The Khovanov{Lauda{Rouquier algebras Rn are a relatively new family of Z-graded algebras. Their cyclotomic quotients R n are intimately connected to a smaller family of algebras, the cyclotomic Hecke algebras H n of type A, via Brundan and Kleshchev's Graded Isomorphism Theorem. The study of representation theory of H n is well developed, partly inspired by the remaining open questions about the modular representations of the symmetric group Sn. There is a profound interplay between the representations for Sn and combinatorics, whereby each irreducible representation in characteristic zero can be realised as a Specht module whose basis is constructed from combinatorial objects. For R n , we can similarly construct their representations as analogous Specht modules in a combinatorial fashion. Many results can be lifted through the Graded Isomorphism Theorem from the symmetric group algebras, and more so from H n , to the cyclotomic Khovanov{Lauda{Rouquier algebras, providing a foundation for the representation theory of R n . Following the introduction of R n , Brundan, Kleshchev and Wang discovered that Specht modules over R n have Z-graded bases, giving rise to the study of graded Specht modules. In this thesis we solely study graded Specht modules and their irreducible quotients for R n . One of the main problems in graded representation theory of R n , the Graded Decomposition Number Problem, is to determine the graded multiplicities of graded irreducible R n -modules arising as graded composition factors of graded Specht modules. We rst consider R n in level one, which is isomorphic to the Iwahori{Hecke algebra of type A, and research graded Specht modules labelled by hook partitions in this context. In quantum characteristic two, we extend to R n a result of Murphy for the symmetric groups, determining graded ltrations of Specht modules labelled by hook partitions, whose factors appear as Specht modules labelled by two-part partitions. In quantum characteristic at least three, we determine an analogous R n -version of Peel's Theorem for the symmetric groups, providing an alternative approach to Chuang, Miyachi and Tan. We then study graded Specht modules labelled by hook bipartitions for R n in level two, which is isomorphic to the Iwahori{Hecke algebra of type B. In quantum characterisitic at least three, we completely determine the composition factors of Specht modules labelled by hook bipartitions for R n , together with their graded analogues.

Les algèbres de Temperley–Lieb de type B, aussi appelées algèbres de Temperley–Lieb à une frontière, sont une famille d’algèbres associatives unitaires de dimension finie généralisant les algèbres de Temperley–Lieb. Elles ont été introduites en 1992 par P.P. Martin et H. Saleur pour la résolution de modèles en mécanique statistique [MS94], mais elles ont rapidement pris de l’importance en théorie de la représentation suite aux travaux de P.P. Martin et D. Woodcock [MW00] [MW03], qui montrent qu’elles s’obtiennent comme quotient d’al- gèbres de Hecke cyclotomiques et qui observent des liens profonds avec la théorie de Lie. Ces quotients sont liés aux algèbres de Khovanov–Lauda–Rouquier (KLR) par les travaux de Brundan et Kleshchev [BK09]; c’est à l’aide des algèbres KLR et de leur formulation diagrammatique que les résultats de ce mémoire seront obtenus. Elles seront maintenant appelées algèbres blob. Ce mémoire porte sur la théorie de la représentation de certains blocs des algèbres blob. Plus précisément, nous trouvons les carquois et relations décrivant les catégories de modules des blocs réguliers en caractéristique nulle. Les résultats sont obtenus par calcul diagram- matique, en utilisant la base cellulaire construite par Plaza–Ryom-Hansen [PRH14] et les idempotents primitifs de Hazi–Martin–Parker [HMP21]. Structure du mémoire: Le premier chapitre rappelle brièvement les notions algébriques qui seront utilisées. Le deuxième chapitre présente les algèbres blob de façon algébrique et diagrammatique, puis plusieurs résultats connus sur celles-ci. Les troisième et quatrième chapitres contiennent tous les résultats originaux, c’est-à-dire le calcul du carquois et relations pour les blocs réguliers. / The Temperley–Lieb algebras of type B, also known as one-boundary Temperley–Lieb al- gebras, are a family of unitary associative algebras of finite dimension that generalize the Temperley–Lieb algebras. They were introduced in 1992 by P.P Martin and H. Saleur for solving models in statistical mechanics [MS94] but they quickly became important in rep- resentation theory following the work of P.P. Martin and D. Woodcock [MW00] [MW03], who showed that they can be realized as quotients of cyclotomic Hecke algebras and observed deep connections with Lie theory. These quotients are related to Khovanov–Lauda–Rouquier (KLR) algebras through the work of Brundan and Kleshchev [BK09]; it is with the help of KLR algebras and their diagrammatic presentation that the results of this thesis will be obtained. They will now be referred to as blob algebras. This thesis focuses on the representation theory of certain blocks of blob algebras. Specif- ically, we find the quivers and relations describing the module categories of regular blocks in characteristic zero. The results are obtained through diagrammatic calculus, using the cellular basis constructed by Plaza–Ryom-Hansen [PRH14] and the primitive idempotents of Hazi–Martin–Parker [HMP21]. Structure: The first chapter briefly recalls the algebraic concepts that will be used. The second chapter presents blob algebras in both algebraic and diagrammatic ways, along with several known results about them. The third and fourth chapters contain all the original results, namely the calculation of quivers and relations for regular blocks.