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 $d \leq 5$. In particular, we prove that if $\sigma _1$ and $\sigma _2$ denote the two generating twists of $B_3$, then a simple representation $\rho :B_3 \to \operatorname {GL} (V)$ (for $\dim V \leq 5$) is unitarizable if and only if the eigenvalues $\lambda _1, \lambda _2, \ldots , \lambda _d$ of $\rho (\sigma _1)$ are distinct, satisfy $|\lambda _i|=1$ and $\mu ^{(d)}_{1i} > 0$ for $2 \leq i \leq d$, where the $\mu ^{(d)}_{1i}$ are functions of the eigenvalues, explicitly described in this paper.