Representations of the braid group B_3 and of SL(2,Z)

Abstract

We give a complete classification of simple representations of the braid group B_3 with dimension $\leq 5$ over any algebraically closed f ield. In particular, we prove that a simple d-dimensional representation $ρ: B_3 \to GL(V)$ is determined up to isomorphism by the eigenvalues $λ_1, λ_2, ..., λ_d$ of the image of the generators for d=2,3 and a choice of a $δ=\sqrt{\det ρ(σ_1)}$ for d=4 or a choice of $δ=\sqrt[5]{\det ρ(σ_1)}$ for d=5. We also s howed that such representations exist whenever the eigenvalues and $δ$ are not roots of certain polynomials $Q_{ij}^{(d)}$, which are explicitly given. In this case, we construct the matrices via which the generators act on V. As an application of our techniques, we also obtain nontrivial q-versions of some of Deligne's formulas for dimensions of representations of exceptional Lie groups.