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

Abstract

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