The cubic Hecke algebra on at most 5 strands

TL;DR

This paper proves that a specific algebra related to the braid group on five strands has finite rank, confirming a long-standing conjecture and showing it deforms the group algebra of a complex reflection group.

Contribution

It establishes the finite rank property of the cubic Hecke algebra on at most 5 strands, confirming a conjecture from 1998 and linking it to complex reflection groups.

Findings

The quotient algebra has finite rank.

Confirmed the algebra is a flat deformation of G_{32}.

Resolved a conjecture from 1998.

Abstract

We prove that the quotient of the group algebra of the braid group on 5 strands by a generic cubic relation has finite rank. This was conjectured in 1998 by Brou\'e, Malle and Rouquier and has for consequence that this algebra is a flat deformation of the group algebra of the complex reflection group $G_{32}$, of order 155,520.