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

TL;DR

This paper classifies simple representations of the braid group B_3 of dimension up to 5 over algebraically closed fields, providing explicit criteria and constructions based on eigenvalues and determinants.

Contribution

It offers a complete classification of low-dimensional simple representations of B_3, including explicit construction methods and conditions for existence.

Findings

Classified simple B_3 representations of dimension ≤ 5.

Provided explicit eigenvalue and determinant conditions for existence.

Constructed matrices for generator actions and extended to q-versions of Deligne's formulas.

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 $\rho: B_3 \to GL(V)$ is determined up to isomorphism by the eigenvalues $\lambda_1, \lambda_2, ..., \lambda_d$ of the image of the generators for d=2,3 and a choice of a $\delta=\sqrt{\det \rho(\sigma_1)}$ for d=4 or a choice of $\delta=\sqrt[5]{\det \rho(\sigma_1)}$ for d=5. We also s howed that such representations exist whenever the eigenvalues and $\delta$ 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.