Nikolaus Conference 2018

Exactly twenty years ago, Broué, Malle and Rouquier associated to every complex reflection group two objects classically associated to real reflection groups: a braid group and a Hecke algebra. Trying to generalize the properties of these objects from the real to complex case, they stated a number of conjectures concerning both braid groups and Hecke algebras. An example of such a conjecture is the BMR freeness conjecture. It states that the Hecke algebra associated with a complex reflection group is a free module over its ring of definition of rank equal to the order of the group. This conjecture is now a theorem, and in this talk we focus on its proof for the exceptional reflection groups of the tetrahedral and octahedral families, by providing specific bases for these cases. These particular bases not only yield the proof of the BMR freeness conjecture for these two families, but also have a similar description to the standard bases of the Hecke algebras associated to real reflection groups.