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ED Mathématiques, Information, Ingénierie des Systèmes NORMANDIE_UNIVERSITE


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Soutenance Georges NEAIME

Interval Structures, Hecke Algebras, and Krammer's Representations for the Complex Braid Groups B(e,e,n)


26 juin 2018 à 14h00
Monsieur Georges NEAIME – laboratoire LMNO
Spécialité : Mathématiques
Directeur de thèse : Eddy GODELLE et Ivan MARIN
Lieu : salle des thèses, UFR sciences, bâtiment sciences 3, campus 2
Titre de la thèse : Interval Structures, Hecke Algebras, and Krammer's Representations for the Complex Braid Groups B(e,e,n)

Résumé : We define geodesic normal forms for the general series of complex reflection groups G(de,e,n). This requires the elaboration of a combinatorial technique in order to determine minimal word representatives and to compute the length of the elements of G(de,e,n) over some generating set. Using these geodesic normal forms, we construct intervals in G(e,e,n) that give rise to Garside groups. Some of these groups correspond to the complex braid group B(e,e,n). For the other Garside groups that appear, we study some of their properties and compute their second integral homology groups. Inspired by the geodesic normal forms, we also define new presentations and new bases for the Hecke algebras associated to the complex reflection groups G(e,e,n) and G(d,1,n) which lead to a new proof of the BMR (Broué-Malle-Rouquier) freeness conjecture for these two cases. Next, we define a BMW (Birman-Murakami-Wenzl) and Brauer algebras for type (e,e,n). This enables us to construct explicit Krammer's representations for some cases of the complex braid group B(e,e,n). We conjecture that these representations are faithful. Finally, based on our heuristic computations, we propose a conjecture about the structure of the BMW algebra.

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