In this thesis, we investigate various problems in the representation theory of finite groups of Lie type. In Chapter 2, we hope to make sense of the last statement - we will introduce some background and notation that will be useful for the remainder of the thesis. In Chapter 3, we find bounds for the largest irreducible representation degree of a finite unitary group. In Chapter 4, we describe the block distribution and Brauer characters in cross characteristic for Sp₆(2ᵃ) in terms of the irreducible ordinary characters. This will be useful in Chapter 5 and Chapter 7, which focus primarily on the group Sp₆(2ᵃ) and contain the main results of this thesis, which we now summarize. Given a subgroup H ≤ G and a representation V for G, we obtain the restriction V|H of V to H by viewing V as an FH-module. However, even if V is an irreducible representation of G, the restriction V|H may (and usually does) fail to remain irreducible as a representation of H. In Chapter 5, we classify all pairs (V, H), where H is a proper subgroup of G = Sp₆(q) or Sp₄(q) with q even, and V is an l-modular representation of G for l ≠ 2 which is absolutely irreducible as a representation of H. This problem is motivated by the Aschbacher-Scott program on classifying maximal subgroups of finite classical groups. The local-global philosophy plays an important role in many areas of mathematics. In the representation theory of finite groups, the so-called "local-global" conjectures would relate the representation theory of G to that of certain proper subgroups, such as the normalizer of a Sylow subgroup. One might hope that these conjectures could be proven by showing that they are true for all simple groups. Though this turns out not quite to be the case, some of these conjectures have been reduced to showing that a finite set of stronger conditions hold for all finite simple groups. In Chapter 7, we show that Sp₆(q) and Sp₄(q), q even, are "good" for these reductions.

Dans cette thèse, on démontre que certaines représentations du groupe de Galois absolu d'une extension finie de $Q_p$ à coefficients dans $\bar{F_p}$ se relèvent en des représentations cristallines à coefficients dans $\bar{Z_p}$.

Cette thèse s'inscrit dans le cadre de la correspondance de Langlands locale $p$-adique, imaginée par Breuil et établie par Colmez pour GL_2(Q_p). Soit L une extension finie de Q_p et soit V une L-représentation irréductible du groupe de Galois absolu de Q_p, de dimension 2. En utilisant la théorie des (phi,Gamma)-modules de Fontaine, Colmez associe à V une GL_2(Q_p)-représentation de Banach Pi(V), unitaire, admissible, topologiquement irréductible. On donne une nouvelle preuve, nettement plus simple, d'un théorème de Colmez, qui permet de décrire les vecteurs localement analytiques Pi(V)^an de Pi(V) en fonction du (phi,\Gamma)-module surconvergent attaché à V. Le résultat principal de cette thèse est une description simple de l'action infinitésimale de GL_2(Q_p) sur Pi(V)^an. En particulier, on montre que Pi(V)^an admet un caractère infinitésimal, que l'on peut calculer en fonction des poids de Hodge-Tate de V, ce qui répond à une question de Harris. En utilisant ces résultats, on montre aussi l'absence d'un analogue p-adique d'un théorème classique de Tunnell et Saito, répondant à une autre question de Harris. Nous étendons et précisons certains résultats de Colmez concernant le modèle de Kirillov des vecteurs U-finis de Pi(V) (U est l'unipotent supérieur de GL_2(Q_p)). En combinant cette étude avec la description de l'action infinitésimale, on obtient une démonstration simple d'un des résultats principaux de Colmez, caractérisant les représentations V telles que Pi(V) possède des vecteurs localement algébriques non nuls. Ce résultat permet de faire le pont avec la correspondance classique et est un des ingrédients clés de la preuve d'Emerton de la conjecture de Fontaine-Mazur en dimension 2. On étend nos méthodes pour démontrer l'analogue de ce résultat pour les déformations infinitésimales de V. Cela répond à une question de Paskunas et a des applications à la conjecture de Breuil-Mézard. Une autre application est l'étude du module de Jacquet de Pi(V)^an. On montre qu'il est non nul si et seulement si V est trianguline, ce qui permet de donner une preuve simple des conjectures de Berger, Breuil et Emerton. Enfin, dans un travail en collaboration avec Benjamin Schraen, nous démontrons le lemme de Schur pour les représentations de Banach et localement analytiques topologiquement irréductibles d'un groupe de Lie p-adique. Ce résultat basique n'était connu que pour des groupes de Lie commutatifs et pour GL_2(Q_p).

L'étude des représentations d'un groupe algébrique complexe semi-simple connexe G est généralement menée en choisissant un sous-groupe de Borel B de G et un tore maximal T inclus dans B. Étant donnée une représentation de G sur un espace vectoriel V, il est dès lors naturel de vouloir étudier les bases de V compatibles avec ce choix de (B,T). Différents travaux de Zelevinsky, Berenstein, Lusztig et Kashiwara ont conduit aux notions de " base canonique ", de " bonne base ", de " base parfaite ", de " base en cordes ", ... , et à la construction de telles bases. Le but de ce mémoire est de présenter succintement cette théorie, d'exposer quelques propriétés remarquables de ces bases et de la combinatoire qu'elles définissent, et de proposer quelques perspectives.

The main focus of this thesis is algebraic modules---modules that satisfy a polynomial equation with integer co-efficients in the Green ring---in various finite groups, as well as their general theory. In particular, we ask the question `when are all the simple modules for a finite group G algebraic?' We call this the (p-)SMA property. The first chapter introduces the topic and deals with preliminary results, together with the trivial first results. The second chapter provides the general theory of algebraic modules, with particular attention to the relationship between algebraic modules and the composition factors of a group, and between algebraic modules and the Heller operator and Auslander--Reiten quiver. The third chapter concerns itself with indecomposable modules for dihedral and elementary abelian groups. The study of such groups is both interesting in its own right, and can be applied to studying simple modules for simple groups, such as the sporadic groups in the final chapter. The fourth chapter analyzes the groups PSL(2,q); here we determine, in characteristic 2, which simple modules for PSL(2,q) are algebraic, for any odd q. The fifth chapter generalizes this analysis to many groups of Lie type, although most results here are in defining characteristic only. Notable exceptions include the small Ree groups, which have the 2-SMA property for all q. The sixth and final chapter focuses on the sporadic groups: for most groups we provide results on some simple modules, and some of the groups are completely analyzed in all characteristics. This is normally carried out by restricting to the Sylow p-subgroup. This thesis develops the current state of knowledge concerning algebraic modules for finite groups, and particularly for which simple groups, and for which primes, all simple modules are algebraic.

It is impossible to overstate the importance of symmetry in physics and mathematics. Symmetry arguments play a central role in a broad range of problems from simplifying a system of linear equations to a deep role in organizing the fundamental principles of physics. They are used, for instance, in Noether’s theorem to find the consequences of symmetry of a dynamics. For many systems of interest, the dynamics are sufficiently complicated that one cannot hope to characterize their evolution completely, whereas by appealing to the symmetries of the dynamical laws one can easily infer many useful results. In part I of this thesis we study the problem of finding the consequences of symmetry of a (possibly open) dynamics from an information-theoretic perspective. The study of this problem naturally leads us to the notion of asymmetry of quantum states. The asymmetry of a state relative to some symmetry group specifies how and to what extent the given symmetry is broken by the state. Characterizing these is found to be surprisingly useful to constrain which final states of the system can be reached from a given initial state. Another motivation for the study of asymmetry comes from the field of quantum metrology and relatedly the field of quantum reference frames. It turns out that the degree of success one can achieve in many metrological tasks depends only on the asymmetry properties of the state used for metrology. We show that some ideas and tools developed in the field of quantum information theory are extremely useful to study the notion of asymmetry of states and therefore to find the consequences of symmetry of an open or closed system dynamics. In part II of this thesis we present a novel application of symmetry arguments in the field of quantum estimation theory. We consider a family of multi-copy estimation problems wherein one is given n copies of an unknown quantum state according to some prior distribution and the goal is to estimate certain parameters of the given state. In particular, we are interested to know whether collective measurements are useful and if so to find an upper bound on the amount of entanglement which is required to achieve the optimal estimation. We introduce a new approach to this problem by considering the symmetries of the prior and the symmetries of the parameters to be estimated. We show that based on these symmetries one can find strong constraints on the amount of entanglement required to implement the optimal measurement. In order to infer properties of the optimal estimation procedure from the symmetries of the parameters and the prior we come up with a generalization of Schur-Weyl duality. Just as Schur-Weyl duality has many applications to quantum information theory and quantum algorithms so too does this generalization. In this thesis we explore some of these applications.

The graphical modelling of processes is of growing popularity and high relevance to organisations that seek to document, analyse and improve their business operations. This research investigates the phenomenon of continued user acceptance of the grammars that are used to build process models. It develops and tests a theory that can be used to explain and predict why users would opt to continue working with certain grammars in their process modelling efforts. This study builds on established theories, including the Technology Acceptance Model, Expectation-Confirmation Theory, Task-Technology Fit Theory and Representation Theory. These theories suggest that end users typically strive for tools that are useful and easy to use, which confirm their expectations through firsthand utility, and which match task requirements and individual abilities. Representation theory suggests that modelling grammars should be complete and clear in their capabilities to represent real-world domains. The research model has been designed by combining conceptual studies of acceptance and continuance theories with a representational analysis of the BPMN grammar, which is a recently ratified industry standard for process modelling and thereby of high practical relevance to process modelling practice. It further incorporates findings from nineteen semi-structured interviews with process modellers in Australia. The research model has been tested and validated by means of a web-based survey with 590 process modellers world-wide. This thesis contributes to the body of knowledge in a number of ways: First, it presents an empirically validated model of the factors determining a user's intention to continue using a process modelling grammar. Second, it measures the impact that grammar characteristics as well as user and task characteristics have on user evaluations of a process modelling grammar. Third, it presents empirical evidence on the consequences that perceived representational deficiencies entail on user perceptions of a process modelling grammar.

Representation theory is concerned with the ways of writing elements of abstract algebraic structures as linear transformations of vector spaces. Typical structures amenable to representation theory are groups, associative algebras and Lie algebras. In this thesis we study linear representations of finite groups. The study focuses on character theory and how character theory can be used to extract information from a group. Prior to that, concepts needed to treat character theory, and some of their ramifications, are investigated. The study is based on existing literature, with excessive use of examples to illuminate important aspects. An example treating a class of p-groups is also discussed.

In this thesis, we explore the interface between symbolic and dynamical system computation, with particular regard to dynamical system models of neuronal networks. In doing so, we adhere to a definition of computation as the physical realization of a formal system, where we say that a dynamical system performs a computation if a correspondence can be found between its dynamics on a vectorial space and the formal system’s dynamics on a symbolic space. Guided by this definition, we characterize computation in a range of neuronal network models. We first present a constructive mapping between a range of formal systems and Recurrent Neural Networks (RNNs), through the introduction of a Versatile Shift and a modular network architecture supporting its real-time simulation. We then move on to more detailed models of neural dynamics, characterizing the computation performed by networks of delay-pulse-coupled oscillators supporting the emergence of heteroclinic dynamics. We show that a correspondence can be found between these networks and Finite-State Transducers, and use the derived abstraction to investigate how noise affects computation in this class of systems, unveiling a surprising facilitatory effect on information transmission. Finally, we present a new dynamical framework for computation in neuronal networks based on the slow-fast dynamics paradigm, and discuss the consequences of our results for future work, specifically for what concerns the fields of interactive computation and Artificial Intelligence.

abstract: Since Duffin and Schaeffer's introduction of frames in 1952, the concept of a frame has received much attention in the mathematical community and has inspired several generalizations. The focus of this thesis is on the concept of an operator-valued frame (OVF) and a more general concept called herein an operator-valued frame associated with a measure space (MS-OVF), which is sometimes called a continuous g-frame. The first of two main topics explored in this thesis is the relationship between MS-OVFs and objects prominent in quantum information theory called positive operator-valued measures (POVMs). It has been observed that every MS-OVF gives rise to a POVM with invertible total variation in a natural way. The first main result of this thesis is a characterization of which POVMs arise in this way, a result obtained by extending certain existing Radon-Nikodym theorems for POVMs. The second main topic investigated in this thesis is the role of the theory of unitary representations of a Lie group G in the construction of OVFs for the L^2-space of a relatively compact subset of G. For G=R, Duffin and Schaeffer have given general conditions that ensure a sequence of (one-dimensional) representations of G, restricted to (-1/2,1/2), forms a frame for L^{2}(-1/2,1/2), and similar conditions exist for G=R^n. The second main result of this thesis expresses conditions related to Duffin and Schaeffer's for two more particular Lie groups: the Euclidean motion group on R^2 and the (2n+1)-dimensional Heisenberg group. This proceeds in two steps. First, for a Lie group admitting a uniform lattice and an appropriate relatively compact subset E of G, the Selberg Trace Formula is used to obtain a Parseval OVF for L^{2}(E) that is expressed in terms of irreducible representations of G. Second, for the two particular Lie groups an appropriate set E is found, and it is shown that for each of these groups, with suitably parametrized unitary duals, the Parseval OVF remains an OVF when perturbations are made to the parameters of the included representations. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2014