Thesis (M. Sc.)--University of Alberta, 2009. / Title from pdf file main screen (viewed on Dec. 10, 2009). "A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science in Pure Mathematics, Department of Mathematical and Statistical Sciences, University of Alberta." Includes bibliographical references.

published_or_final_version / Mathematics / Master / Master of Philosophy

Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. It turns out that the finite Coxeter groups are precisely the finite Euclidean reflection groups. Coxeter studied these groups and classified all finite ones in 1935, however they were known as reflection groups until J. Tits coined the term Coxeter groups for them in the sixties. Finite crystallographic Coxeter groups, also known as finite Weyl groups, play a prominent role in many branches of mathematics like combinatorics, Lie theory, number theory, and geometry. The computational aspects of these groups are of great interests and play a very important role in representation theory. Since it’s enough to study only the irreducible class of groups in order to understand any Coxeter group, we discuss irreducible crystallographic Coxeter groups here. Our goal is to try to deal with some of the fundamental computational problems that arise in working with the structures such as Weyl groups, root system, Weyl characters. For the classical cases, especially type A, many of these problems are not very subtle and have been solved completely. However, these solutions often do not generalize. In this report, our emphasis is on algorithms which do not really depend on the classifications of root systems. The canonical example, we always keep in mind is E8. In chapter 1, we fix the notations and give some basic results which have been used in this report. In chapter 2, we explain algorithms to various Weyl group problems like membership problem; how to find the length of an element; how to check if two words in a Weyl group represent the same element or not; finding the coset representative for an element for a given parabolic subgroup; and list all the expressions possible for an element. In chapter 3, the main goal is to write an algorithm to compute the weight multiplicities of the irreducible representations using Freudenthal’s formula. For this, we first compute the positive roots and dominant weights for a given root system and then finally find the weight multiplicities. We argue this mathematically using the results given in chapter 1. The crystallographic hypothesis is unnecessary for much of what is discussed in chapter 2. In the last chapter, we give codes of the computer programs written in C++ which implement the algorithms described in the previous chapters in this report.

For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in Pic(X)⊗C. We identify the lattice part of this groupoid with the operators of quantum difference equation for X. The cases of quivers of finite and affine type are illustrated by explicit examples.

Thesis (Ph.D.) - University of Queensland, 2005. / Includes bibliography.

This thesis is about extended affine Lie algebras and extended affine Weyl groups. In Chapter I, we provide the basic knowledge necessary for the study of extended affine Lie algebras and related objects. In Chapter II, we show that the well-known twisting phenomena which appears in the realization of the twisted affine Lie algebras can be extended to the class of extended affine Lie algebras, in the sense that some extended affine Lie algebras (in particular nonsimply laced extended affine Lie algebras) can be realized as fixed point subalgebras of some other extended affine Lie algebras (in particular simply laced extended affine Lie algebras) relative to some finite order automorphism. We show that extended affine Lie algebras of type A₁, B, C and BC can be realized as twisted subalgebras of types A_§¤(l ¡Ã 2) and D algebras. Also we show that extended affine Lie algebras of type BC can be realized as twisted subalgebras of type C algebras. In Chapter III, the last chapter, we study the Weyl groups of reduced extended affine root systems. We start by describing the extended affine Weyl group as a semidirect product of a finite Weyl group and a Heisenberg-like normal subgroup. This provides a unique expression for the Weyl group elements which in turn leads to a presentation of the Weyl group, called a presentation by conjugation. Using a new notion, called the index, which is an invariant of the extended affine root systems, we show that one of the important features of finite and affine root systems (related to Weyl group) holds for the class of extended affine root systems. We also show that extended affine Weyl groups (of index zero) are homomorphic images of some indefinite Weyl groups where the homomorphism and its kernel are given explicitly.

The purpose of this dissertation is to elaborate, with specific examples and calculations, on a new refinement of triangular factorization for the loop group of a simple, compact Lie group K, first appearing in Pickrell & Pittman-Polletta 2010. This new factorization allows us to write a smooth map from the unit circle into K (having a triangular factorization) as a triply infinite product of loops, each of which depends on a single complex parameter. These parameters give a set of coordinates on the loop group of K.The order of the factors in this refinement is determined by an infinite sequence of simple generators in the affine Weyl group associated to K, having certain properties. The major results of this dissertation are examples of such sequences for all the classical Weyl groups.We also produce a variation of this refinement which allows us to write smooth maps from the unit circle into the special unitary group of n by n matrices as products of 2n+1 infinite products. By analogy with the semisimple analog of our factorization, we suggest that this variation of the refinement has simpler combinatorics than that appearing in Pickrell & Pittman-Polletta 2010.

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.

Actuellement, l’étude des systèmes multi-antennaires MIMO (Multiple Input Multiple Output) est orientée dans beaucoup de cas vers l’augmentation considérable du nombre d’antennes de la station de base (« massive MIMO », « large-scale MIMO »), afin notamment d’augmenter la capacité de transmission, réduire l’énergie consommée par bit transmis, exploiter la dimension spatiale du canal de propagation, diminuer l’influence des évanouissements, etc. Pour les systèmes MIMO à bande étroite ou ceux utilisant la technique OFDM (Orthogonal Frequency Division Multiplex), le canal de propagation (ou les sous-canaux correspondants à chaque sous-porteuse d’un système OFDM) sont pratiquement plats (non-sélectifs en fréquence), ce qui revient à considérer la réponse fréquentielle de chaque canal SISO invariante par rapport à la fréquence mais variante dans le temps. Ainsi, le canal de propagation MIMO peut être caractérisé en bande de base par une matrice dont les coefficients sont des nombres complexes. Les systèmes MIMO cohérents nécessitent pour pouvoir démoduler le signal en réception de disposer de la connaissance de cette matrice de canal, donc le sondage périodique, en temps réel, du canal de propagation. L’augmentation du nombre d’antennes et la variation dans le temps, parfois assez rapide, du canal de propagation, rend ce sondage de canal difficile, voire impossible. Il est donc intéressant d’étudier des systèmes MIMO différentiels qui n’ont pas besoin de connaître la matrice de canal. Pour un bon fonctionnement de ces systèmes, la seule contrainte est que la matrice de canal varie peu pendant la transmission de deux matrices d’information successives. Le sujet de cette thèse concerne l’étude et l’analyse de nouveaux systèmes MIMO différentiels. On considère des systèmes à 2, 4 et 8 antennes d’émission, mais la méthode utilisée peut être étendue à des systèmes MIMO avec 2n antennes d’émission, le nombre d’antennes de réception étant quelconque. Pour les systèmes MIMO avec 2 antennes d’émission qui ont été étudiés dans le cadre de cette thèse, les matrices d’information sont des éléments du groupe de Weyl. Pour les systèmes avec 2n antennes d’émission, (n ≥ 2), les matrices utilisées sont obtenues en effectuant des produits de Kronecker des matrices unitaires du groupe de Weyl. Pour chaque nombre d’antennes d’émission on identifie d’abord le nombre de matrices disponibles et on détermine la valeur maximale de l’efficacité spectrale. Pour chaque valeur de l’efficacité spectrale on détermine les meilleurs sous-ensembles de matrices d’information à utiliser (selon le spectre des distances ou le critère du produit de diversité). On optimise ensuite la correspondance ou mapping entre les vecteurs binaires et les matrices d’information. Enfin, on détermine par simulation les performances des systèmes MIMO différentiels ainsi obtenus et on les compare avec celles des systèmes similaires existants. […] / At present, the study of multi-antenna systems MIMO (Multiple Input Multiple Output) is developed in many cases to intensively increase the number of base station antennas («massive MIMO», «largescale MIMO»), particularly in order to increase the transmission capacity, reduce energy consumed per bit transmitted, exploit the spatial dimension of the propagation channel, reduce the influence of fading, etc. For MIMO systems with narrowband or those using OFDM technique (Orthogonal Frequency Division Multiplex), the propagation channel (or the sub-channels corresponding to each sub-carrier of an OFDM system) are substantially flat (frequency non-selective). In this case the frequency response of each SISO channel is invariant with respect to frequency, but variant in time. Furthermore, the MIMO propagation channel can be characterized in baseband by a matrix whose coefficients are complex numbers. Coherent MIMO systems need to have the knowledge of the channel matrix to be able to demodulate the received signal. Therefore, periodic pilot should be transmitted and received to estimate the channel matrix in real time. The increase of the number of antennas and the change of the propagation channel over time, sometimes quite fast, makes the channel estimation quite difficult or impossible. It is therefore interesting to study differential MIMO systems that do not need to know the channel matrix. For proper operation of these systems, the only constraint is that the channel matrix varies slightly during the transmission of two successive information matrices. The subject of this thesis is the study and analysis of new differential MIMO systems. We consider systems with 2, 4 and 8 transmit antennas, but the method can be extended to MIMO systems with 2n transmit antennas, the number of receive antennas can be any positive integer. For MIMO systems with two transmit antennas that were studied in this thesis, information matrices are elements of the Weyl group. For systems with 2n (n ≥ 2) transmit antennas, the matrices used are obtained by performing the Kronecker product of the unitary matrices in Weyl group. For each number of transmit antennas, we first identify the number of available matrices and the maximum value of the spectral efficiency. For each value of the spectral efficiency, we then determine the best subsets of information matrix to use (depending on the spectrum of the distances or the diversity product criterion). Then we optimize the correspondence or mapping between binary vectors and matrices of information. Finally, the performance of differential MIMO systems are obtained by simulation and compared with those of existing similar systems. […]

We consider the problem of testing whether r (>=2) samples are drawn from the same continuous distribution F(x). The test statistic we will study in some detail is defined as the maximum of the circular differences of the empirical distribution functions, a generalization of the classical 2-sample Kolmogorov-Smirnov test to r (>=2) independent samples. For the case of equal sample sizes we derive the exact null distribution by counting lattice paths confined to stay in the scaled alcove $\mathcal{A}_r$ of the affine Weyl group $A_{r-1}$. This is done using a generalization of the classical reflection principle. By a standard diffusion scaling we derive also the asymptotic distribution of the test statistic in terms of a multivariate Dirichlet series. When the sample sizes are not equal the reflection principle no longer works, but we are able to establish a weak convergence result even in this case showing that by a proper rescaling a test statistic based on a linear transformation of the circular differences of the empirical distribution functions has the same asymptotic distribution as the test statistic in the case of equal sample sizes. / Series: Research Report Series / Department of Statistics and Mathematics