On some finite dimensional complex representations of mapping class groups and Fox derivation

Abstract

We study the finite dimensional complex representations of the mapping class group $\mathcal{M}_{g,1}$ that are derived from some finite Galois coverings of the compact oriented surface with one boundary component $\Sigma_{g,1}$. The key ingredients are Fox derivation, Magnus modules and the Skolem-Noether theorem, which enable us to compute the $\mathcal{M}_{g,1}$-action on the module $L_g^{\eta}$ very explicitly, where $\eta$ is a primitive $p$the root of unity for an odd prime $p$.