The broad topic of this dissertation is the study of algebraic structure arising from polyhedral geometric objects. There are three distinct topics covered over three main chapters. However, each of these topics are further linked by a connection to the Eulerian polynomials. Chapter 2 studies Euler-Mahonian identities arising from both the symmetric group and generalized permutation groups. Specifically, we study the algebraic structure of unit cube semigroup algebra using Gröbner basis methods to acquire these identities. Moreover, this serves as a bridge between previous methods involving polyhedral geometry and triangulations with descent bases methods arising in representation theory. In Chapter 3, the aim is to characterize Hilbert basis elements of certain 𝒔-lecture hall cones. In particular, the main focus is the classification of the Hilbert bases for the 1 mod 𝑘 cones and the 𝓁-sequence cones, both of which generalize a previous known result. Additionally, there is much broader characterization of Hilbert bases in dimension ≤ 4 for 𝒖-generated Gorenstein lecture hall cones. Finally, Chapter 4 focuses on certain algebraic and geometric properties of 𝒔-lecture hall polytopes. This consists of partial classification results for the Gorenstein property, the integer-decomposition property, and the existence of regular, unimodular triangulations.

Tout au long de cette thèse, l’évaluation musicale est traitée comme un objet d’étude latent visant à donner des outils pour l’apprentissage de la lecture musicale. Grâce à l’analyse des mouvements oculaires et les variables provenant de la performance, nous avons défini certaines variables qui rendent compte de l’expertise et des interactions entre différents groupes de niveaux d’expertise musicale lors de l’apprentissage d’un nouveau morceau de musique. De façon plus détaillée, nous avons observé la mise en œuvre de différentes stratégies de prise d’information, de traitement et de récupération de l’information musicale en fonction du niveau pianistique et souligné l’importance d’apprendre en interaction avec la classe et le professeur. Les stratégies sont en même temps ajustées par rapport à la confiance acquise au cours du processus d'acquisition de compétences (Bandura, 1997 ; McPherson et McCormick, 2006). En référence au débat actuel concernant la nature de la lecture de partitions, nous avons comparé les traitements musicaux et verbaux pendant une tâche de lecture « compréhensive » de textes et de partitions. Dans l’ensemble et au regard du modèle de Baddeley (1990), les ressources cognitives des musiciens pendant la lecture musicale seraient mobilisées en fonction de l’expertise et du type de style musical. / Throughout this thesis, evaluation of music performance is viewed as a latent object of study in order to provide tools for learning to read music. We have defined some variables from eye movements and music performance accounting for expert performance and interactions between skill groups when learning a new piece of music. In more details, we have observed the use of different strategies for music information intake, processes and information retrieval depending on musicians’ expertise and we have stressed the importance of learning through interaction. In the process of skill acquisition, when self-confidence is gained strategies are simultaneously adjusted (Bandura, 1997; McPherson and McCormick, 2006). In reference to the current debate about the nature of music reading, we have compared musical and verbal processing during comprehensive reading of texts and scores. On the whole, considering the model of Baddeley (1990), musicians’ cognitive resources during music reading would be mobilized depending on the expertise and the music style.