We give natural correspondences of odd-degree characters of the symmetric groups and some of their subgroups, which can be described easily by restriction of characters, degrees and multiplicities.

A determinação dos invariantes numéricos associados a germes de aplicações diferenciáveis é uma ferramenta muito útil no estudo de problemas de equisingularidade em famílias. Em geral, estes invariantes são obtidos algebricamente através de esquemas r-dimensionais, que surgem nos tipos estáveis de uma perturbação estável do germe. Neste trabalho é feito um estudo sobre estes invariantes nos tipos estáveis de germes de aplicações holomorfas f : (Cn,0) em (Cn,0) finitamente determinados de coposto 1. Inicialmente é feita uma caracterização completa de todos os tipos estáveis, bem como de sua geometria. Como aplicações são estudados os invariantes no discriminante de germes quase homogêneos. São descritas fórmulas para os invariantes 0-stáveis de germes de (Cn,0) em (Cn,0). Estes resultados são aplicados para o cálculo das multiplicidades polares do discriminante de germes quase homogêneos de (C3,0) em (C3,0). / The determination of the numerical invariants associated to map germs is a helpful tool in the study of problems of equisingularity in families. In general, these invariants are given as zero schemes, that appear in the stable types of a stable perturbation of the germ. In this work we study the invariants in the stable types of corank one finitely determined holomorphic map germs f : (Cn,0) to (Cn,0). First we completely characterize all stable types and study their geometry. As applications are studied the invariants in the discriminant of weighted homogeneous germs. Formulas are described for the 0-stable invariants of map germs of (Cn,0) to (Cn,0) and these results are applied to compute the polar multiplicities of the discriminant of weighted homogeneous germes of (C3,0) to (C3,0).

A determinação dos invariantes numéricos associados a germes de aplicações diferenciáveis é uma ferramenta muito útil no estudo de problemas de equisingularidade em famílias. Em geral, estes invariantes são obtidos algebricamente através de esquemas r-dimensionais, que surgem nos tipos estáveis de uma perturbação estável do germe. Neste trabalho é feito um estudo sobre estes invariantes nos tipos estáveis de germes de aplicações holomorfas f : (Cn,0) em (Cn,0) finitamente determinados de coposto 1. Inicialmente é feita uma caracterização completa de todos os tipos estáveis, bem como de sua geometria. Como aplicações são estudados os invariantes no discriminante de germes quase homogêneos. São descritas fórmulas para os invariantes 0-stáveis de germes de (Cn,0) em (Cn,0). Estes resultados são aplicados para o cálculo das multiplicidades polares do discriminante de germes quase homogêneos de (C3,0) em (C3,0). / The determination of the numerical invariants associated to map germs is a helpful tool in the study of problems of equisingularity in families. In general, these invariants are given as zero schemes, that appear in the stable types of a stable perturbation of the germ. In this work we study the invariants in the stable types of corank one finitely determined holomorphic map germs f : (Cn,0) to (Cn,0). First we completely characterize all stable types and study their geometry. As applications are studied the invariants in the discriminant of weighted homogeneous germs. Formulas are described for the 0-stable invariants of map germs of (Cn,0) to (Cn,0) and these results are applied to compute the polar multiplicities of the discriminant of weighted homogeneous germes of (C3,0) to (C3,0).

Un des buts de la collaboration COMPASS est l'étude de la structure de spin du nucléon. La question de la polarisation des quarks de la mer est un sujet capital en physique hadronique, en particulier pour la polarisation du quark étrange. En vue de mieux contraindre la polarisation des quarks, une connaissance précise des fonctions de fragmentation (FFs), qui expriment l'hadronisation d'un quark q en un hadron h dans l'état final, est nécessaire. Les FFs peuvent être extraites depuis les multiplicités de hadrons produites en diffusion inélastique profonde semi-inclusive (SIDIS). Les données ont été prises à COMPASS avec un faisceau de muons de 160 GeV/c diffusant sur une cible de protons pure (1H₂). La présente thèse présente les mesures des multiplicités de hadrons chargés (pions, kaons et protons) faites à partir des données SIDIS collectées en 2016. Elle détaille aussi les améliorations apportées au générateur d'événement DJANGOH dans le but d'améliorer la description des corrections radiative inclusive et semi-inclusive qui sont ensuite utilisées comme facteurs de corrections aux multiplicités. Les données couvrent un large spectre cinématique : Q² > 1 (GeV/c)², y ε [0.1,0.7], x ε [0.004,0.4], W ε [5,17] GeV et z ε [0.2,0.85]. Ces multiplicités, qui représentent un total d'environ 1800 points de données, apportent une contribution importante aux fit QCD globaux des données mondiales à NLO, visant à la détermination des FFs. Les FFs de quarks en kaons sont particulièrement attendues car elles pourront mieux contraindre la polarization du quark étrange. / One of the goals of the COMPASS collaboration is the study of the nucleon spin structure. The question of the polarization of the sea quark is a burning issue in the hadronic physics, especially for the strange quark polarization. In order to better constrain the quark polarization, a precise knowledge of the quark Fragmentation Functions (FFs) into hadrons, which are the final state hadronisation of quark q into hadron h, is mandatory. The FFs can be extracted from hadron multiplicities produced in Semi-Inclusive Deep Inelastic Scattering (SIDIS). Data were taken at COMPASS from a 160 GeV/c muon beam scattering off a pure proton target (1H₂). This thesis presents the measurement of charged hadrons (pions, kaons and protons) multiplicities from SIDIS data collected in 2016. It also details the improvements brought to the DJANGOH event generator to better describe the inclusive and semi-inclusive radiative corrections in DIS that are then used as correction factors to the multiplicities. The data cover a large kinematical range : Q² > 1 (GeV/c)², y ε [0.1,0.7], x ε [0.004,0.4], W ε [5,17] GeV et z ε [0.2,0.85]. These multiplicities, which represent about 1800 data points in total, provide an important input for global QCD fit of world data at NLO, aiming at the FFs determination. The quark FFs into kaons are particularly awaited as they can better constrain the strange quark polarization.

Understanding the nucleon structure is currently one of the main challenges encountered in nuclear physics. The present work represents a contribution to the study of the nucleon structure and deals, in particular, with the study of the role of strange quarks in the nucleon. The latter can be investigated by determing the strange quark distribution in the nucleon as well as the contribution of the spins of strange quarks to the nucleon spin ($\Delta s$). This work first presents a measurement of $\Delta s$ performed via Deeply Inelastic Scattering of a muon beam off polarized proton and deuterium targets. The result is found to be strongly dependent on the quark fragmentation functions into hadrons (FFs), which define the probability that a quark of a given flavour fragments into a final state hadron. The FFs are poorly known, in particular, the FF of strange quark into kaons, which play an important role in the determination of $\Delta s$. In deep inelastic scattering process, the access to the FFs is provided by the hadron multiplicities which, in turn, define the average number of hadrons produced per DIS event. Pion and kaon multiplicities have been extracted versus different kinematic variables, using DIS data collected by deeply inelastic scattering of a $160$ GeV muons off a deuterium target. A first LO extraction of the fragmentation functions has then been performed using the measured pion and kaon multiplicities.

The direct sum decomposition of tensor products for SU(3) has many applications in physics, and the problem has been studied extensively. This has resulted in many decomposition methods, each with its advantages and disadvantages. The description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities. In addition to providing decompositions of specific tensor products, this approach is very well suited to studying tensor products as the parameters vary, and drawing general conclusions. After a description and proof of the method, several applications are discussed and proved. The decompositions are also studied further for the special cases of tensor products of an irreducible representation with itself or with its conjugate. In particular, questions regarding multiplicities are considered. As an extension of this diagrammatic method, the repeated tensor product of N copies of the fundamental representation is studied, and a method for its decomposition is provided. Again, questions regarding multiplicities are considered.

In 1976, Uhlenbeck used transversality theory to show that for certain families of elliptic operators, the property of having only simple eigenvalues is generic. As one application, she proved that on a closed Riemannian manifold, the eigenvalues of the Laplace-Beltrami operator Δg are all simple for a residual set of Cr metrics. In 2012, Enciso and Peralta-Salas established an analogue of Uhlenbeck's theorem for differential forms, showing that on a closed 3-manifold, there exists a residual set of Cr metrics such that the nonzero eigenvalues of the Hodge Laplacian Δg(k) on k-forms are all simple for 0 ≤ k ≤ 3. In this dissertation, we continue to address the question of whether Uhlenbeck's theorem can be extended to differential forms. In particular, we prove that for a residual set of Cr metrics, the nonzero eigenvalues of the Hodge Laplacian Δg(2) acting on coexact 2-forms on a closed 5-manifold have multiplicity 2. To prove our main result, we structure our argument around a study of the Beltrami operator *gd, which is related to the Hodge Laplacian by Δg(2) = -(*gd)2 when the operators are restricted to coexact 2-forms on a 5-manifold. We use techniques from perturbation theory to show that the Beltrami operator has only simple eigenvalues for a residual set of metrics. We further establish even eigenvalue multiplicities for the Hodge Laplacian acting on coexact k-forms in the more general setting n = 4 ℓ + 1 and k = 2 ℓ for ℓ ϵ N.

The direct sum decomposition of tensor products for SU(3) has many applications in physics, and the problem has been studied extensively. This has resulted in many decomposition methods, each with its advantages and disadvantages. The description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities. In addition to providing decompositions of specific tensor products, this approach is very well suited to studying tensor products as the parameters vary, and drawing general conclusions. After a description and proof of the method, several applications are discussed and proved. The decompositions are also studied further for the special cases of tensor products of an irreducible representation with itself or with its conjugate. In particular, questions regarding multiplicities are considered. As an extension of this diagrammatic method, the repeated tensor product of N copies of the fundamental representation is studied, and a method for its decomposition is provided. Again, questions regarding multiplicities are considered.

[pt] Uma questão central percorre a filosofia de Gilles Deleuze, a da determinação das multiplicidades nelas mesmas: o que são; qual a sua matéria; seus elementos; o tipo de relação que elas implicam; sua realidade singular e seus acontecimentos. Nossa pretensão foi explorar essa questão a partir do mapeamento de três registros fundamentais: o primeiro, correspondente à emergência das multiplicidades no livro Bergsonismo; o segundo, correspondente à elaboração de uma teoria das multiplicidades, propriamente deleuziana, presente em Diferença e Repetição e; por fim, em uma projeção prático-política a que o conceito é alçado em Mil Platôs. / [en] A central question runs through the Gilles Deleuze s philosophy, the determination of multiplicities in themselves: what they are; what the substance is; their elements; the type of relationship they imply; their singular reality and their events. Our intention was to explore this issue the mapping of three fundamental registers: the first corresponds to the emergency of multiplicities in the Bergsonism s book; the second corresponds to the elaboration of a theory of multiplicities, properly deleuzian, present in Difference and Repetition and, finally, in its pratical-political projection to which the concept is raised in a Thousand Plateaus.

One of the goals of the COMPASS experience is the study of the nucleon spin structure. Data were taken from a polarized muon beam (160 GeV/c) scattering off a polarized target (6LiD or NH3). In this context, the need of a precise knowledge of quark Fragmentation Functions (final-state hadronisation of quarks q into hadrons h, FFs) was raised. The FFs can be extracted from hadron multiplicities produced in Semi-Inclusive Deep Inelastic Scattering (SIDIS). This thesis presents the measurement of charged hadrons (pions and kaons) multiplicities from SIDIS data collected in 2006. The data cover a large kinematical range: Q2>1 (GeV/c)2, y є [0.1,0.9], x є [0.004,0.7] and W є [5,17] GeV. These multiplicities provide an important input for global QCD analyses of world data at NLO, aiming at the FFs determination. / L'un des objectifs de l'expérience COMPASS est l'étude de la structure du nucléon de spin. Les données ont été prises à partir d'un faisceau de muons polarisée (160 GeV/c) diffuse sur une cible polarisée (6LiD ou NH3). Dans ce contexte, la nécessité d'une connaissance précise des fonctions de fragmentation des quarks (état final du hadronisation de quarks q en hadrons h, FFs) a été soulevée. Le FFs peuvent être extraites de multiplicités de hadrons produits en Semi-Inclusive diffusion profondément inélastique (SIDIS). Cette thèse présente la mesure de la multiplicité de hadrons charge (pions et kaons) à partir de données SIDIS collectées en 2006. Les données couvrent un large domaine cinématique : Q2>1 (GeV/c)2, y є [0.1,0.9], x є [0.004,0.7] and W є [5,17] GeV. Ces multiplicités fournissent un apport important pour l'analyse des données mondiales au 2ème ordre de QCD, visant la détermination de FFs.