The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group G26

We review the state-of-the-art concerning the freeness conjecture stated in the 1990s by Broué, Malle and Rouquier for generic Hecke algebras associated to complex reflection groups, and in particular we expose in detail one of the main differences with the ordinary case, namely the lack of 0-Hecke algebras. We end the paper by proving a new case of this conjecture, the exceptional group called G26 in the Shephard–Todd classification, namely the largest linear group of automorphisms of the Hessian configuration.