A concrete description of Hochschild cohomology is the first step toward exploring associative deformations of algebras. In this dissertation, deformation theory, geometry, combinatorics, invariant theory, representation theory, and homological algebra merge in an investigation of Hochschild cohomology of skew group algebras arising from complex reflection groups. Given a linear action of a finite group on a finite dimensional vector space, the skew group algebra under consideration is the semi-direct product of the group with a polynomial ring on the vector space. Each representation of a group defines a different skew group algebra, which may have its own interesting deformations. In this work, we explicitly describe all graded Hecke algebras arising as deformations of the skew group algebra of any finite group acting by the regular representation. We then focus on rank two exceptional complex reflection groups acting by any irreducible representation. We consider in-depth the reflection representation and a nonfaithful rotation representation. Alongside our study of cohomology for the rotation representation, we develop techniques valid for arbitrary finite groups acting by a representation with a central kernel. Additionally, we consider combinatorial questions about reflection length and codimension orderings on complex reflection groups. We give algorithms using character theory to compute reflection length, atoms, and poset relations. Using a mixture of theory, explicit examples, and calculations using the software GAP, we show that Coxeter groups and the infinite family G(m,1,n) are the only irreducible complex reflection groups for which the reflection length and codimension orders coincide. We describe the atoms in the codimension order for the groups G(m,p,n). For arbitrary finite groups, we show that the codimension atoms are contained in the support of every generating set for cohomology, thus yielding information about the degrees of generators for cohomology.

Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references (leaves 49-50). Also available in electronic version. Access restricted to campus users.

The invariant theory of finite reflection groups has rich connections to geometry, topology, representation theory, and combinatorics. We consider finite reflection groups acting on vector spaces over fields of arbitrary characteristic, where many arguments of classical invariant theory break down. When the characteristic of the underlying field is positive, reflections may be nondiagonalizable. A group containing these so-called transvections has order which is divisible by the characteristic of the underlying field, so is in the modular setting. In this thesis, we examine the action on differential derivations, which include products of differential forms and derivations, and identify the structure of the set of invariants under the action of groups fixing a single hyperplane, groups with maximal transvection root spaces acting on vector spaces over prime fields, as well as special linear groups and general linear groups over finite fields.

Suppose that G is a finite, unitary reflection group acting on a complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in G, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of G-invariant polynomial functions on V to the algebra of C-invariant functions on X. In my thesis, I extend earlier work by Douglass and Röhrle for Coxeter groups to the case where G is a complex reflection group of type G(r,p,n) in the notation of Shephard and Todd and X is in the lattice of the reflection arrangement of G. The main result characterizes when the restriction mapping is surjective in terms of the exponents of G and C and their reflection arrangements.

Cette thèse traite des invariants cohomologiques en cohomologie galoisienne des groupes de Coxeter finis en caractéristique nulle. On établit d'abord un principe général d'annulation vérifié par tout invariant cohomologique d'un groupe de Coxeter fini sur un corps de caractéristique nulle suffisamment grand. On utilise ensuite ce principe pour déterminer tous les invariants cohomologiques des groupes de Weyl de type classique à coefficients modulo 2 sur un corps de caractéristique nulle. / This PhD thesis deals with cohomological invariants in Galois cohomology of finite Coxeter groups in characteristic zero. We first state a general vanishing principle for the cohomological invariants of a finite Coxeter group over a sufficiently large field of characteristic zero. We then use this principle to determine all the cohomological invariants of the Weyl groups of classical type with coefficients modulo 2 over a field of characteristic zero.

Rational Catalan combinatorics connects various Catalan numbers to the representation theory of rational Cherednik algebras for Coxeter and complex reflection groups. Lewis, Reiner, and Stanton seek a theory of rational Catalan combinatorics for the general linear group over a finite field. The finite general linear group is a modular reflection group that behaves like a finite Coxeter group. They conjecture a Hilbert series for a space of invariants under the action of this group using (q,t)-binomial coefficients. They consider the finite general linear group acting on the quotient of a polynomial ring by iterated powers of the irrelevant ideal under the Frobenius map. Often conjectures about reflection groups are solved by considering the local case of a group fixing one hyperplane and then extending via the theory of hyperplane arrangements to the full group. The Lewis, Reiner and Stanton conjecture had not previously been formulated for groups fixing a hyperplane. We formulate and prove their conjecture in this local case.

No description available.

Neste trabalho, temos o objetivo de introduzir o Problema de Noether Clássico e sua versão não- comutativa introduzida por J. Alev e F. Dumas em [AD06]. Discutiremos os principais casos co- nhecidos nos quais os problemas têm solução positiva, observando um forte paralelo entre os casos comutativo e não-comutativo. Cobriremos os tópicos preliminares necessários para entendimento dos enunciados: álgebras de Weyl, anéis de operadores diferenciais, extensões de Ore, localização em domínios não-comutativos, e corpos de Weyl. No Capítulo 5 deste trabalho, o aluno apresenta duas contribuições originais, obtidas em colaboração com seu orientador V. Futorny e F. Eshmatov: o Teorema 5.5, que é um resultado folclórico sobre invariantes de ações livres de grupos finitos no anel de operadores diferenciais de variedades afins; e o Teorema 5.6, que até onde sabemos é iné- dito, sobre invariantes dos Corpos de Weyl sob a ação de grupos de pseudo-reflexão. Todo material algébrico preliminar para a demonstração destes dois teoremas é incluído no texto da dissertação: um básico de teoria de invariantes, vários resultados da teoria de grupos de pseudo-reflexão, alguns conceitos básicos de geometria algébrica e álgebra comutativa, e uma discussão detalhada do quo- ciente de variedades afins sob ação de grupos finitos. / In this work we aim to introduce the Classical Noether´s Problem, and its noncommutative version introduced by J. Alev and F. Dumas in [AD06]. We discuss the most well known cases of positive solution of these problems, pointing out a strong similarity between the cases of positive solution for the classical and noncommutative versions of the Problem. We cover the preliminary topics to understand the statement and solutions of these problems: Weyl algebras, differential operators rings, Ore extensions, noncommutative localization, and Weyl Skew-Fields. In the Chapter 5 of this dissertation, the student shows two original contributions, obtained in collaboration with his advisor V. Futorny and F. Eshmatov: Theorem 5.5, a result belonging to the folklore of the area of differential operators, describing its invariants under the free action of a finite group on an affine variety; and Theorem 5.6, about the invariants of the Weyl skew-fields under the action of pseudo-reflection groups. As far as we know, this result is new. All preliminary algebraic facts to prove these two facts are included in the body of this text. It includes some basic facts on invariant theory, many results about pseudo-reflection groups, some basic concepts of algebraic geometry and commutative algebra, and a detailed discussion of the quotient of an affine variety under the action of a finite group.

Cette thèse présente quelques résultats de la théorie des représentations des algèbres de Cherednikrationnelles en t=0 et traite en particulier des différents ordres construits sur la partition de Calogero-Moserdes groupes imprimitifs.On commence par généraliser au cas abélien certains résultats obtenus par M. Chlouveraki concernant lesblocs d'algèbres en système de Clifford pour un groupe cyclique, puis on construit un ordre sur les C*-pointsfixes d'une variété complexe quasi-projective normale, en utilisant la décomposition de Bialynicki-Birula.Dans la deuxième partie, on s'intéresse à la description des partitions de Calogero-Moser de deux groupesde réflexions complexes K et W quand K est un sous-groupe distingué de W et on généralise au cas abélienles résultats obtenus par G. Bellamy dans le cas d'un quotient W/K cyclique.Dans la troisième partie, on présente les différents ordres, construits par I. Gordon, sur la partition deCalogero-Moser des groupes G(l,1,n) pour certains paramètres : les ordres des a et c-fonctions, un ordrecombinatoire et l'ordre géométrique, qui est défini grâce aux C*-points fixes de certaines variétés decarquois, ces points fixes paramétrant les blocs de la partition de Calogero-Moser de G(l,1,n). On donneensuite les relations entre ces ordres, puis on étend ces constructions ainsi que ces liens à l'ensemble desparamètres.Enfin, dans la dernière partie, on tente de généraliser ces propriétés aux groupes G(l,e,n). On cherche alors,pour construire l'ordre géométrique sur la partition de Calogero-Moser de G(l,e,n), une variété dont les C*-points fixes décrivent les blocs de la partition de G(l,e,n). Dans le cas où e ne divise pas n, on construit lavariété qui nous permet de définir l'ordre géométrique et de le relier aux autres ordres. Pour le cas e divise n,on propose une variété qui pourrait décrire par ses points fixes les blocs de Calogero-Moser de G(l,e,n) etnous permettre de construire l'ordre géométrique. / This work is a contribution to the representation theory of Rational Cherednik Algebras for t=0 and deals inparticular with different orders on the Calogero-Moser partition of imprimitive reflection groups.In the first part, we generalize to the abelian case some results about blocs of algebras in Clifford systemobtained by M. Chlouveraki in the cyclic case, and then we build an order on the C*-fixed points of acomplex, quasi-projective and normal variety, using the Bialynicki-Birula decomposition.The second part deals with the Calogero-Moser partition of two groups K and W, when K is a normalsubgroup of W, and generalize to the abelian case the results that G. Bellamy obtained when the quotientW/K is cyclic.In the third part, we present the different orders that I. Gordon built in the Calogero-Moser partition of thegroups G(l,1,n) and for some parameters : the orders of the a and c-functions, a combinatorial order and thegeometric order, defined using the C*-fixed points of some quiver varieties which parametrise the blocs of theCalogero-Moser partition of G(l,1,n). Then we give some relations between these orders and we extendthese constructions and these links for all parameters.Finally, in the last part, we try to generalize these properties for the groups G(l,e,n). We are looking for avariety whose C*-fixed points describe blocs of G(l,e,n) to construct the geometric order on the Calogero-Moser partition of G(l,e,n). When n is not divided by e, we build this variety that enables us to define thegeometric order and to show all the links with the other orders. When e don't divide n, we suggest a varietywhich could describe the blocs of G(l,e,n) and allow us to build the geometric order.

Cette thèse porte sur l'étude des données $mathbb{Z}$-modulaires et leur catégorification, et particulièrement sur des données $mathbb{Z}$-modulaires reliées aux groupes de réflexions complexes, ainsi que sur la notion de caractère cellulaire pour ces derniers. Dans sa classification des caractères des groupes finis de type de Lie, Lusztig décrit une transformée de Fourier non abélienne et définit des données $mathbb{N}$-modulaires pour chaque famille de caractères unipotents. Dans des tentatives de généralisation aux Spetses, Broué, Malle et Michel introduisent des données $mathbb{Z}$-modulaires. On commence par donner une explication catégorique de certaines de ces données via la catégorie des représentations du double de Drinfeld d'un groupe fini, que l'on munit d'une structure pivotale non sphérique. Une étude approfondie de la notion de catégorie de fusion pivotale et légèrement dégénérée montre que l'on peut ainsi produire des données $mathbb{Z}$-modulaires. Afin de construire des exemples de telles catégories, on considère des extensions des catégories de fusion associées à $qgrroot{mathfrak{g}}$, où $mathfrak{g}$ est une algèbre de Lie simple, et $xi$ une racine de l'unité. Ces dernières sont construites comme des semi-simplifications de la catégorie des modules basculants de l'algèbre $qdblroot{mathfrak{g}}$, qui est une extension centrale de $qgrroot{mathfrak{g}}$. Dans le cas où $mathfrak{g}=mathfrak{sl}_{n+1}$, on relie cette catégorie à une des données $mathbb{Z}$-modulaires associée au groupe de réflexions complexes $Gleft(d,1,frac{n(n+1)}{2}right)$. Les groupes de réflexions exceptionnels sont également étudiés, et les catégorifications des données $mathbb{Z}$-modulaires associées font apparaître diverses catégories : des catégories de représentations de doubles de Drinfeld tordus ainsi que des sous-catégories des catégories de fusion des modules basculants en $qdblroot{mathfrak{g}}$ en type $A$ et $B$. / This work is a contribution to the categorification of $mathbb{Z}$-modular data and deals mainly with $mathbb{Z}$-modular data arising from complex reflection groups, as well as cellular characters for these groups. In his classification of representations of finite groups of Lie type, Lusztig defines a nonabelian Fourier transform, and associate a $mathbb{N}$-modular datum to each family of unipotent characters. In a generalization of Lusztig's theory to Spetses, Broué, Malle and Michel construct $mathbb{Z}$-modular data associated to some complex reflection groups. We first give a categorical explanation of some of these $mathbb{Z}$-modular data in terms of representation of the Drinfeld double of a finite group. We had to endow the category of representations with a non-spherical structure. The study of slightly degenerate categories shows that they naturally give rise to $mathbb{Z}$-modular data. In order to construct some examples, we consider an extension of the fusion categories associated to $qgrroot{mathfrak{g}}$, where $mathfrak{g}$ is a simple Lie algebra and $xi$ a root of unity. These categories are constructed as semisimplification of the category of tilting modules of $qdblroot{mathfrak{g}}$, which is a central extension of $qgrroot{mathfrak{g}}$. If $mathfrak{s}=mathfrak{sl}_{n+1}$, we show that this category is related to some $mathbb{Z}$-modular data associated to the complex reflection group $Gleft(d,1,frac{n(n+1)}{2}right)$. Exceptional complex reflection groups are also considered and many different categories appear in the categorification of the associated $mathbb{Z}$-modular data : modules categories over twisted Drinfeld doubles as well as some subcategories of fusion categories of tilting modules over $qdblroot{mathfrak{g}}$ in type $A$ and $B$.