This paper forms an introductory account of the irreducible representations of the permutation group using Young Tableaux as the tool to achieve this. The basics of C*-Algebra theory and Young Tableaux are provided including a brief history of the two subjects. This paper provides a straightforward development of the subject up to the main result which says that restricting the irreducible representations of S_n corresponding to the Young diagrams of shape ? to S_n-1 decomposes as the direct sum of the irreducible representations of S_n-1 corresponding to the Young diagrams formed by removing one box from ?. / Graduate

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In this paper, we discuss the need for quantum error correction. We also describe some basic techniques used in quantum error correction which includes decoherence-free subspaces and subsystems. These subspaces and subsystems are described in detail. We also introduce a numerical algorithm that was used previously to search for these decoherence-free subspaces and subsystems under collective error. It is useful to search for them as they can be used to store quantum information. We use this algorithm in some specific examples involving qubits and qutrits. The results of these algorithm are then compared with the error algebra obtained using Young tableaux. We use these results to describe how the specific numerical algorithm can be used for the search of approximate decoherence-free subspaces and subsystems and minimal noise subsystems.

We study a quantum mechanical $\sigma$-model whose target space is a hyperKähler cone. As shown by Singleton, [184], such a theory has superconformal invariance under the algebra $\mathfrak{osp}(4^*|4)$. One can formally define a superconformal index that counts the short representations of the algebra. When the hyperKähler cone has a projective symplectic resolution, we define a regularised superconformal index. The index is defined as the equivariant Hirzebruch index of the Dolbeault cohomology of the resolution, hereafter referred to as the index. In many cases, the index can be explicitly calculated via localisation theorems. By limiting to zero the fugacities in the index corresponding to an isometry, one forms the index of the submanifold of the target space invariant under that isometry. There is a limit of the fugacities that gives the Hilbert series of the target space, and often there is another limit of the parameters that produces the Poincaré polynomial for $\mathbb C^\times$-equivariant Borel-Moore homology of the space. A natural class of hyperKähler cones are Nakajima quiver varieties. We compute the index of the $A$-type quiver varieties by making use of the fact that they are submanifolds of instanton moduli space invariant under an isometry. Every Nakajima quiver variety arises as the Higgs branch of a three dimensional $\mathcal N =4$ quiver gauge theory, or equivalently the Coulomb branch of the mirror dual theory. We show the equivalence between the descriptions of the Hilbert series of a line bundle on the ADHM quiver variety via localisation, and via Hanany's monopole formula. Finally, we study the action of the Poisson algebra of the coordinate ring on the Hilbert series of line bundles. We restrict to the case of looking at the Coulomb branch of balanced $ADE$-type quivers in a certain infinite rank limit. In this limit, the Poisson algebra is a semiclassical limit of the Yangian of $ADE$-type. The space of global sections of the line bundle is a graded representation of the Poisson algebra. We find that, as a representation, it is a tensor product of the space of holomorphic functions with a finite dimensional representation. This finite dimensional representation is a tensor product of two irreducible representations of the Yangian, defined by the choice of line bundle. We find a striking duality between the characters of these finite dimensional representations and the generating function for Poincaré polynomials.

The limiting law of the length of the longest increasing subsequence, LI_n, for sequences (words) of length n arising from iid letters drawn from finite, ordered alphabets is studied using a straightforward Brownian functional approach. Building on the insights gained in both the uniform and non-uniform iid cases, this approach is then applied to iid countable alphabets. Some partial results associated with the extension to independent, growing alphabets are also given. Returning again to the finite setting, and keeping with the same Brownian formalism, a generalization is then made to words arising from irreducible, aperiodic, time-homogeneous Markov chains on a finite, ordered alphabet. At the same time, the probabilistic object, LI_n, is simultaneously generalized to the shape of the associated Young tableau given by the well-known RSK-correspondence. Our results on this limiting shape describe, in detail, precisely when the limiting shape of the Young tableau is (up to scaling) that of the iid case, thereby answering a conjecture of Kuperberg. These results are based heavily on an analysis of the covariance structure of an m-dimensional Brownian motion and the precise form of the Brownian functionals. Finally, in both the iid and more general Markovian cases, connections to the limiting laws of the spectrum of certain random matrices associated with the Gaussian Unitary Ensemble (GUE) are explored.

Une approche inductive est développée pour la théorie des représentations de la chaîne des algèbres de Hecke cyclotomiques de type G(m,1,n). Cette approche repose sur l'étude du spectre d'une famille commutative maximale, formée par les analogues des éléments de Jucys--Murphy.Les représentations irréductibles, paramétrées par les multi-partitions, sont construites avec l'aide d'une nouvelle algèbre associative, dont l'espace vectoriel sous-jacent est le produit tensoriel de l'algèbre de Hecke cyclotomique avec l'algèbre associative libre engendrée par les multi-tableaux standards.L'analogue de cette approche est présentée pour la limite classique, c'est-à-dire la chaîne des groupes de réflexions complexes de type G(m,1,n).Dans une seconde partie, une base des algèbres de Hecke cyclotomiques est donnée et la platitude de la déformation est montrée sans utiliser la théorie des représentations. Ces résultats sont généralisés aux algèbres de Hecke affines de type A.Ensuite, une procédure de fusion est présentée pour les groupes de réflexions complexes et les algèbres de Hecke cyclotomiques de type G(m,1,n). Dans les deux cas, un ensemble complet d'idempotents primitifs orthogonaux est obtenu par évaluation consécutive d'une fonction rationnelle.Dans une troisième partie, une nouvelle présentation est obtenue pour les sous-groupes alternés de tous les groupes de Coxeter. Les générateurs sont reliés aux arêtes orientées du graphe de Coxeter. Cette présentation est ensuite étendue, pour tous les types, aux extensions spinorielles des groupes alternés, aux algèbres de Hecke alternées et aux sous-groupes alternés des groupes de tresses. / An inductive approach to the representation theory of the chain of the cyclotomic Hecke algebras of type G(m,1,n) is developed. This approach relies on the study of the spectrum of a maximal commutative family formed by the analogues of the Jucys--Murphy elements.The irreducible representations, labelled by the multi-partitions, are constructed with the help of a new associative algebra, whose underlying vector space is the tensor product of the cyclotomic Hecke algebra with the free associative algebra generated by the standard multi-tableaux.The analogue of this approach is presented for the classical limit, that is for the chain of complex reflection groups of type G(m,1,n).In a second part, a basis of the cyclotomic Hecke algebras is given and the flatness of the deformation is proved without using the representation theory. These results are extended to the affine Hecke algebras of type A.Then a fusion procedure is presented for the complex reflection groups and the cyclotomic Hecke algebras of type G(m,1,n). In both cases, a complete set of primitive orthogonal idempotents is obtained by successive evaluations of a rational fonction.In a third part, a new presentation is obtained for the alternating subgroups of all Coxeter groups. The generators are related to oriented edges of the Coxeter graph. This presentation is then extended, for all types, to the spinor extensions of the alternating groups, the alternating Hecke algebras and the alternating subgroups of braid groups.

Cette thèse est consacrée à l’étude des monoïdes plaxiques par une nouvelle approche utilisant des méthodes issues de la réécriture. Ces méthodes sont appliquées à des présentations de monoïdes plaxiques décrites en termes de tableaux de Young, de bases cristallines de Kashiwara et de modèle des chemins de Littelmann. On étudie le problème des syzygies pour la présentation de Knuth des monoïdes plaxiques. En utilisant la procédure de complétion homotopique basée sur les procédures de complétion de Squier et de Knuth–Bendix, on construit des présentations cohérentes de monoïdes plaxiques de type A. Une telle présentation cohérente étend la notion de présentation convergente d’un monoïde par une famille génératrice de syzygies, décrivant toutes les relations entre les relations. On explicite une présentation cohérente finie des monoïdes plaxiques de type A avec les générateurs colonnes. Cependant, cette présentation n’est pas minimale dans le sens que plusieurs de ses générateurs sont superflus. En appliquant la procédure de réduction homotopique, on réduit cette présentation en une présentation cohérente finie qui étend la présentation de Knuth, donnantainsi toutes les syzygies des relations de Knuth. D’une manière plus générale, on étudie des présentations de monoïdes plaxiques généralisés du point de vue de la réécriture. On construit des présentations convergentes finies de ces monoïdes en utilisant les chemins de Littelmann. De plus, on étudie ces présentations pour le type C en termes de bases cristallines de Kashiwara. En introduisant les générateurs colonnes admissibles, on construit une présentation convergente finie du monoïde plaxique de type C avec des relations explicites. Cette approche nous permettrait d’étudier le problème des syzygies des présentations de monoïdes plaxiques en tout type / This thesis focuses on the study of plactic monoids by a new approach using methods issued from rewriting theory. These methods are applied on presentations of plactic monoids given in terms of Young tableaux, Kashiwara’s crystal bases and Littelmann path model. We study the syzygy problem for the Knuth presentation of the plactic monoids. Using the homotopical completion procedure that extends Squier’s and Knuth–Bendix’s completions procedure, we construct coherent presentations of plactic monoids of type A. Such a coherent presentation extends the notion of a presentation of a monoid by a family of generating syzygies, taking into account all the relations among the relations. We make explicit a finite coherent presentation of plactic monoids of type A with the column generators. However, this presentation is not minimal in the sense that many of its generators are superfluous. After applying the homotopical reduction procedure on this presentation, we reduce it to a finite coherent one that extends the Knuth presentation, giving then all the syzygies of the Knuth relations. More generally, we deal with presentations of plactic monoids of any type from the rewriting theory perspective. We construct finite convergent presentations for these monoids in a general way using Littelmann paths. Moreover, we study the latter presentations in terms of Kashiwara’s crystal graphs for type C. By introducing the admissible column generators, we obtain a finite convergent presentation of the plactic monoid of type C with explicit relations. This approach should allow us to study the syzygy problem for the presentations of plactic monoids for any type

Recent efforts to classify off-shell representations of supersymmetry without a central charge have focused upon directed, supermultiplet graphs of hypercubic topology known as Adinkras. These encodings of Super Poincare algebras, depict every generator of a chosen supersymmetry as a node-pair transformtion between fermionic bosonic component fields. This research thesis is a culmination of investigating novel diagrammatic sums of gauge-quotients by supersymmetric images of other Adinkras, and the correlated building of field theoretic worldline Lagrangians to accommodate both classical and quantum venues. We find Ref [40], that such gauge quotients do not yield other stand alone or "proper" Adinkras as afore sighted, nor can they be decomposed into supermultiplet sums, but are rather a connected "Adinkraic network". Their iteration, analogous to Weyl's construction for producing all finite-dimensional unitary representations in Lie algebras, sets off chains of algebraic paradigms in discrete-graph and continuous-field variables, the links of which feature distinct, supersymmetric Lagrangian templates. Collectively, these Adiankraic series air new symbolic genera for equation to phase moments in Feynman path integrals. Guided in this light, we proceed by constructing Lagrangians actions for the N = 3 supermultiplet YI /(iDI X) for I = 1, 2, 3, where YI and X are standard, Salam-Strathdee superfields: YI fermionic and X bosonic. The system, bilinear in the component fields exhibits a total of thirteen free parameters, seven of which specify Zeeman-like coupling to external background (magnetic) fluxes. All but special subsets of this parameter space describe aperiodic oscillatory responses, some of which are found to be surprisingly controlled by the golden ratio, [phi] = 1.61803, Ref [52]. It is further determined that these Lagrangians allow an N = 3 - > 4 supersymmetric extension to the Chiral-Chiral and Chiral-twistedChiral multiplet, while a subset admits two inequivalent such extensions. In a natural proiii gression, a continuum of observably and usefully inequivalent, finite-dimensional off-shell representations of worldline N = 4 extended supersymmetry are explored, that are variate from one another but in the value of a tuning parameter, Ref [53]. Their dynamics turns out to be nontrivial already when restricting to just bilinear Lagrangians. In particular, we find a 34-parameter family of bilinear Lagrangians that couple two differently tuned supermultiplets to each other and to external magnetic fluxes, where the explicit parameter dependence is unremovable by any field redefinition and is therefore observable. This offers the evaluation of X-phase sensitive, off-shell path integrals with promising correlations to group product decompositions and to deriving source emergences of higher-order background flux-forms on 2-dimensional manifolds, the stacks of which comprise space-time volumes. Application to nonlinear sigma models would naturally follow, having potential use in M- and F- string theories.