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Between 1994 and 1998, the work of M. Brou\'e, G. Malle, and R. Rouquiergeneralized in a natural way the definition of the Hecke algebra associated toa finite Coxeter group, for the case of an arbitrary complex reflection group.Attempting to also generalize the properties of the Coxeter case, they stated anumber of conjectures concerning these Hecke algebras. One specific example ofimportance regarding those yet unsolved conjectures is the so-called BMRfreeness conjecture. This conjecture is known to be true apart from 16 cases,that are almost all the exceptional groups of rank 2. These exceptional groupsof rank 2 fall into three families: the tetrahedral, octahedral and icosahedralfamily. We prove the validity of the BMR freeness conjecture for theexceptional groups belonging to the first two families, using a case-by-caseanalysis and we give a nice description of the basis, similar to the classicalcase of the finite Coxeter groups. We also give a new consequence of thisconjecture, by obtaining the classification of irreducible representations ofthe braid group on 3 strands in dimension at most 5, recovering results of Tubaand Wenzl.

The Brou\'e-Malle-Rouquier conjecture for the exceptional groups of rank 2