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

TL;DR

This paper characterizes when irreducible representations of the modular group extend to the loop braid group, providing conditions that determine the extendability of these representations across all dimensions.

Contribution

It offers a necessary and sufficient condition for the extension of irreducible modular group representations to the loop braid group in all dimensions.

Findings

Identifies conditions for representation extension across all dimensions.

Provides a complete characterization of extendability for irreducible representations.

Shows that most irreducible representations extend to the loop braid group under these conditions.

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 $\leq 5$ 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 $\Gamma$ 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 $LB_3$.