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The lattice of simples in B ⊕ (3, 3, 3). To prove Theorem 3.7, we will need the two lemmas below, which make use of the notation of the height of an element of B ⊕ (e, e, r) :  

The lattice of simples in B ⊕ (3, 3, 3). To prove Theorem 3.7, we will need the two lemmas below, which make use of the notation of the height of an element of B ⊕ (e, e, r) :  

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Figure 2. The new diagram of type (e, e, r): there are e nodes on the...
Figure 9. The lattice of simples in B ⊕ (3, 3, 3). To prove Theorem...
Figure 10. Proof of Theorem 3.7 with q = q ′ k−1 · · · q ′ 2 and Ht(q)...
A new Garside structure for braid groups of type (e,e,r)
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We describe a new presentation for the complex reflection groups of type (e, e, r) and their braid groups. A diagram for this presentation is proposed. The presentation is a monoid presentation that is shown to give rise to a Garside structure. This structure has since been used in understanding periodic elements, calculating homology and determini...
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... However, in the case of irreducible complex reflection group of rank two, it is Garside. Moreover, the braid groups associated to the groups G(e, e, r) are Garside, as shown in [5] and [13]. In [26], Neaime realised the braid groups associated to the groups G(e, e, r) as interval groups. ...
... In this section, we define the classical braid monoid associated to J-reflection groups and give another presentation for their braid groups (see Proposition 4.6). This presentation coincides with the dual presentation given in [3] when the group is a real reflection group (in our case, a dihedral group), which in turn corresponds to the presentations given in [5], [13] and [26] for groups of the form G(e, e, r). Moreover, whenever the group is a toric reflection group, we retrieve the dual presentation defined in [3] and [23] We call these monoids classical and say that they are associated to the braid groups with the respective presentations. ...
... Relabeling z i by a 2i−1 and x i by a 2i , one retrieves the dual presentation of B * (1, m) = B(I 2 (2m)) given in [3], [5] and [13]. ...
Braid groups of J-reflection groups and associated classical and dual Garside structures
Preprint
  • Oct 2024
The family of J-reflection groups can be seen as a combinatorial generalisation of irreducible rank two complex reflection groups and was introduced by the author in a previous article. In this article, we define the braid groups associated to J-reflection groups, which coincide with the complex braid group when the J-reflection group is finite. We show that the isomorphism type of the braid groups only depend on the reflection isomorphism types of the corresponding J-reflection groups. Moreover, we show that these braid groups are always abstractly isomorphic to circular groups. At the same time, we show that the center of the braid groups is cyclic and sent onto the center of the corresponding J-reflection groups under the natural quotient. Finally, we exhibit two Garside structures for each braid group of J-reflection group. These structures generalise the classical and dual Garside structures (when defined) of rank two irreducible complex reflection groups. In particular, the dual Garside structure of J-reflection groups provides candidates for dual monoids associated to the irreducible complex reflection groups of rank two which do not already have one.
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... An axiomatic setting up was provided in [DP99,Deh02], to study groups that share a similar structure as a class, called Garside groups. Since then, other important classes of groups were proven to be Garside groups, including but not limited to some semi-direct products [CP05], some complex braid groups [Bes15,CP11,CLL15], structure groups of non-degenerate, involutive and braided set-theoretical solutions of the quantum Yang-Baxter equation [Cho10], crystallographic braid groups [MS17] etc. Garside groups are also known to be closed under certain kind of amalgamation products and HNN extensions [Pic22], as well as Zappa-Szép products [GT16]. There are also a number of variations and generalizations of Garside groups, applying to more natural examples -we refer to the book [Deh15] for a comprehensive review. ...
New Garside structures and applications to Artin groups
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Garside groups are combinatorial generalizations of braid groups which enjoy many nice algebraic, geometric, and algorithmic properties. In this article we propose a method for turning the direct product of a group G by Z\mathbb{Z} into a Garside group, under simple assumptions on G. This method gives many new examples of Garside groups, including groups satisfying certain small cancellation condition (including surface groups) and groups with a systolic presentation. Our method also works for a large class of Artin groups, leading to many new group theoretic, geometric and topological consequences for them. In particular, we prove new cases of K(π,1)K(\pi,1)-conjecture for some hyperbolic type Artin groups.
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... s t x y u In this 'steering wheel' diagram, all edges represent Artin relations, that is sus = usu, tut = utu, xuxu = uxux, uyuy = yuyu, and the oriented circle has the same meaning as for the Corran-Picantin presentations of the groups G(e, e, n) (see [12]), namely it symbolizes the relation st = tx = xy = ys, originating from the dual braid monoid of dihedral type I 2 (4) = B 2 . ...
Hecke Algebras of Normalizers of Parabolic Subgroups
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  • ALGEBR REPRESENT TH
In the context of Hecke algebras of complex reflection groups, we prove that the generalized Hecke algebras of normalizers of parabolic subgroups are semidirect products, under suitable conditions on the parameters involved in their definition.
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... Thus, this presentation does not give rise to a Garside structure for B(e, e, n). In [11], Corran and Picantin described another presentation for B(e, e, n) and showed that it gives rise to a Garside structure for B(e, e, n). This presentation consists in attaching a dual presentation of the dihedral group I 2 (e) to standard presentations of type A. ...
... · · · · · · It is also shown in [11] that if one adds the quadratic relations for all the generators of the presentation of Corran-Picantin of B(e, e, n), one obtains a presentation of a group isomorphic to G(e, e, n). It is called the presentation of Corran-Picantin of G(e, e, n). ...
Interval Garside Structures Related to the Affine Artin Groups of Type $\tilde{A}
Preprint
  • Sep 2019
Garside theory emerged from the study of Artin groups and their generalizations. Finite-type Artin groups admit two types of interval Garside structures corresponding to their standard and dual presentations. Concerning affine Artin groups, Digne established interval Garside structures for two families of these groups by using their dual presentations. Recently, McCammond established that none of the remaining dual presentations (except for one additional case) correspond to interval Garside structures. In this paper, shifting attention from dual presentations to other nice presentations for the affine Artin group of type A~\tilde{A} discovered by Shi and Corran--Lee--Lee, I will construct interval Garside structures related to this group. This construction is the first successful attempt to establish interval Garside structures not related to the dual presentations in the case of affine Artin groups.
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... We use the presentation of Corran and Picantin for G(e, e, n) obtained in [26]. The generators and relations of this presentation can be described by the following diagram. ...
... Details about this presentation can be found in the next chapter. In [26], it is shown that if we remove the quadratic relations of this presentation, we get a presentation of the complex braid group B(e, e, n) that we call the presentation of Corran and Picantin of B(e, e, n). The first step is to define geodesic normal forms (words of minimal length) for all elements of G(e, e, n) over the generating set of the presentation of Corran and Picantin. ...
... Let e ≥ 1 and n > 1. We recall the presentation of the complex reflection group G(e, e, n) given in [26]. Definition 2.1.1. ...
Interval Structures, Hecke Algebras, and Krammer's Representations for the Complex Braid Groups B(e,e,n)
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We define geodesic normal forms for the general series of complex reflection groups G(de,e,n). This requires the elaboration of a combinatorial technique in order to determine minimal word representatives and to compute the length of the elements of G(de,e,n) over some generating set. Using these geodesic normal forms, we construct intervals in G(e,e,n) that give rise to Garside groups. Some of these groups correspond to the complex braid group B(e,e,n). For the other Garside groups that appear, we study some of their properties and compute their second integral homology groups. Inspired by the geodesic normal forms, we also define new presentations and new bases for the Hecke algebras associated with the complex reflection groups G(e,e,n) and G(d,1,n) which lead to a new proof of the BMR (Brou\'e-Malle-Rouquier) freeness conjecture for these two cases. Next, we define a BMW (Birman-Murakami-Wenzl) and Brauer algebras for type (e,e,n). This enables us to construct explicit Krammer's representations for some cases of the complex braid groups B(e,e,n). We conjecture that these representations are faithful. Finally, based on our heuristic computations, we propose a conjecture about the structure of the BMW algebra.
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... Let B(de, e, n) denotes the complex braid group attached to G(de, e, n), as defined in [6]. We establish nice presentations for the Hecke algebras H(de, e, n) by using the presentations of Corran-Picantin [10] and Corran-Lee-Lee [9] of the complex braid groups B(e, e, n) and B(de, e, n) for d > 1, respectively. ...
Geodesic Normal Forms and Hecke Algebras for the Complex Reflection Groups G(de,e,n)
Preprint
Full-text available
  • Oct 2018
We establish geodesic normal forms for the general series of complex reflection groups G(de,e,n) by using the presentations of Corran-Picantin and Corran-Lee-Lee of G(e,e,n) and G(de,e,n) for d > 1, respectively. This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of G(de,e,n). Using these geodesic normal forms, we construct natural bases for the Hecke algebras associated with the complex reflection groups G(e,e,n) and G(d,1,n). As an application, we obtain a new proof of the BMR freeness conjecture for these groups.
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... Therefore, it is interesting to search for (possibly various) Garside structures for these groups. For instance, it is shown by Bessis and Corran [4] in 2006, and by Corran and Picantin [26] in 2009 that B(e, e, n) admits Garside structures. It is also shown in [25] that B(de, e, n) admits quasi-Garside structures (the set of divisors of the fundamental element is infinite). ...
... We use the presentation of Corran and Picantin for G(e, e, n) obtained in [26]. The generators and relations of this presentation can be described by the following diagram. ...
... Details about this presentation can be found in the next chapter. In [26], it is shown that if we remove the quadratic relations of this presentation, we get a presentation of the complex braid group B(e, e, n) that we call the presentation of Corran and Picantin of B(e, e, n). The first step is to define geodesic normal forms (words of minimal length) for all elements of G(e, e, n) over the generating set of the presentation of Corran and Picantin. ...
Interval structures, Hecke algebras, and Krammer’s representations for the complex braid groups B(e,e,n)
Thesis
  • Jun 2018
We define geodesic normal forms for the general series of complex reflection groups G(de,e,n). This requires the elaboration of a combinatorial technique in order to determine minimal word representatives and to compute the length of the elements of G(de,e,n) over some generating set. Using these geodesic normal forms, we construct intervals in G(e,e,n) that give rise to Garside groups. Some of these groups correspond to the complex braid group B(e,e,n). For the other Garside groups that appear, we study some of their properties and compute their second integral homology groups. Inspired by the geodesic normal forms, we also define new presentations and new bases for the Hecke algebras associated to the complex reflection groups G(e,e,n) and G(d,1,n) which lead to a new proof of the BMR (Broué-Malle-Rouquier) freeness conjecture for these two cases. Next, we define a BMW (Birman-Murakami-Wenzl) and Brauer algebras for type (e,e,n). This enables us to construct explicit Krammer's representations for some cases of the complex braid groups B(e,e,n). We conjecture that these representations are faithful. Finally, based on our heuristic computations, we propose a conjecture about the structure of the BMW algebra.
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... The homomorphism H i pBpd, d, nqq Ñ H i pBpd, d, n`1qq induced by the natural inclusion Bpd, d, nq ãÑ Bpd, d, n`1q is an epimorphism for i ď n 2 and an isomorphism for i ă n 2 . Notice that there were very few cohomological computations about the homology of complex braid groups of type Bpd, d, nq; in fact, the only known computations (see [CM14], not more than the second homology groups) used methods based on a resolution given in [DL03] for a Garside monoid introduced in [CP11]. This method seems too complicated to be used for higher homology groups. ...
Families of superelliptic curves, complex braid groups and generalized Dehn twists
Preprint
  • May 2018
We consider the universal family EndE_n^d of superelliptic curves: each curve Σnd\Sigma_n^d in the family is a d-fold covering of the unit disk, totally ramified over a set P of n distinct points; Σnd↪End→Cn\Sigma_n^d\hookrightarrow E_n^d\to C_n is a fibre bundle, where CnC_n is the configuration space of n distinct points. We find that EndE_n^d is the classifying space for the complex braid group of type B(d,d,n) and we compute a big part of the integral homology of End,E_n^d, including a complete calculation of the stable groups by means of Poincar\`e series. The computation of the main part of the above homology reduces to the computation of the homology of the classical braid group with coefficients in the first homology group of Σnd,\Sigma_n^d, endowed with the monodromy action. While giving a geometric description of such monodromy of the above bundle, we introduce generalized 1d1\over d-twists, associated to each standard generator of the braid group, which reduce to standard Dehn twists for $d=2.
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... In this section, we recall some presentations by generators and relations of G(e, e, n) and B(e, e, n) based on the results of [3] and [6]. ...
... It is shown in [6] that the group defined by this presentation is isomorphic to B(e, e, n). We call it the presentation of CP (Corran Picantin) of B(e, e, n). ...
... We call it the presentation of CP (Corran Picantin) of B(e, e, n). It is also shown in [6] that: Proposition 2.1. The presentation of CP gives rise to a Garside structure for B(e, e, n) with ...
Interval structures for braid groups B(e,e,n)
Article
  • Jul 2017
Complex braid groups are a generalization of Artin-Tits groups. The general goal is to extend what is known for Artin-Tits groups to other complex braid groups. We are interested in Garside structures that derive from intervals. Actually, we construct intervals in the complex reflection group G(e,e,n) which gives rise to Garside groups. Some of these groups correspond to the complex braid group B(e,e,n). For the other Garside groups that appear, we give some of their properties in order to understand these new structures.
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... For the general series of the G(e, e, n) a nice presentation for B has been obtained by R. Corran and M. Picantin in [5], with generators t i , i ∈ Z/eZ, s 3 , s 4 , . . . , s n and relations t i+1 t i = t j+1 t j , s 3 t i s 3 = t i s 3 t i and ordinary braid relations between the s 3 , . . . ...
... ) = Br 4 we notice that the two actions almost coincide when (λ, f ) belong to the subgroup IΓ ⊂ GT defined in [18]. Indeed, recall from [18] that, when (λ, f ) ∈ IΓ , its natural action on the image of Br 4 modulo center 1 inside the profinite mapping class group Γ [5] 0 composed with Adf (x 45 , x 34 ) satisfies s 2 → f (s ...
Galois actions on complex braid groups
Article
  • Feb 2012
We establish the faithfulness of a geometric action of the absolute Galois group of the rationals that can be defined on the discriminantal variety associated to a finite complex reflection group, and review some possible connections with the profinite Grothendieck-Teichm\"uller group.
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