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.

Thesis (Ph. D.)--University of Oregon, 2007. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 107-108). Also available for download via the World Wide Web; free to University of Oregon users.

Universiẗat, Diss., 1999--Freiburg.

viii, 50 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We study a certain quotient of the Iwahori-Hecke algebra of the symmetric group Sd , called the super Temperley-Lieb algebra STLd. The Alexander polynomial of a braid can be computed via a certain specialization of the Markov trace which descends to STLd. Combining this point of view with Ocneanu's formula for the Markov trace and Young's seminormal form, we deduce a new state-sum formula for the Alexander polynomial. We also give a direct combinatorial proof of this result. / Committee in charge: Arkady Vaintrob, Co-Chairperson, Mathematics Jonathan Brundan, Co-Chairperson, Mathematics; Victor Ostrik, Member, Mathematics; Dev Sinha, Member, Mathematics; Paul van Donkelaar, Outside Member, Human Physiology

xii, 76 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We study the graded representation theory of the Iwahori-Hecke algebra, denoted by Hd , of the symmetric group over a field of characteristic zero at a root of unity. More specifically, we use graded Specht modules to calculate the graded decomposition numbers for Hd . The algorithm arrived at is the Lascoux-Leclerc-Thibon algorithm in disguise. Thus we interpret the algorithm in terms of graded representation theory. We then use the algorithm to compute several examples and to obtain a closed form for the graded decomposition numbers in the case of two-column partitions. In this case, we also precisely describe the 'reduction modulo p' process, which relates the graded irreducible representations of Hd over [Special characters omitted.] at a p th -root of unity to those of the group algebra of the symmetric group over a field of characteristic p. / Committee in charge: Alexander Kleshchev, Chairperson, Mathematics; Jonathan Brundan, Member, Mathematics; Boris Botvinnik, Member, Mathematics; Victor Ostrik, Member, Mathematics; William Harbaugh, Outside Member, Economics

Let (W, S) be an arbitrary Coxeter system, and let J be the asymptotic Hecke algebra associated to (W, S) via Kazhdan-Lusztig polynomials by Lusztig. We study a subalgebra J_C of J corresponding to the subregular cell C of W . We prove a factorization theorem that allows us to compute products in J_C without inputs from Kazhdan-Lusztig theory, then discuss two applications of this result. First, we describe J_C in terms of the Coxeter diagram of (W, S) in the case (W, S) is simply- laced, and deduce more connections between the diagram and J_C in some other cases. Second, we prove that for certain specific Coxeter systems, some subalgebras of J_C are free fusion rings, thereby connecting the algebras to compact quantum groups arising in operator algebra theory.

Este trabalho compõe-se de três partes istintas. A primeira contém uma extensão da fórmula de Funk-Hecke `a esfera unitária de Cq, e aplicaçães dela no estudo das propriedades anuladoras de operadores gerados por convolução esférica. A segunda introduz um método indutivo para construir bases para os espaços dos harmônicos esféricos complexos. A terceira apresenta um estudo de aproximações para soluções de equações definidas por transformações multplicativas sobre a esfera. O estudo engloba a construção das funções aproximadoras e estimativas do erro na aproximação, incluindo estimativas em casos específicos.

This thesis establishes a framework for cellularity of algebras related to the Jones basic construction. The framework allows a uniform proof of cellularity of Brauer algebras, BMW algebras, walled Brauer algebras, partition algebras, and others. In this setting, the cellular bases are labeled by paths on certain branching diagrams rather than by tangles. Moreover, for this class of algebras, the cellular structures are compatible with restriction and induction of modules.

Le but de cette thèse est d'analyser et de développer les objets mathématiques apparus dans l'article "New classical limits of quantum theories" de S. G. Rajeev, notamment les (loc.sit) "limites néoclassiques" dans le contexte de la théorie des formes modulaires. Afin de voir au mieux quels sont les objets en jeu dans l'étude de Rajeev, on a dans une première étape construit certains modèles-jouets afin de mener dedans des calculs similaires que ceux exposés dans l'article en question tout en essayant d'étudier le rapprochement avec des objets et théories mathématiques rigoureuses, notamment la quantification kählerienne, la géométrie algébrique arithmétique et la formule pour la trace des opérateurs de Hecke. Dans une deuxième étape, on a développé un cadre mathématique rigoureux où vivent naturellement les objets de l'étude de Rajeev. Ce cadre devrait servir dans la suite afin de faire de manière rigoureuse les calculs de "limite néoclassique" dans ce contexte ci. Ainsi les objets développés devraient servir aux mathématiciens de mieux comprendre les idées des physiciens et aux physiciens de pouvoir pousser plus loin les calculs de perturbations / The goal of this thesis is to analyze and to develop the mathematical objects that appeared in "New classical limits of quantum theories" of S. G. Rajeev, especially the (loc. sit) "neoclassical limits" in the context of the contexte of the theory of modular forms. To see what are the objects involved in the study of Rajeev, we constructed certain toy models where could develop similar calculations as those done in the article mentioned above. This was done by trying to compare these toy models with rigorous mathematical theories, for example Kähler quantization, algebric geometry and the trace formula for Hecke operators. After that we developed a rigorous mathematical frame where the objects introduced by Rajeev naturally live. This frame should be used in the futur to do the "neoclassical limit" calculations in this context. So the objects developed could be used by the mathematicians to understand the physical ideas and by the physicists to push further the calculations of perturbation

This thesis deals with the study of Schubert varieties, which are subsets of flag varieties indexed by elements of Weyl groups. We start by defining Lascoux elements in the Hecke algebra, and showing that they coincide with the Kazhdan-Lusztig basis elements in certain cases. We then construct a resolution (Zw, π) of the Schubert variety Xw for which Rπ*(C[l(w)]) is a sheaf on Xw whose expression in the Hecke algebra is closely related to the Lascoux element. We also define two new polynomials which coincide with the intersection cohomology Poincar\'e polynomial in certain cases. In the final chapter, we discuss some interesting combinatorial results concerning Bell and Catalan numbers which arose throughout the course of this work.