Universal deformations of the finite quotients of the braid group on 3 strands

TL;DR

This paper proves that certain quotients of the braid group on 3 strands have finite rank, confirming a special case of a broader conjecture and enabling complete classification of low-dimensional irreducible representations.

Contribution

It establishes the finite rank property for quotients with quartic and quintic relations, advancing the understanding of the representation theory of braid groups and Hecke algebras.

Findings

Quotients of the group algebra have finite rank for generic quartic and quintic relations.

Complete classification of irreducible representations up to dimension 5.

Reproves and generalizes the Tuba-Wenzl classification.

Abstract

We prove that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation respectively, have finite rank. This is a special case of a conjecture by Brou\'{e}, Malle and Rouquier for the generic Hecke algebra of an arbitrary complex reflection group. Exploring the consequences of this case, we will prove that we can determine completely the irreducible representations of this braid group for dimension at most 5, thus reproving a classification of Tuba and Wenzl in a more general framework.