BMR Freeness for Icosahedral Family

ABSTRACT

We verify the Broué–Malle–Rouquier (BMR) freeness for cyclotomic Hecke algebras associated with complex reflection groups G₁₇, G₁₈, G₁₉ in the Shephard–Todd classification. Together with results of Ariki, Ariki–Koike, Broué–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.

Keywords:

• complex reflection groups
• cyclotomic Hecke algebras
• BMR freeness conjecture
• noncommutative Gröbner basis
• Bergman’s diamond lemma

2010 Mathematics Subject Classification:

• Primary 20C08
• Secondary 68W30

5. Appendix: The sequences

Acknowledgments

The major part of the work was carried out at Caffe Strada when the author visited University of California Berkeley for Berkeley-Tokyo Summer School “Geometry, Representation Theory and Mathematical Physics” in August, 2017. The author thanks Toshitake Kohno for giving him an opportunity.

Additional information

Funding

The first author was supported in part by JSPS Kakenhi Grants 17K14154.