Report on the Broué–Malle–Rouquier Conjectures

The BMR freeness conjecture for the tetrahedral and octahedral families

“…Attempting to also generalize the properties of the Coxeter case, they stated a number of conjectures concerning the Hecke algebras, which haven't been proven yet. Even without being proven, those conjectures have been used by a number of papers in the last two decades as assumptions, and are still being used in various subjects, such as representation theory of finite reductive groups, Cherednik algebras, and usual braid groups (more details about these conjectures and their applications can be found in [24]).…”

The Broué-Malle-Rouquier conjecture for the exceptional groups of rank 2

“…En tentant de généraliser les propriétés dans le cas des groupes de Coxeter, ils ont formulé plusieurs conjectures concernant les algèbres de Hecke, qui n'ont pas encore été prouvées. Mêmes sans être prouvées, ces conjectures ont été utilisées dans de nombreux articles de ces dernières décennies comme hypothèses, et sont toujours utilisées dans de nombreux domaines, tels que la théorie des représentations des groupes réductifs finis, les algèbres de Cherednik et les groupes de tresses usuels (plus de détails sur ces conjectures et leurs applications peuvent être trouvées dans [30]).…”

“…Attempting to also generalize the properties of the Coxeter case, they stated a number of conjectures concerning the Hecke algebras, which haven't been proven yet. Even without being proven, those conjectures have been used by a number of papers in the last decades as assumptions, and are still being used in various subjects, such as representation theory of finite reductive groups, Cherednik algebras, and usual braid groups (more details about these conjectures and their applications can be found in [30]).…”

The BMR freeness conjecture for the first two families of the exceptional groups of rank 2

“…Attempting to also generalize the properties of the Coxeter case, they stated a number of conjectures concerning the Hecke algebras, which had not been proven yet. Even without being proven, those conjectures have been used by a number of papers in the last decades as assumptions, and are still being used in various subjects, such as representation theory of finite reductive groups, Cherednik algebras, and usual braid groups (more details about these conjectures and their applications can be found in [14]).…”