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

Abstract

In a recent paper here arXiv:1508.0005 it is shown that irreducible representations of the three string braid group $B_3$ of dimensions $\le 5$ extend to representations of the 3-component loop braid group $L{B}_3$. 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 $\Gamma$ such that sufficiently general representations extend to $\Gamma {\ast }_{C_3}{S}_3$. As a consequence, the corresponding irreducible $B_3$-representations do extend to $L{B}_3$.

Corrected the statement of the main result and streamlined the exposition

References in corpus (4)

• Low-dimensional representations of the three component loop braid group
• Exotic Statistics for Strings in 4d BF Theory
• Most irreducible representations of the 3-string braid group
• Trees of semi-simple algebras

Cited by in corpus (1)

• Topological Quantum Computation from the 3-dimensional Bordism 2-Category