New Permutation Representations of the Braid Group

TL;DR

This paper introduces a new infinite family of permutation representations of the braid group, expanding known representations, proving their properties, and refuting a prior conjecture in the field.

Contribution

The authors present a novel infinite family of non-cyclic, transitive permutation representations of braid groups, including all known types and addressing previous conjectures.

Findings

Most known permutation representations are included in the new family.

Homomorphisms are proven to be non-cyclic and transitive.

Refutes a prior conjecture on permutation representations of braids.

Abstract

We give a new infinite family of group homomorphisms from the braid group B_k to the symmetric group S_{mk} for all k and m \geq 2. Most known permutation representations of braids are included in this family. We prove that the homomorphisms in this family are non-cyclic and transitive. For any divisor l of m, 1\leq l < m, we prove in particular that if \frac{m}{l} is odd then there are 1 + \frac{m}{l} non-conjugate homomorphisms included in our family. We define a certain natural restriction on homomorphisms B_k to S_n, common to all homomorphisms in our family, which we term 'good', and of which there are two types. We prove that all good homomorphisms B_k to S_{mk} of type 1 are included in the infinite family of homomorphisms we gave. For m=3, we prove that all good homomorphisms B_k to S_{3k} of type 2 are also included in this family. Finally, we refute a conjecture made by Matei and Suciu regarding permutation representations of braids and give an updated conjecture.