The BMR freeness conjecture for the tetrahedral and octahedral families

Abstract

We prove the validity of the freeness conjecture of Broué, Malle and Rouquier for the generic Hecke algebras associated to the exceptional complex reflection groups of rank 2 belonging to the tetrahedral and octahedral families, and we give a description of the basis similar to the classical case of the finite Coxeter groups.

The BMM symmetrising trace conjecture for the exceptional 2-reflection groups of rank 2

“…Note that, since the generators of G 13 are of order 2 and belong to two different conjugacy classes, H(G 13 ) depends on four parameters a, b, c and d; among them, only b and d are units in R(G 13 ). In [Cha1], the second author proved the BMR freeness conjecture for G 13 by providing the following explicit basis for H(G 13 ):…”

“…It is entirely possible to do all these calculations by hand -examples are provided in the Appendix. Actually, the second author has done all these calculations by hand in order to prove the BMR freeness conjecture for G 13 in [Cha1], without however keeping track of the explicit coefficients. Since the calculations are time-consuming and mistakes can be made when computing the coefficients, we created a computer program in the language C++ whose purpose was to produce the desired linear combination for every product b i b j , with b i ∈ {u, s, t}, using the methodology described below and illustrated in the examples found in the Appendix.…”

“…Remark 4.7. As we also remarked in [BCCK], this does not mean that we have obtained a computerised proof of the BMR freeness conjecture for G 13 , because the "special cases" incorporated in the C++ program are the exact calculations made by hand by the second author for the proof of the BMR freeness conjecture in [Cha1].…”

“…As we can see from the examples, doing all the calculations by hand is possible (it has already been done in [Cha1]), but time-consuming. Moreover, simple mistakes can be made, like forgetting a coefficient.…”

Proof of the Broué–Malle–Rouquier Conjecture in Characteristic Zero (After I. Losev and I. Marin—G. Pfeiffer)

“…This conjecture is currently known for all irreducible complex reflection groups except G 17 , ..., G 21 (according to the Shephard-Todd classification), and there is a hope that these cases can be proved as well using a sufficiently powerful computer (see [Cha1,Cha2] and [M3] for more details). Also, it is shown in [BMR] that to prove the conjecture, it suffices to show that H(W ) is spanned by |W | elements.…”