Representations of the necklace braid group {NB}}_n of dimension (n=2,3,4)

TL;DR

This paper classifies and analyzes the irreducible 2-dimensional representations of the necklace braid group for n=2,3,4, including conditions for their irreducibility and unitarity.

Contribution

It provides a complete classification of 2D irreducible representations of 3n, including tensor product criteria and unitarity conditions.

Findings

Necessary and sufficient conditions for irreducibility of tensor product representations.

Conditions for irreducibility of 2D irreducible representations.

Criteria for unitarity of these representations.

Abstract

We consider the irreducible representations each of dimension 2 of the necklace braid group $\mathcal{NB}_n$ ($n=2,3,4$). We then consider the tensor product of the representations of $\mathcal{NB}_n$ ($n=2,3,4$) and determine necessary and sufficient condition under which the constructed representations are irreducible. Finally, we determine conditions under which the irreducible representations of $\mathcal{NB}_n$ ($n=2,3,4$) of degree 2 are unitary relative to a hermitian positive definite matrix