Low-Dimensional Unitary Representations of B_3

Abstract

We characterize all simple unitarizable representations of the braid group $B_3$ on complex vector spaces of dimension $d\le 5$. In particular, we prove that if ${\sigma }_1$ and ${\sigma }_2$ denote the two generating twists of $B_3$, then a simple representation $ρ:B_3 \to \gl(V)$ (for $\dim V\le 5$) is unitarizable if and only if the eigenvalues ${\lambda }_1,{\lambda }_2,...,{\lambda }_d$ of $\rho ({\sigma }_1)$ are distinct, satisfy $|{\lambda }_i|=1$ and ${\mu }_{1i}^{(d)}>0$ for $2\le i\le d$, where the ${\mu }_{1i}^{(d)}$ are functions of the eigenvalues, explicitly described in this paper.

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