High-dimensional representations of the 3-component loop braid group

In a recent paper here arXiv:1508.0005 it is shown that irreducible representations of the three string braid group $B_3$ of dimensions $\leq 5$ extend to representations of the 3-component loop braid group $LB_3$. Further, an explicit $6$-dimensional irreducible $B_3$-representation is given that does not extend. In this note we give a necessary and sufficient condition, in all dimensions, on the components of irreducible representations of the modular group $Γ$ such that sufficiently general representations extend to $Γ\ast_{C_3} S_3$. As a consequence, the corresponding irreducible $B_3$-representations do extend to $LB_3$.

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