The BMR freeness conjecture for the tetrahedral and octahedral families

TL;DR

This paper proves the freeness conjecture for certain complex reflection groups' Hecke algebras, providing a basis description akin to classical Coxeter groups, advancing understanding in algebraic structures.

Contribution

It establishes the conjecture's validity for rank 2 exceptional groups in the tetrahedral and octahedral families, with a basis description similar to Coxeter groups.

Findings

Freeness conjecture proven for specific complex reflection groups

Provides a basis description analogous to Coxeter groups

Advances algebraic understanding of Hecke algebras

Abstract

We prove the validity of the freeness conjecture of Brou\'e, 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.