Low-dimensional unitary representations of 𝐵₃

Abstract

We characterize all simple unitarizable representations of the braid group B 3 B_3 on complex vector spaces of dimension d ≤ 5 d \leq 5 . In particular, we prove that if σ 1 \sigma _1 and σ 2 \sigma _2 denote the two generating twists of B 3 B_3 , then a simple representation ρ : B 3 → GL ⁡ ( V ) \rho :B_3 \to \operatorname {GL} (V) (for dim ⁡ V ≤ 5 \dim V \leq 5 ) is unitarizable if and only if the eigenvalues λ 1 , λ 2 , … , λ d \lambda _1, \lambda _2, \ldots , \lambda _d of ρ ( σ 1 ) \rho (\sigma _1) are distinct, satisfy | λ i | = 1 |\lambda _i|=1 and μ 1 i ( d ) > 0 \mu ^{(d)}_{1i} > 0 for 2 ≤ i ≤ d 2 \leq i \leq d , where the μ 1 i ( d ) \mu ^{(d)}_{1i} are functions of the eigenvalues, explicitly described in this paper.