BMR freeness for icosahedral family

TL;DR

This paper confirms the BMR freeness conjecture for certain complex reflection groups by computational verification, advancing understanding of cyclotomic Hecke algebras and their algebraic properties.

Contribution

It provides the first computational proof of BMR freeness for groups G_{17}, G_{18}, G_{19}, completing the conjecture's validation for these cases.

Findings

BMR freeness verified for G_{17}, G_{18}, G_{19}

Uses Bergman's diamond lemma for verification

Requires significant computational resources (1GB memory, 5 days)

Abstract

We verify the Brou\'e-Malle-Rouquier (BMR) freeness for cyclotomic Hecke algebras associated with complex reflection groups $G_{17}$, $G_{18}$, $G_{19}$ in the Shephard-Todd classification. Together with results of Ariki, Ariki-Koike, Brou\'e-Malle, Marin, Marin-Pfeiffer and Chavli, this settled affirmatively the BMR freeness conjecture. Our verification is inspired by Bergman's diamond lemma and requires 1GB memory and 5 days calculation on a PC.