Doctor of Philosophy / This thesis presents a study of the cyclotomic BMW algebras, introduced by Haring-Oldenburg as a generalization of the BMW (Birman-Murakami-Wenzl) algebras related to the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. The motivation behind the definition of the BMW algebras may be traced back to an important problem in knot theory; namely, that of classifying knots (and links) up to isotopy. The algebraic definition of the BMW algebras uses generators and relations originally inspired by the Kauffman link invariant. They are intimately connected with the Artin braid group of type A, Iwahori-Hecke algebras of type A, and with many diagram algebras, such as the Brauer and Temperley-Lieb algebras. Geometrically, the BMW algebra is isomorphic to the Kauffman Tangle algebra. The representations and the cellularity of the BMW algebra have now been extensively studied in the literature. These algebras also feature in the theory of quantum groups, statistical mechanics, and topological quantum field theory. In view of these relationships between the BMW algebras and several objects of "type A", several authors have since naturally generalized the BMW algberas for other types of Artin groups. Motivated by knot theory associated with the Artin braid group of type B, Haring-Oldenburg introduced the cyclotomic BMW algebras B_n^k as a generalization of the BMW algebras such that the Ariki-Koike algebra h_{n,k} is a quotient of B_n^k, in the same way the Iwahori-Hecke algebra of type A is a quotient of the BMW algebra. In this thesis, we investigate the structure of these algebras and show they have a topological realization as a certain cylindrical analogue of the Kauffman Tangle algebra. In particular, they are shown to be R-free of rank k^n (2n-1)!! and bases that may be explicitly described both algebraically and diagrammatically in terms of cylindrical tangles are obtained. Unlike the BMW and Ariki-Koike algebras, one must impose extra so-called "admissibility conditions" on the parameters of the ground ring in order for these results to hold. This is due to potential torsion caused by the polynomial relation of order k imposed on one of the generators of B_n^k. It turns out that the representation theory of B_2^k is crucial in determining these conditions precisely. The representation theory of B_2^k is analysed in detail in a joint preprint with Wilcox in [45] (http://arxiv.org/abs/math/0611518). The admissibility conditions and a universal ground ring with admissible parameters are given explicitly in Chapter 3. The admissibility conditions are also closely related to the existence of a non-degenerate Markov trace function of B_n^k which is then used together with the cyclotomic Brauer algebras in the linear independency arguments contained in Chapter 4. Furthermore, in Chapter 5, we prove the cyclotomic BMW algebras are cellular, in the sense of Graham and Lehrer. The proof uses the cellularity of the Ariki-Koike algebras (Graham-Lehrer [16] and Dipper-James-Mathas [8]) and an appropriate "lifting" of a cellular basis of the Ariki-Koike algebras into B_n^k, which is compatible with a certain anti-involution of B_n^k. When k = 1, the results in this thesis specialize to those previously established for the BMW algebras by Morton-Wasserman [30], Enyang [9], and Xi [47]. REMARKS: During the writing of this thesis, Goodman and Hauschild-Mosley also attempt similar arguments to establish the freeness and diagram algebra results mentioned above. However, they withdrew their preprints ([14] and [15]), due to issues with their generic ground ring crucial to their linear independence arguments. A similar strategy to that proposed in [14], together with different trace maps and the study of rings with admissible parameters in Chapter 3, is used in establishing linear independency of our basis in Chapter 4. Since the submission of this thesis, new versions of these preprints have been released in which Goodman and Hauschild-Mosley use alternative topological and Jones basic construction theory type arguments to establish freeness of B_n^k and an isomorphism with the cyclotomic Kauffman Tangle algebra. However, they require their ground rings to be an integral domain with parameters satisfying the (slightly stronger) admissibility conditions introduced by Wilcox and the author in [45]. Also, under these conditions, Goodman has obtained cellularity results. Rui and Xu have also obtained freeness and cellularity results when k is odd, and later Rui and Si for general k, under the assumption that \delta is invertible and using another stronger condition called "u-admissibility". The methods and arguments employed are strongly influenced by those used by Ariki, Mathas and Rui [3] for the cyclotomic Nazarov-Wenzl algebras and involve the construction of seminormal representations; their preprints have recently been released on the arXiv. It should also be noted there are slight differences between the definitions of cyclotomic BMW algebras and ground rings used, as explained partly above. Furthermore, Goodman and Rui-Si-Xu use a weaker definition of cellularity, to bypass a problem discovered in their original proofs relating to the anti-involution axiom of the original Graham-Lehrer definition. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007.

Doctor of Philosophy / This thesis presents a study of the cyclotomic BMW algebras, introduced by Haring-Oldenburg as a generalization of the BMW (Birman-Murakami-Wenzl) algebras related to the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. The motivation behind the definition of the BMW algebras may be traced back to an important problem in knot theory; namely, that of classifying knots (and links) up to isotopy. The algebraic definition of the BMW algebras uses generators and relations originally inspired by the Kauffman link invariant. They are intimately connected with the Artin braid group of type A, Iwahori-Hecke algebras of type A, and with many diagram algebras, such as the Brauer and Temperley-Lieb algebras. Geometrically, the BMW algebra is isomorphic to the Kauffman Tangle algebra. The representations and the cellularity of the BMW algebra have now been extensively studied in the literature. These algebras also feature in the theory of quantum groups, statistical mechanics, and topological quantum field theory. In view of these relationships between the BMW algebras and several objects of "type A", several authors have since naturally generalized the BMW algberas for other types of Artin groups. Motivated by knot theory associated with the Artin braid group of type B, Haring-Oldenburg introduced the cyclotomic BMW algebras B_n^k as a generalization of the BMW algebras such that the Ariki-Koike algebra h_{n,k} is a quotient of B_n^k, in the same way the Iwahori-Hecke algebra of type A is a quotient of the BMW algebra. In this thesis, we investigate the structure of these algebras and show they have a topological realization as a certain cylindrical analogue of the Kauffman Tangle algebra. In particular, they are shown to be R-free of rank k^n (2n-1)!! and bases that may be explicitly described both algebraically and diagrammatically in terms of cylindrical tangles are obtained. Unlike the BMW and Ariki-Koike algebras, one must impose extra so-called "admissibility conditions" on the parameters of the ground ring in order for these results to hold. This is due to potential torsion caused by the polynomial relation of order k imposed on one of the generators of B_n^k. It turns out that the representation theory of B_2^k is crucial in determining these conditions precisely. The representation theory of B_2^k is analysed in detail in a joint preprint with Wilcox in [45] (http://arxiv.org/abs/math/0611518). The admissibility conditions and a universal ground ring with admissible parameters are given explicitly in Chapter 3. The admissibility conditions are also closely related to the existence of a non-degenerate Markov trace function of B_n^k which is then used together with the cyclotomic Brauer algebras in the linear independency arguments contained in Chapter 4. Furthermore, in Chapter 5, we prove the cyclotomic BMW algebras are cellular, in the sense of Graham and Lehrer. The proof uses the cellularity of the Ariki-Koike algebras (Graham-Lehrer [16] and Dipper-James-Mathas [8]) and an appropriate "lifting" of a cellular basis of the Ariki-Koike algebras into B_n^k, which is compatible with a certain anti-involution of B_n^k. When k = 1, the results in this thesis specialize to those previously established for the BMW algebras by Morton-Wasserman [30], Enyang [9], and Xi [47]. REMARKS: During the writing of this thesis, Goodman and Hauschild-Mosley also attempt similar arguments to establish the freeness and diagram algebra results mentioned above. However, they withdrew their preprints ([14] and [15]), due to issues with their generic ground ring crucial to their linear independence arguments. A similar strategy to that proposed in [14], together with different trace maps and the study of rings with admissible parameters in Chapter 3, is used in establishing linear independency of our basis in Chapter 4. Since the submission of this thesis, new versions of these preprints have been released in which Goodman and Hauschild-Mosley use alternative topological and Jones basic construction theory type arguments to establish freeness of B_n^k and an isomorphism with the cyclotomic Kauffman Tangle algebra. However, they require their ground rings to be an integral domain with parameters satisfying the (slightly stronger) admissibility conditions introduced by Wilcox and the author in [45]. Also, under these conditions, Goodman has obtained cellularity results. Rui and Xu have also obtained freeness and cellularity results when k is odd, and later Rui and Si for general k, under the assumption that \delta is invertible and using another stronger condition called "u-admissibility". The methods and arguments employed are strongly influenced by those used by Ariki, Mathas and Rui [3] for the cyclotomic Nazarov-Wenzl algebras and involve the construction of seminormal representations; their preprints have recently been released on the arXiv. It should also be noted there are slight differences between the definitions of cyclotomic BMW algebras and ground rings used, as explained partly above. Furthermore, Goodman and Rui-Si-Xu use a weaker definition of cellularity, to bypass a problem discovered in their original proofs relating to the anti-involution axiom of the original Graham-Lehrer definition. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007.

Soit $W$ un groupe de Weyl fini et soit $H$ l'algèbre de Hecke correspondante, définie sur l'anneau $A:=Z[v,v^(-1)]$ où $v$ est une indéterminée. Soit $K$ le corps des fractions de $A$ et soit $\theta$ une spécialisation dans un corps $L$ de ``bonne'' caractéristique. Dans une série d'articles récents, M.Geck et R.Rouquier ont présenté une méthode pour déterminer l'ensemble des $H_L$-modules simples $\Irr(H_L)$. Celle-ci consiste à construire un ``ensemble basique canonique'' $B$ contenu dans $\Irr(H_K)$ défini grace à la $a$-fonction de Lusztig et en bijection avec $\Irr(H_L)$. Le but de ce travail est de déterminer explicitement $B$ pour tout groupe de Weyl et pour toute spécialisation puis d'étendre la méthode ci-dessus aux algèbres de Ariki-Koike. Comme conséquences, nous obtenons un algorithme pour le calcul des matrices de décompositions des algèbres de Ariki-Koike et une caractérisation des modules simples pour certaines algèbres cyclotomiques de type $G(l,l,n)$.

Cette thèse est consacrée à l'étude des algèbres de Hecke carquois et de certaines généralisations des algèbres d'Iwahori-Hecke. Dans un premier temps, nous montrons deux résultats concernant les algèbres de Hecke carquois, dans le cas où le carquois possède plusieurs composantes connexes puis lorsqu'il possède un automorphisme d'ordre fini. Ensuite, nous rappelons un isomorphisme de Brundan-Kleshchev et Rouquier entre algèbres d'Ariki-Koike et certaines algèbres de Hecke carquois cyclotomiques. D'une part nous en déduisons qu'une équivalence de Morita importante bien connue entre algèbres d'Ariki-Koike provient d'un isomorphisme, d'autre part nous donnons une présentation de type Hecke carquois cyclotomique pour l'algèbre de Hecke de G(r,p,n). Nous généralisons aussi l'isomorphisme de Brundan-Kleshchev pour montrer que les algèbres de Yokonuma-Hecke cyclotomiques sont des cas particuliers d'algèbres de Hecke carquois cyclotomiques. Finalement, nous nous intéressons à un problème de combinatoire algébrique, relié à la théorie des représentations des algèbres d'Ariki-Koike. En utilisant la représentation des partitions sous forme d'abaque et en résolvant, via un théorème d'existence de matrices binaires, un problème d'optimisation convexe sous contraintes à variables entières, nous montrons qu'un multi-ensemble de résidus qui est bégayant provient nécessairement d'une multi-partition bégayante. / This thesis is devoted to the study of quiver Hecke algebras and some generalisations of Iwahori-Hecke algebras. We begin with two results concerning quiver Hecke algebras, first when the quiver has several connected components and second when the quiver has an automorphism of finite order. We then recall an isomorphism of Brundan-Kleshchev and Rouquier between Ariki-Koike algebras and certain cyclotomic quiver Hecke algebras. From this, on the one hand we deduce that a well-known important Morita equivalence between Ariki--Koike algebras comes from an isomorphism, on the other hand we give a cyclotomic quiver Hecke-like presentation for the Hecke algebra of type G(r,p,n). We also generalise the isomorphism of Brundan-Kleshchev to prove that cyclotomic Yokonuma-Hecke algebras are particular cases of cyclotomic quiver Hecke algebras. Finally, we study a problem of algebraic combinatorics, related to the representation theory of Ariki-Koike algebras. Using the abacus representation of partitions and solving, via an existence theorem for binary matrices, a constrained optimisation problem with integer variables, we prove that a stuttering multiset of residues necessarily comes from a stuttering multipartition.

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.