Published online by Cambridge University Press: 20 November 2018
Given a faithful action of a finite group $G$ on an algebraic curve $X$ of genus $gx\,\ge \,2$, we give explicit criteria for the induced action of $G$ on the Riemann–Roch space ${{H}^{0}}\left( X,\,{{\mathcal{O}}_{X}}\left( D \right) \right)$ to be faithful, where $D$ is a $G$-invariant divisor on $X$ of degree at least ${{2}_{gX}}\,-\,2$. This leads to a concise answer to the question of when the action of $G$ on the space ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$ of global holomorphic polydifferentials of order $m$ is faithful. If $X$ is hyperelliptic, we provide an explicit basis of ${{H}^{0}}\left( X,\,\Omega _{X}^{\otimes m} \right)$. Finally, we give applications in deformation theory and in coding theory and discuss the analogous problem for the action of $G$ on the first homology ${{H}_{1}}\left( X,\,\mathbb{Z}/m\mathbb{Z} \right)$ if $X$ is a Riemann surface.