  The necklace braid group $\mathcal{NB}_n$ is the motion group of the $n+1$
component necklace link $\mathcal{L}_n$ in Euclidean $\mathbb{R}^3$. Here
$\mathcal{L}_n$ consists of $n$ pairwise unlinked Euclidean circles each linked
to an auxiliary circle. Partially motivated by physical considerations, we
study representations of the necklace braid group $\mathcal{NB}_n$, especially
those obtained as extensions of representations of the braid group
$\mathcal{B}_n$ and the loop braid group $\mathcal{LB}_n$. We show that any
irreducible $\mathcal{B}_n$ representation extends to $\mathcal{NB}_n$ in a
standard way. We also find some non-standard extensions of several well-known
$\mathcal{B}_n$-representations such as the Burau and LKB representations.
Moreover, we prove that any local representation of $\mathcal{B}_n$ (i.e.
coming from a braided vector space) can be extended to $\mathcal{NB}_n$, in
contrast to the situation with $\mathcal{LB}_n$. We also discuss some
directions for future study from categorical and physical perspectives.
