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

## TL;DR

This paper investigates finite dimensional complex representations of the mapping class group derived from Galois coverings of surfaces, utilizing Fox derivation, Magnus modules, and algebraic theorems for explicit computation.

## Contribution

It introduces a novel approach to compute mapping class group actions on modules using Fox derivation and algebraic tools, providing explicit representations from surface coverings.

## Key findings

- Explicit computation of the $ ext{M}_{g,1}$-action on $L_g^{ exteta}$
- Application of Fox derivation and Magnus modules in representation theory
- New insights into surface group representations at roots of unity

## 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 $p$.