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The paper comprises a graphical calculus which is designed to deal with the Coxeter-Dynkin series of type E and some generalizations. Temperley-Lieb algebras of type E are defined as quotients of Hecke algebras and the module structure of the algebra associated to E 6 is determined. The graphical calculus is a refinement of the calculus for the ordinary Temperley-Lieb algebra: a planar strip is decomposed by the arcs of a diagram into domains and the domains are used to incorporate additional information into the figure.

Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving operator algebra, as poset algebra, as commutant algebra, ...). In type A, the Hecke-group algebra can be described as the algebra generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. It turns out to be closely related to the monoid algebras of respectively nondecreasing functions and nondecreasing parking functions, the representation theory of which we describe as well. This defines three towers of algebras, and we give explicitly the Grothendieck algebras and coalgebras given respectively by their induction products and their restriction coproducts. This yields some new interpretations of the classical bases of quasi-symmetric and noncommutative symmetric functions as well as some new bases., v2: 30 pages, 2 figures, extended proof of Prop. 3.25, update of citations, typo and grammar fixes v3: final version: typos and encoding fixes

This work contributes to the general study of linear representations of Artin's Braid group $B_n$ that arise as monodromy of KZ-systems. We consider these systems as representations of the Hopf algebra of infinitesimal braids, and apply the technique of Gelfand-Tsetlin basis. The purpose is twofold : this technique gives a good insight into the representation theory of this algebra, and we show that it helps in the explicit construction of the corresponding braid group representations. We give a complete classification of KZ-systems that are irreducible for the action of the symmetric group, and build the new representations of $B_n$ that arise at this stage. Among other results that are useful in this setting, we obtain irreducibility criteria on tensor products and related constructions, and get a partial decomposition of the Lie algebra generated by transpositions in the group algebra of the symmetric group. This partial decomposition involves summands of the Jones representation., L'objet de ce travail est l'étude générale des représentations linéaires dugroupe de tresses $B_n$ qui proviennent de l'intégration de systèmes de Knizhnik-Zamolodchikov (KZ), vus comme représentations de l'algèbre des tressesinfinitésimales. Nous utilisons la technique des bases de Gelfand-Tsetlin pour étudier certaines représentations de cette algèbre, et montrons comment construire explicitement les représentations du groupe d'Artin correspondantes. Nous classifions complètement les systèmes KZ qui sont irréductibles pour l'action du groupesymétrique et construisons les nouvelles représentations de $B_n$ qui apparaissent àcette occasion. Nous obtenons d'autre part des critères d'irréductibilité sur les représentations de $B_n$ obtenues par construction tensorielle. Nous obtenons enfin d'autres résultats utiles dans ce cadre, notamment une décomposition partielle de l'algèbre de Lie engendrée par les transpositions dansl'algèbre de groupe du groupe symétrique. Cette décomposition partielle est en rapport avec les composantes irréductibles de la représentation de Jones.