$N$-Dimensional Representations of the Braid Groups $B_{N}$

TL;DR

This paper introduces a new class of representations for braid groups, explicitly constructing an N-dimensional irreducible representation that encompasses known trivial and Burau representations.

Contribution

It constructs a novel N-dimensional irreducible representation of braid groups, expanding the understanding of their representation theory.

Findings

Contains trivial, Burau, and N-dimensional irreducible representations

Provides explicit form of the N-dimensional irreducible representation

Enhances the classification of braid group representations

Abstract

In this note, a new class of representations of the braid groups $B_{N}$ is constructed. It is proved that those representations contain three kinds of irreducible representations: the trivial (identity) one, the Burau one, and an $N$-dimensional one. The explicit form of the $N$-dimensional irreducible representation of the braid group $B_{N}$ is given here.