This thesis investigates the geometric and topological constraints placed on the quotient space of a Fuchsian or Kleinian group by requiring that the group admits a fundamental domain which is simultaneously a Ford domain and a Dirichlet domain. In the case of Fuchsian groups, a direct correspondence with reflection groups is proved, and this result is used to first find explicitly the 23 non-cocompact arithmetic maximal hyperbolic reflection groups in the group of isometries of the hyperbolic plane, and subsequently to test whether these groups are all congruence. In the case of Kleinian groups, similar results are shown, and some examples of reflection groups are considered. / text

Humphreys (1990) defines a simple root system for a finite reflection group which is an important concept fundamental to the understanding of reflection groups as well as Coxeter complexes and Coxeter groups. The proof that Humphreys uses to establish the existence and uniqueness of these systems follows an indirect method that left portions of the proof as exercises to the reader. I present a more complete and direct proof using different organization and methods.

Crystallographic complex reflection groups are generated by reflections about affine hyperplanes in complex space and stabilize a full rank lattice. These analogs of affine Weyl groups have infinite order and were classified by V.L. Popov in 1982. The classical Braid theorem (first established by E. Artin and E. Brieskorn) asserts that the Artin group of a reflection group (finite or affine Weyl) gives the fundamental group of regular orbits. In other words, the fundamental group of the space with reflecting hyperplanes removed has a presentation mimicking that of the Coxeter presentation; one need only remove relations giving generators finite order. N.V Dung used a semi-cell construction to prove the Braid theorem for affine Weyl groups. Malle conjectured that the Braid theorem holds for all crystallographic complex reflection groups after constructing Coxeter-like reflection presentations. We show how to extend Dung's ideas to crystallographic complex reflection groups and then extend the Braid theorem to some groups in the infinite family [G(r,p,n)]. The proof requires a new classification of crystallographic groups in the infinite family that fail the Steinberg theorem.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / A presente tese se propõe descrever e analisar as relações interpessoais entre mulheres, homens e profissionais das áreas do direito, psicologia e serviço social envolvidos na institucionalização da Lei 11.340/06, conhecida como Lei Maria da Penha (LMP), que rege hoje no Brasil os crimes de violência doméstica e familiar contra a mulher. Inicialmente é apresentada uma concisa contextualização da LMP e do campo de debate em que se insere, além das principais mudanças introduzidas por ela em relação às antigas políticas. As controvérsias que a lei vem levantando e as modificações sofridas em pouco tempo de existência, apontam para as dificuldades em se estabelecer um consenso por parte dos operadores e formuladores da lei quanto à percepção da violência doméstica e familiar contra a mulher como um crime e quanto a sua justa punição. Não só os operadores, mas as feministas também se envolveram em controvérsias teóricas em torno da distinção entre as definições de violência contra a mulher e de crime de violência contra a mulher. O esforço de se avançar na análise dessas categorias se justifica pelas dificuldades e impasses que se observam nas práticas institucionais na implementação da LMP. Essas práticas são descritas e analisadas a partir da incursão etnográfica em dois campos. No Juizado de Violência Doméstica e Familiar contra a Mulher participei de encontros de um grupo de reflexão para homens autores de violência, assisti audiências, entrevistei profissionais e dezoito homens envolvidos com a LMP. Os sentidos em disputa que os vários atores sociais constroem relativos aos conflitos violentos da intimidade ali julgados e suas relações com o exercício da(s) masculinidade(s) são discutidos. As informações do outro campo, um Centro de Referência da Mulher, provêm das observações de cenas do cotidiano institucional, do acompanhamento de atendimentos às usuárias, da participação em grupos de reflexão para as mulheres vítimas de violência e de entrevistas com duas profissionais e dezessete mulheres. É enfatizado o caráter de intervenção pedagógica das instituições que objetivam promover mudanças em caracteres considerados como de gênero de homens (a agressividade) e mulheres (a passividade) que estariam influenciando o engendramento e manutenção das violências. Nas entrevistas é ressaltado o que ecoa, corrobora, complementa, destoa ou mesmo mostra novos ângulos do que é apreendido nos grupos (confronto entre os sentidos da violência e suas relações com o que é ser homem e o que é ser mulher) e nas audiências (tendência à vitimização e à relativização dos papéis de vítima e acusado). Independente dos embates e controvérsias suscitadas, pode-se afirmar que a violência contra a mulher ingressou no mundo da lei nacional trazendo com sua institucionalização uma intensa circulação de diferentes sentidos, lógicas e moralidades que (re)modelam convenções sobre as relações de gêneros e sua influência sobre a citada violência. / This paper is designed to describe and analyze the interpersonal relationships among women, men and professionals in the fields of law, psychology and social service involved in the institutionalization of Law 11.340/06. Also known as the Maria da Penha Law (MPL), it currently governs in Brazil crimes of domestic violence against women. First, the paper presents a concise contextualization of the LMP and the field of debate in which it operates, besides the main changes introduced by the law in relation to the previous policy. The controversies the law has been raising and the modifications it underwent in a short period of existence bring obstacles to establish a consensus among operators and law makers in regards to the perception of domestic and family violence against women as a crime as well as their just punishment. Not only operators, but feminists were also involved in theoretical controversies surrounding the distinction between the definition of "violence against women" and "crime of violence against women." The effort to advance in the analysis of these categories is justified by the difficulties and dilemmas observed in institutional practices during the implementation of MPL. These practices are described and analyzed from the ethnographic incursion into two fields. In the Court of Domestic and Familiar Violence against Women, I attended meetings of a reflection group for men who commit violence, I watched hearings, and I interviewed professionals and eighteen men involved with MPL. This research discusses the senses in dispute that the various social actors construct concerning violent conflicts of the intimacy there judged, and their relations with the exercise of masculinity. Information collected from the other field, a Reference Center for Women, come from observations of everyday institutional scenes, from the follow up with female user of the Reference Center, and from participation in reflection groups for women victims of violence, besides interviews with two professionals and seventeen women. This paper emphasizes the character of pedagogical intervention of the institutions that aim to promote changes in characteristics considered typical of males (aggressiveness) and females (passive) that would be influencing the engendering and maintenance of violence. In the interviews it is emphasized what echoes, corroborates, complements, distunes or even shows new angles of what is comprehended in the groups (confrontation between the senses of violence and its relations with what means to be a man or to be a women) and in the hearings (tendency to victimization and relativization of the roles of the victim and the assaulter). Regardless the conflicts and controversies that are raised, it can be concluded that "violence against women" entered the world of national law with its institutionalization bringing an intense circulation of different meanings, logic and morals that are shaping and reshaping conventions on gender relations and its influence on the above-mentioned violence. Keywords: Maria da Penha Law. Violence against women. Reflection group of men perpetrators of violence. Reflection group of women victims of violence.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / A presente tese se propõe descrever e analisar as relações interpessoais entre mulheres, homens e profissionais das áreas do direito, psicologia e serviço social envolvidos na institucionalização da Lei 11.340/06, conhecida como Lei Maria da Penha (LMP), que rege hoje no Brasil os crimes de violência doméstica e familiar contra a mulher. Inicialmente é apresentada uma concisa contextualização da LMP e do campo de debate em que se insere, além das principais mudanças introduzidas por ela em relação às antigas políticas. As controvérsias que a lei vem levantando e as modificações sofridas em pouco tempo de existência, apontam para as dificuldades em se estabelecer um consenso por parte dos operadores e formuladores da lei quanto à percepção da violência doméstica e familiar contra a mulher como um crime e quanto a sua justa punição. Não só os operadores, mas as feministas também se envolveram em controvérsias teóricas em torno da distinção entre as definições de violência contra a mulher e de crime de violência contra a mulher. O esforço de se avançar na análise dessas categorias se justifica pelas dificuldades e impasses que se observam nas práticas institucionais na implementação da LMP. Essas práticas são descritas e analisadas a partir da incursão etnográfica em dois campos. No Juizado de Violência Doméstica e Familiar contra a Mulher participei de encontros de um grupo de reflexão para homens autores de violência, assisti audiências, entrevistei profissionais e dezoito homens envolvidos com a LMP. Os sentidos em disputa que os vários atores sociais constroem relativos aos conflitos violentos da intimidade ali julgados e suas relações com o exercício da(s) masculinidade(s) são discutidos. As informações do outro campo, um Centro de Referência da Mulher, provêm das observações de cenas do cotidiano institucional, do acompanhamento de atendimentos às usuárias, da participação em grupos de reflexão para as mulheres vítimas de violência e de entrevistas com duas profissionais e dezessete mulheres. É enfatizado o caráter de intervenção pedagógica das instituições que objetivam promover mudanças em caracteres considerados como de gênero de homens (a agressividade) e mulheres (a passividade) que estariam influenciando o engendramento e manutenção das violências. Nas entrevistas é ressaltado o que ecoa, corrobora, complementa, destoa ou mesmo mostra novos ângulos do que é apreendido nos grupos (confronto entre os sentidos da violência e suas relações com o que é ser homem e o que é ser mulher) e nas audiências (tendência à vitimização e à relativização dos papéis de vítima e acusado). Independente dos embates e controvérsias suscitadas, pode-se afirmar que a violência contra a mulher ingressou no mundo da lei nacional trazendo com sua institucionalização uma intensa circulação de diferentes sentidos, lógicas e moralidades que (re)modelam convenções sobre as relações de gêneros e sua influência sobre a citada violência. / This paper is designed to describe and analyze the interpersonal relationships among women, men and professionals in the fields of law, psychology and social service involved in the institutionalization of Law 11.340/06. Also known as the Maria da Penha Law (MPL), it currently governs in Brazil crimes of domestic violence against women. First, the paper presents a concise contextualization of the LMP and the field of debate in which it operates, besides the main changes introduced by the law in relation to the previous policy. The controversies the law has been raising and the modifications it underwent in a short period of existence bring obstacles to establish a consensus among operators and law makers in regards to the perception of domestic and family violence against women as a crime as well as their just punishment. Not only operators, but feminists were also involved in theoretical controversies surrounding the distinction between the definition of "violence against women" and "crime of violence against women." The effort to advance in the analysis of these categories is justified by the difficulties and dilemmas observed in institutional practices during the implementation of MPL. These practices are described and analyzed from the ethnographic incursion into two fields. In the Court of Domestic and Familiar Violence against Women, I attended meetings of a reflection group for men who commit violence, I watched hearings, and I interviewed professionals and eighteen men involved with MPL. This research discusses the senses in dispute that the various social actors construct concerning violent conflicts of the intimacy there judged, and their relations with the exercise of masculinity. Information collected from the other field, a Reference Center for Women, come from observations of everyday institutional scenes, from the follow up with female user of the Reference Center, and from participation in reflection groups for women victims of violence, besides interviews with two professionals and seventeen women. This paper emphasizes the character of pedagogical intervention of the institutions that aim to promote changes in characteristics considered typical of males (aggressiveness) and females (passive) that would be influencing the engendering and maintenance of violence. In the interviews it is emphasized what echoes, corroborates, complements, distunes or even shows new angles of what is comprehended in the groups (confrontation between the senses of violence and its relations with what means to be a man or to be a women) and in the hearings (tendency to victimization and relativization of the roles of the victim and the assaulter). Regardless the conflicts and controversies that are raised, it can be concluded that "violence against women" entered the world of national law with its institutionalization bringing an intense circulation of different meanings, logic and morals that are shaping and reshaping conventions on gender relations and its influence on the above-mentioned violence. Keywords: Maria da Penha Law. Violence against women. Reflection group of men perpetrators of violence. Reflection group of women victims of violence.

Le but de cette thèse est d’étudier les structures obtenues à partir des groupes de réflexion finis. Ce travail consiste en quatre articles publiés, un article soumis et un article en préparation dont les résultats partiels constituent un chapitre de cette thèse. Dans le premier article, nous présentons une réduction des orbites des groupes de Coxeter finis vers leurs sous-groupes. Nous utilisons des matrices de projection, c’est-à-dire, des applications qui transforment les racines simples d’un groupe de réflexion en les racines simples du sous-groupe associé. Les résultats présentés dans ce papier se concentrent sur les groupes finis de réflexion non crystallographiques. De plus, nous utilisons les polytopes engendrés par le groupe non crystallographique H3 pour illustrer les lois de ramification (branching rules), c’est-à-dire une réduction des orbites des groupes finis de Coxeter. Dans le deuxième article, nous étudions les polytopes avec 60 sommets engendrés par le groupe non crystallographique H3. Nous utilisons la méthode de décoration des diagrammes de Coxeter–Dynkin pour décrire leurs structures en détails et décomposer les sommets en somme des orbits de symétries de dimension inférieure. Le troisième article compare deux notations utilisées pour décrire le polyèdre engendré par le groupe de réflexion. Il s’agit du symbole de Schläfli et de la notation des points dominants. Nous y présentons les avantages de chaque méthode, expliquons les deux approches et nous les illustrons par des exemples. Dans le quatrième article, nous nous concentrons sur le graphène, c’est-à-dire un pavement d’hexagones sur le plan, qui possède de remarquables propriétés quand les sommets sont modélisés par des atomes de carbone. Dans ce travail, nous présentons différentes méthodes pour obtenir du graphène à partir de réseaux (lattices) et des orbites de dimension 3 des groupes finis de réflexion. De plus, une technique de coloriage des hexagones au moyen d’un nombre fini de couleurs est donnée avec une méthode systématique pour raffiner le graphène. Dans le cinquième article, nous utilisons des v fonctions spéciales et les transformations de Fourier pour traiter les données échantillonnées sur un réseau de carrés du groupe de Lie SU(2)×SU(2), relié au groupe de symétrie A1×A1. / The goal of this thesis is to study structures obtained from finite reflection groups. The work is contained in four published papers, one submitted article and a research paper currently in preparation, with partial results presented as a chapter of this thesis. In the first article, we present a reduction of the orbits of finite Coxeter groups to their subgroups. We use projection matrices, that is, mappings that transform the simple roots of a reflection group to the simple roots of the appropriate subgroup. The results presented in this paper focus on non-crystallographic finite reflection groups. Moreover, we use polytopes generated by the non-crystallographic group H3 to illustrate the obtained branching rules, i.e., reductions of orbits of the finite Coxeter groups. In the second article, we study polytopes with 60 vertices, generated by the non-crystallographic group H3. We use a method of decoration of the Coxeter–Dynkin diagram to describe their structure in detail, and decompose their vertices into sums of orbits of lower-dimensional symmetries. The third article compares two notations used to describe polyhedra generated by reflection groups, namely the Schläfli symbol, and the dominant point notation. Here, we present the advantages of each method, we explain the two approaches, and we illustrate them through examples. In the fourth article, we focus on graphene, i.e., a hexagonal tiling of the plane that possesses remarkable properties when the vertices are modelled with carbon atoms. In this work, we present different methods to obtain graphene from lattices and three-dimensional orbits of finite reflection groups. Moreover, a technique to colour the hexagons by a finite number of colours is provided, along with a systematic method to refine the graphene. In the fifth article, we use special functions and Fourier transforms to process data sampled on a square lattice of the Lie group SU(2) × SU(2), related to the A1 × A1 symmetry group.

Cette thèse est consacrée à l’étude des représentations modulaires des algèbres d’Ariki-Koike, et des liens avec la théorie des cristaux et des bases canoniques de Kashiwara via le théorème de catégorification d’Ariki. Dans un premier temps, on étudie, grâce à des outils combinatoires, les matrices de décomposition de ces algèbres en généralisant les travaux de Geck et Jacon. On classifie entièrement les cas d’existence et de non-existence d’ensembles basiques, en construisant explicitement ces ensembles lorsqu’ils existent. On explicite ensuite les isomorphismes de cristaux pour les représentations de Fock de l’algèbre affine quantique Uq(sle). On construit alors un isomorphisme particulier, dit canonique, qui permet entre autres une caractérisation non-récursive de n’importe quelle composante connexe du cristal. On souligne également les liens avec la combinatoire des mots sous-jacente à la structure cristalline des espaces de Fock, en décrivant notamment un analogue de la correspondance de Robinson-Schensted-Knuth pour le type A affine. / This thesis is devoted to the study of modular representations of Ariki-Koike algebras, and of the connections with Kashiwara’s crystal and canonical bases theory via Ariki’s categorification theorem. First, we study, using combinatorial tools, the decomposition matrices associated to these algebras, generalising the works of Geck and Jacon. We fully classify the cases of existence and non-existence of canonical basic sets, and we explicitely construct these sets when they exist. Next, we make explicit the crystal isomorphisms for Fock spaces representations of the quantum affine algebra Uq(sle). We then construct of a particular isomorphism, so-called canonical, which gives, inter alia, a non-recursive description of any connected component of the crystal. We also stress the links with the combinatorics of words underlying the crystal structure of Fock spaces, by describing notably an analogue of the Robinson-Schensted-Knuth correspondence for affine type A.