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 $σ_1$ and $σ_2$ denote the two generating twists of $B_3$, then a simple representation $ρ:B_3 \to \gl(V)$ (for $\dim V \leq 5$) is unitarizable if and only if the eigenvalues $λ_1, λ_2, ..., λ_d$ of $ρ(σ_1)$ are distinct, satisfy $|λ_i|=1$ and $μ^{(d)}_{1i} > 0$ for $2 \leq i \leq d$, where the $μ^{(d)}_{1i}$ are functions of the eigenvalues, explicitly described in this paper.