Low-Dimensional Unitary Representations of B₃

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We characterize all simple unitarizable representations of the braid group B₃ on complex vector spaces of dimension d ≤ 5. In particular, we prove that if σ₁ and σ₂ denote the two generating twists of B₃, then a simple representation ρ:B₃ → (V) (for V ≤ 5) is unitarizable if and only if the eigenvalues λ₁, λ₂,..., λd of ρ(σ₁) are distinct, satisfy |λᵢ|=1 and μ⁽ᵈ⁾₁ᵢ > 0 for 2 ≤ i ≤ d, where the μ⁽ᵈ⁾₁ᵢ are functions of the eigenvalues, explicitly described in this paper.