s 16 u 4 = s 7 s 8 s 14 s 15 = s 3 s 7 s 11 s 12 u 6 = s 3 s 4 s 7 s 11 u 7 = s 3 s 4 s 7 s 17 u 8 = s 3 s 7 s 12 s 17.As we want links between the different presentations we obtain for each representative, we give explicit isomorphisms between the different groups B 31 (u i , u i ). For this we use the following graph in B 31

	by the following elements of W .      u 1      u 3 u 1 o o O O / / u 5 u 8 O O u 5 u 2 u 4 u 6	/ / u 7	15 2 is given
	   	   

The remainder of this section is devoted to the study of presentations of B(G 31 ). Let u be an object of C 31 , we have an isomorphism B 31 (u, u) ≃ B(G 31 ), which sends atomic loops to braided reflections.

On the one hand, we build a conjectural presentation of B 31 (u, u) with atomic loops as generators. On the other hand, the Reidemeister-Schreier method for groupoids (cf. Annex A.2) gives a presentation of B 31 (u, u) which we know holds. We then prove that these two presentations are equivalent.

4.1. The method. Let u be an object of B 31 . We start by considering the submonoid L + u of C 31 (u, u) generated by atomic loops of u. Note that we have L + u = C 31 (u, u) in general. Our first goal is to construct a (conjectural) group presentation using L + u . We consider the following algorithm. Output: If λ(s), λ(t) admit a common right-multiple in L + u , then the output is a pair of words θ(λ(s), λ(t)), θ(λ(t), λ(s)) in L + u such that λ(s)θ(λ(s), λ(t)) = λ(t)θ(λ(t), λ(s)). No output otherwise. put i := 1 compute the set S i of words of length i in L + u while λ(s)m 1 = λ(s)m 2 in C 31 for all (m 1 , m 2 ) ∈ S i × S i do put i := i + 1 compute the set S i of words of length i in L + u end while

) ∈ S such that m 1 is the lowest possible in the lexicographic order, and m 2 is the lowest possible in the lexicographic order among the words m such that (m 1 , m) ∈ S.

Algorithm 4.1, running on two atomic loops of u, terminates if and only if they admit a common right-multiple in L + u . The fact that it terminates on every pair of atomic loops of u (which we checked by computer using the data of Section 4.2) proves that all pair of atomic loops of u admit a common right-multiple. We now consider the following presentation:

-The set of generators is a set X u := {λ ′ (s)} in bijection with atomic loops of u in L + u ⊂ C 31 (u, u). -The set R u is given by relations of the form

where θ(λ ′ (s), λ ′ (t)) is the same word as θ(λ(s), λ(t)) but with letters in X u . We define The goal of this section is to show the following theorem Theorem 4.2. Let u ∈ Ob(C 31 ). The natural map from X u to the set of atomic loops of u induces a group isomorphism H u ≃ B 31 (u, u). In particular, X u | R u gives a positive homogeneous presentations of B(G 31 ) with braided reflections as generators.

We first notice that, by definition of R u and X u , the natural map from X u to the set of atomic loops of u induces a group morphism f u : H u → B 31 (u, u). We only have to show that the said morphism is an isomorphism. We show this by case by case analysis on the objects of C 31 .

Lemma 4.3. Let u ∈ Ob(C 31 ). If f u is an isomorphism, then f φ 2 (u) is also an isomorphism. In particular we only need to show that Theorem 4.2 holds for a system of representative of φ 2 -orbits in B 31 .

Proof. The automorphism φ 2 gives an isomorphism between C 31 (u, u) and C 31 (φ(u), φ(u)). Because of Lemma 3.5, φ 2 induces a bijection between the sets of atomic loops of u and of φ(u). We obtain that φ 2 induces bijections between X u and X φ(u) , and between R u and R φ(u) , respectively. Thus H u ≃ H φ(u) which shows the claim.

Remark 4.4. The atoms of the monoid H + u are in bijection with atomic loops of u. In particular H + u is not isomorphic to the monoid C 31 (u, u) in general. As a matter of fact, we will see that the monoid H + u is never cancellative, and thus cannot be isomorphic to either L + u or C 31 (u, u). In order to prove Theorem 4.2 for the object u, we first compute a presentation of B 31 (u, u) by the Reidemeister-Schreier method for groupoids (cf. Appendix A.2). We start from a presentation of the groupoid B 31 , for instance that of Theorem 3.27.

As we want atomic loops to appear as generators, we need to choose the Schreier transversal accordingly. We use the following lemmas concerning atomic loops in the category C 31 . Lemma 4.5. Let s be an atom of C 31 , the atomic loop λ(s) has length two in C 31 . Proof. Let s := (a, b). We denote by u the source of s, and by v its target. The morphism s # is given by (a, b) # = (a c 8 b -1 , b). We know that

and so a c 8 b -1 = a abc 7 a = a (ba c 8 )c 7 = a vc 7 . The morphism s ## is then given by ((a vc 7 ) avc 7 , b). Because of Lemma 2.8, a and a vc 7 commute, so s ## = (a (vc 7 ) 2 , b). Since (vc 7 ) 2 = c 15 is central, we have s ## = s as claimed.

Lemma 4.6. Let u be an object of B 31 . There is a Schreier transversal T rooted in u and containing all atoms with source u. In particular the atomic loops of u appear as generators of the presentation of B 31 (u, u) induced by T .

Proof. First, note that if a Schreier transversal T contains an atom s with source u, then Lemma 4.5 gives that γ(s # ) = ss # = λ(s) with the notation of Lemma A.12. Now, thanks to Proposition 3.3, all atoms with source u have different targets. We can thus consider a Schreier transversal T rooted in u and containing all atoms with source u.

Let T be a Schreier transversal rooted in u and containing all atoms with source u. Let S * | R * denote the presentation of B 31 (u, u) obtained by the Reidemeister-Schreier method applied to T and to the presentation of Theorem 3.27. Of course, the presentation S * | R * is quite redundant. We first want to show that every element of S * can be expressed as a word in the atomic loops. For this we repeatedly apply Tietze transformations, as in Algorithm 4.3.

Algorithm 4.3 Reduction of generators

Input: A group presentation S | R and a subset S ′ of S Output: A group presentation S ′ | R ′ equivalent to the first by Tietze transformation, or no output. while S ′ = S do choose r ∈ R a relator with only one letter a not belonging to S ′ ∪ S ′-1 replace in R every occurrence of a by its expression in S ′ using the relator r. remove the relator r from R remove the letter a from S. end while return the presentation S ′ | R

The fact that this algorithm terminates for each object of B 31 , which is again checked by computer, proves the following result Proposition 4.7. Let u be an object of B 31 . The atomic loops of u generate the group B 31 (u, u). In particular the natural morphism H u → B 31 (u, u) is surjective.

By applying Algorithm 4.3 to the presentation S * | R * , we obtain a presentation X u | R ′ u of the group B 31 (u, u). In order to prove Theorem 4.2 for the object u, it is sufficient to prove that every relator of R ′ u is in fact trivial in H u . This will prove that the morphism

Because the defining presentation of H + u is right-complemented, we can consider a rightreversing algorithm in the sense of [START_REF] Dehornoy | Foundations of Garside Theory[END_REF]Definition 4.21]. The main idea is that the relation aθ(a, b) = bθ(b, a) implies a -1 b = θ(a, b)θ(b, a) -1 . We can use these type of relation in order to simplify words in In particular, we get that (stuvw) 6 = ∆ 15 2 (u 1 ) generates Z(B 31 (u 1 , u 1 )). 4.3.2. Orbit of u 2 . The object u 2 admits 7 atomic loops a, b, c, s, t, v, w, with relations as follows:

The last line of relations can be omitted, as it is implied by the others. The monoid

given by this presentation is not cancellative: we have bwab = abwa and cbwab = cabwa in H + u 2 . Thus H + u 2 cannot be a Garside monoid. The submonoid L + u 2 of C 31 (u 2 , u 2 ) generated by the atomic loops is cancellative, but it doesn't admit right-lcms: we know that aba = bab is the shortest common multiple of a and b. If right-lcms exists in L + u 2 , then aba must be the right-lcm of a and b. We would then have aba abwa ⇒ a wa Which does not hold in L + u 2 (it doesn't even hold in C 31 (u 2 , u 2 )). The relations defining H + u 2 give in particular b = v -1 sv = s v and c = a v in H u . By deleting these generators, we get that B 31 (u 2 , u 2 ) is generated by a, s, t, v, w, with relations as follows: Again, we know that neither the monoid given by this presentation, neither the submonoid of L + u 2 generated by a, s, t, v, w are Garside monoids. The morphisms ϕ 2,1 and ϕ 2,1 are given by ϕ 2,1 :

The maximal roots of the full-twist are given by Maximal regular number d 20 24 d-th root of ∆ 60 wastvw wastv

In particular, we get that (wastv) 6 = ∆ 15 2 (u 2 ) generates Z(B 31 (u 2 , u 2 )).

4.3.3. Orbit of u 3 . The object u 3 admits 7 atomic loops d, e, f, t, u, v, w, with relations as follows:

The last line of relations can be omitted, as it is implied by the others. The monoid

given by this presentation is not cancellative: we have tudt = udtu and wtudt = wudtu in H + u 3 . Thus H + u 3 cannot be a Garside monoid. The submonoid L + u 3 of C 31 (u 3 , u 3 ) generated by the atomic loops is cancellative, but it doesn't admit right-lcms: we know that utu = tut is the shortest common multiple of t and u. If right-lcms exists in L + u 3 , then utu must be the right-lcm of u and t. We would then have tut tudt ⇒ t dt Which does not hold in L + u 3 (it doesn't even hold in C 31 (u 3 , u 3 )). The relations defining H + u 3 give in particular f := w t and e = f u = w tu in H u 3 . By deleting these generators, we get that B 31 (u 3 , u 3 ) is generated by d, t, u, v, w, with relations as follows:

The monoid given by this presentation is not cancellative: we have tudt = udtu like in H + u 3 . The morphisms ϕ 3,1 and ϕ 1,3 are given by ϕ 3,1 :

The maximal roots of the full-twist are given by Maximal regular number d 20 24 d-th root of ∆ 60 dwtuvd dwtuv

In particular, we get that (dwtuv) 6 = ∆ 15 2 (u 3 ) generates Z(B 31 (u 3 , u 3 )). 4.3.4. Orbit of u 4 . The object u 4 admits 12 atomic loops g, h,k,l,m,n,o,p,s,t,u,v, with relations as follows: The last three lines of relations can be omitted, as they are implied by the others. The monoid H + u 4 given by this presentation is not cancellative: we have hpsh = pshp and mhpsh = mpshp in H + u 4 . Thus H + u 4 cannot be a Garside monoid. The submonoid L + u 4 of C 31 (u 4 , u 4 ) generated by the atomic loops is cancellative, but it doesn't admit right-lcms: we know that hph = php is the shortest common multiple of h and p. If right-lcms exists in L + u 4 , then hph must be the right-lcm of h and p. We would then have hph hpsh ⇒ h sh Which does not hold in L + u 4 (it doesn't even hold in C 31 (u 4 , u 4 )). The relations defining

in H u 4 . By deleting these generators, we get that B 31 (u 4 , u 4 ) is generated by g, h, n, o, v, with relations as follows:

The monoid given by this presentation does not admit right-lcms. We have vgnv = nvgn and vnv = nvn are common right-multiples of v and n, but their longest common divisor is nv,which is not a common righ-multiple of n and v. This also proves that the submonoid of C 31 (u 4 , u 4 ) generated by g, h, n, o, v does not admit right-lcms. The morphisms ϕ 4,1 and ϕ 1,4 are given by ϕ 3,1 :

and ϕ 1,3 :

The maximal roots of the full-twist are given by Maximal regular number d 20 24 d-th root of ∆ 60 hgnvoh hgnvo

In particular, we get that (hgnvo) 6 = ∆ 15 2 (u 4 ) generates Z(B 31 (u 4 , u 4 )). We also have (hgnvo) 2 = ∆ 5 (u 4 ) in this case. 4.3.5. Orbit of u 5 . The object u 5 admits 10 atomic loops b, f, g, n, p, s, t, u, v, w, with relations as follows: The last line of relations can be omitted, as it is implied by the others. The monoid H + u 5

given by this presentation is not cancellative: we have uf su = f suf and puf su = pf suf in H + u 5 . Thus H + u 5 cannot be a Garside monoid.

The submonoid L + u 5 of C 31 (u 5 , u 5 ) generated by the atomic loops is cancellative, but it doesn't admit right-lcms: we know that uf u = f uf is the shortest common multiple of f and u. If right-lcms exists in L + u 5 , then uf u must be the right-lcm of u and f . We would then have uf u uf su ⇒ u su Which does not hold in L + u 5 (it doesn't even hold in C 31 (u 5 , u 5 )). The relations defining H + u 5 give in particular b = s v , f = w t , g = u vt , n = u v , p = u t in H u 5 . By deleting these generators, we get that B 31 (u 5 , u 5 ) is generated by s, t, u, v, w, with the same relations as in H + u 1 : we recover once again the obtained in Section 4.3.1. The morphisms ϕ 5,1 and ϕ 1,5 are given by ϕ 5,1 :

The maximal roots of the full-twist are the same that were given for the object u 1 . 4.3.6. Orbit of u 6 . The object u 6 admits 10 atomic loops b, f, g, n, o, p, q, r, s, v, with relations as follows: given by this presentation is not cancellative: we have nvrn = vrnv and onvrn = ovrnv in H + u 6 . Thus H + u 6 cannot be a Garside monoid. The submonoid L + u 6 of C 31 (u 6 , u 6 ) generated by the atomic loops is cancellative, but it doesn't admit right-lcms: we know that nvn = vnv is the shortest common multiple of n and v. If right-lcms exists in L + u 6 , then nvn must be the right-lcm of n and v. We would then have nvn nvrn ⇒ n rn Which does not hold in L + u 6 (it doesn't even hold in C 31 (u 6 , u 6 )). The relations defining H + u 6 give in particular b = s v , f = q n , o = n s , p = v g , r = q g in H u 6 . By deleting these generators, we get that B 31 (u 6 , u 6 ) is generated by g, n, q, s, v, with relations as follows: A category presentation is a pair (S, R), where S is an oriented graph, and R is a set of relations on S * . We call S the generators and R the relations.

If (S, R) is a category presentation, the quotient category S/ ≡ + R is denoted by S | R + . Remark A.6. If we consider a graph S with one object, we recover the classical notion of monoid presentation.

Let (S, R) be a categorical presentation, and let C be a category. Any morphisms of oriented graphs φ : S → C such that

induces a unique functor from the presented category S | R + to C. A first application of this property is the following lemma.

Lemma A.7. Let C := S | R + be a presented category. If R consists of relations between paths of the same length in S * , then the category C is homogeneous. We then say that S | R + is a homogeneous presentation.

Proof. First, we note that the category S * is homogeneous. The length functor is given by the length of the paths. By assumption, this functor induces a well-defined functor from C to (Z 0 , +), such that elements of S are sent to 1. This functor is a length functor for C since it is generated by S.

The notion of categorical presentation is useful for defining enveloping groupoids: let C = S | R + be a presented category. We consider S a formal copy of S, with source and target reversed. We consider the set I(S) of relations on S ⊔ S defined by This universal property of the enveloping groupoids ensures that it depends only on C and not on its presentation. By convention, if C = S | R + is a presented category, the presentation of the enveloping groupoid of C will be denoted by S | R .

Remark A.9. Every category C admits a standard presentation, where the generators are all the morphisms in C, and the relations are all couples (f g, h) for f, g, h ∈ C satisfying f g = h. Thus the enveloping groupoid can be defined for any category.

A.2. Schreier transversal and presentation. Let G = S | R be a presented connected groupoid. We denote by F(S) := S | ∅ the free groupoid on the graph S. The identity S → S induces a quotient map ϕ : F(S) → G. Just like for free groups, a morphism in F(S) is represented by a unique reduced path , that is a path comporting no subpath of the form ss -1 or s -1 s for s ∈ S.

Definition A.10. Let u be an object of F(S). A Schreier transversal of F(S) rooted in u is a family of reduced paths {m v } v∈Ob(F (S)) satisfying: -For all object v of S, the path m v has source u and target v.

-The family {m v } is stable under prefix. It particular m u = 1 u .

Remark A.11. Since G is connected, it is also the case of S and F(S). In particular, a Schreier transversal of F(S) rooted in u exists for all object u of F(S).

Let u 0 be an object of F(S), and let {m v } be a Schreier transversal in F(S) rooted in u 0 . For s ∈ S(u, v), we define γ(s) := m u sm -1 v ∈ F(S)(u 0 , u 0 ). Let S 1 be the set of all elements γ(s) = 1 u 0 for s ∈ S.

Lemma A.12. The group G(u 0 , u 0 ) is generated by the ϕ(γ(s)) for γ(s) ∈ S 1 .

Proof. Let g ∈ G(u 0 , u 0 ). Since G is generated by S, we can write

We denote by u i the target of

In F(S) we have

Thus, we have g = ϕ(γ(s 1 ))

As G is a groupoid, every relation defining G can be rewritten as a relation of the form

and u is the source of s ε 1 1 . Proposition A.13. (Reidemeister-Schreier method for groupoids) Let S * be a set of elements γ(s) * in one-to-one correspondence with those of S 1 . Let also R * be the set of all relations γ * (s 1 )

The map S * → G(u 0 , u 0 ) sending γ(s) * to ϕ(γ(s)) induces an isomorphism of groups between G * := S * | R * and G(u 0 , u 0 ).

Proof. First, we prove that the map γ(s) * → ϕ(γ(s)) is compatible with the set of relations

u 0 , u 0 ) be the morphism induced by γ(s) * → ϕ(γ(s)). We know that π is surjective by Lemma A.12.

Conversely, the map S → S * sending s to γ(s) * induces a functor φ : F(S) → G * . Let v be an object of F(S). We show that φ sends m v to 1 by induction on the length of m v as an S-path. First if m v = s ε 1 1 has length 1, we have φ(s

k-1 and φ(m v ′ ) = 1 by induction hypothesis. As we also have φ(s

. By definition of the set R * , φ induces a functor φ : G → G * . We call ι the restriction of this functor to G(u 0 , u 0 ). Let γ(s) * be a generator of G * , with s ∈ S(u, v). We have

So ι • π induces the identity on the generators of G * . We get ι • π = 1 G * and π is injective.

Corollary A.14. If G is a finitely presented groupoid, then for every object u of G, the group G(u, u) is finitely presented.

A.3. The particular case of a subgroup of a presented group. We explain rapidly how our result on groupoids can be used to recover the classical result of Reidemeister and Schreier about presentation of subgroups.

Definition A.15. Let G be a group, with H a subgroup. The groupoid of cosets G H of G and H is the groupoid defined by -The objects are the right-cosets of H in G.

-Morphisms between two cosets Hg and Hg ′ are elements x of G such that Hgx = Hg ′ . -Composition is given by the product in G.

The underlying oriented graph of the groupoid of cosets is simply the graph of the action of G on the right-cosets of H in G. For each coset Hg and each element x of G, we denote by x[Hg] the unique morphism Hg → Hgx corresponding to the action of the element x.

By definition of G H , there is a natural functor π : G H → G, sending a morphism x[Hg] to x. For two cosets Hg and Hg ′ , the set of morphisms between Hg and Hg ′ is given by G H (Hg, Hg ′ ) = {x ∈ G | Hgx = Hg ′ } = g -1 Hg ′ .

In particular, we obtain that G H (H, H) = H and H is an automorphism group in G H .

Since the natural action of G on H \ G is transitive, the groupoid G H is connected. In particular we can apply the Reidemeister-Schreier method to obtain a presentation of H from one of G H . Now, if X, R is a presentation of the group G. We denote by F (X) the free group over the set X. We call X the subgraph of G H made of those morphisms in G H whose image under π lie in X. The restriction of π to X induces a functor π : F(X ) → F (X). The inclusion X → G H (resp. X → G) induces a functor ϕ : F(X ) → G H (resp. φ : F (X) → G). We have the following commutative square

k be a word in F (X), and let Hg be a right-coset. There is a unique path in F(X ) which starts at Hg and whose image under π is m. This path is given by Proof. Let G be the groupoid presented by X and R. Since the graph X generates G H , the natural functor G → G H is surjective on morphisms. Consider now two paths in F(X )

By definition, saying that these two paths induce the same morphism in G H amounts to say that they share the same source and that the two words π(m) and π(m ′ ) induce the same element in G.

Suppose that m and m ′ induce the same element in G H . We have π(m) = π(m ′ ) in G, and there is a finite sequence of words m 1 , . . . , m p in F (X) such that -m 1 = π(m), m p = π(m ′ ) -For i ∈ [[1, p -1]], m i is equivalent to m i+1 by the use of one relation in R.