Interval groups related to finite Coxeter groups I

We elaborate presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type $D_n$. This is the only case of the infinite families of finite Coxeter groups that admits proper quasi-Coxeter elements. The presentations we obtain are over a set of generators in bijection with what we call Carter generating set, and the relations are those defined by the related Carter diagram along with a twisted or a cycle commutator relator, depending on whether the quasi-Coxeter element is a Coxeter element or not. In a subsequent work, we complete our analysis to cover all the exceptional cases of finite Coxeter groups, and establish that almost all the interval groups related to proper quasi-Coxeter elements are not isomorphic to the related Artin groups, hence establishing a new family of interval groups with nice presentations. Alongside the proof of the main results of this paper, we establish important properties related to the dual approach to Coxeter and Artin groups

Interval groups related to finite Coxeter groups, Part II

We provide a complete description of the presentations of the interval groups related to quasi-Coxeter elements in finite Coxeter groups. In the simply laced cases, we show that each interval group is the quotient of the Artin group associated with the corresponding Carter diagram by the normal closure of a set of twisted cycle commutators, one for each 4-cycle of the diagram. Our techniques also reprove an analogous result for the Artin groups of finite Coxeter groups, which are interval groups corresponding to Coxeter elements. We also analyse the situation in the non-simply laced cases, where a new Garside structure is discovered. Furthermore, we obtain a complete classification of whether the interval group we consider is isomorphic or not to the related Artin group. Indeed, using methods of Tits, we prove that the interval groups of proper quasi-Coxeter elements are not isomorphic to the Artin groups of the same type, in the case of $D_n$ when $n$ is even or in any of the exceptional cases. In [BHNR22], we show using different methods that this result holds for type $D_n$ for all $n \geq 4$

INTERVAL GARSIDE STRUCTURES FOR THE COMPLEX BRAID GROUPS B(e, e, n)

Neaime G. INTERVAL GARSIDE STRUCTURES FOR THE COMPLEX BRAID GROUPS B(e, e, n). TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. 2019;372(12):8815-8848.We define geodesic normal forms for the general series of complex reflection groups G(e, e, n). This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of G(e, e, n) over the generating set of the presentation of Corran-Picantin. Using these geodesic normal forms, we construct intervals in G(e, e, n) that are proven to be lattices. This gives rise to interval Garside groups. We determine which of these groups are isomorphic to the complex braid group B(e, e, n) and get a complete classification. For the other Garside groups that appear in our construction, we provide some of their properties and compute their second integral homology groups in order to understand these new structures

Structures d'Intervalles, algèbres de Hecke et représentations de Krammer des goupes de tresses complexes B(e,e,n)

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.Nous définissons des formes normales géodésiques pour les séries générales des groupes de réflexions complexes G(de,e,n). Ceci nécessite l'élaboration d'une technique combinatoire afin de déterminer des décompositions réduites et de calculer la longueur des éléments de G(de,e,n) sur un ensemble générateur donné. En utilisant ces formes normales géodésiques, nous construisons des intervalles dans G(e,e,n) qui permettent d'obtenir des groupes de Garside. Certains de ces groupes correspondent au groupe de tresses complexe B(e,e,n). Pour les autres groupes de Garside, nous étudions certaines de leurs propriétés et nous calculons leurs groupes d'homologie sur Z d'ordre 2. Inspirés par les formes normales géodésiques, nous définissons aussi de nouvelles présentations et de nouvelles bases pour les algèbres de Hecke associées aux groupes de réflexions complexes G(e,e,n) et G(d,1,n) ce qui permet d'obtenir une nouvelle preuve de la conjecture de liberté de BMR (Broué-Malle-Rouquier) pour ces deux cas. Ensuite, nous définissons des algèbres de BMW (Birman-Murakami-Wenzl) et de Brauer pour le type (e,e,n). Ceci nous permet de construire des représentations de Krammer explicites pour des cas particuliers des groupes de tresses complexes B(e,e,n). Nous conjecturons que ces représentations sont fidèles. Enfin, en se basant sur nos calculs heuristiques, nous proposons une conjecture sur la structure de l'algèbre de BMW

Geodesic normal forms and Hecke algebras for the complex reflection groups G(de, e, n)

Neaime G. Geodesic normal forms and Hecke algebras for the complex reflection groups G(de, e, n). JOURNAL OF PURE AND APPLIED ALGEBRA. Accepted;225(2): 106500.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. (C) 2020 Elsevier B.V. All rights reserved

Interval Garside structures related to the affine Artin groups of type (A)over-tilde

Neaime G. Interval Garside structures related to the affine Artin groups of type (A)over-tilde. Journal of Algebra. 2022;607(Part B):411-436.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) over tilde 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. (c) 2020 Elsevier Inc. All rights reserved

Isomorphism and non-isomorphism for interval groups of type D

We consider presentations that were derived in [3] for the interval groups associated with proper quasi-Coxeter elements of the Coxeter group W(Dn). We use combinatorial methods to derive alternative presentations for the groups, and use these new presentations to show that the interval group associated with a proper quasi-Coxeter element of W(Dn) cannot be isomorphic to the Artin group of type Dn . While the specific problems we solve arise from the study of interval groups, their solution provides an illustration of how techniques indicated by computational observation can be used to derive properties of all groups within an infinite family