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We define geodesic normal forms for the general series of complex reflectiongroups G(de,e,n). This requires the elaboration of a combinatorial technique inorder to determine minimal word representatives and to compute the length ofthe elements of G(de,e,n) over some generating set. Using these geodesic normalforms, 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 theother Garside groups that appear, we study some of their properties and computetheir second integral homology groups. Inspired by the geodesic normal forms,we also define new presentations and new bases for the Hecke algebrasassociated with the complex reflection groups G(e,e,n) and G(d,1,n) which leadto a new proof of the BMR (Brou\'e-Malle-Rouquier) freeness conjecture forthese two cases. Next, we define a BMW (Birman-Murakami-Wenzl) and Braueralgebras for type (e,e,n). This enables us to construct explicit Krammer'srepresentations for some cases of the complex braid groups B(e,e,n). Weconjecture that these representations are faithful. Finally, based on ourheuristic computations, we propose a conjecture about the structure of the BMWalgebra.

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