A quaternionic braid representation (after Goldschmidt and Jones)

TL;DR

This paper proves that certain braid group representations factor over a finite group using a quaternionic representation, providing a formal proof for a known but unpublished result from the 1980s.

Contribution

It offers the first published proof that the $(3,6)$-quotients of Hecke algebra braid representations factor over a finite group, utilizing an unpublished quaternionic representation.

Findings

Braid group representations factor over finite groups

Use of quaternionic representation by Goldschmidt and Jones

Discussion of topological and categorical generalizations

Abstract

We show that the braid group representations associated with the $(3,6)$-quotients of the Hecke algebras factor over a finite group. This was known to experts going back to the 1980s, but a proof has never appeared in print. Our proof uses an unpublished quaternionic representation of the braid group due to Goldschmidt and Jones. Possible topological and categorical generalizations are discussed.