Pullback of parabolic bundles and covers of ${\mathbb{P}}^1\setminus \{0,1,\infty \}$

Abstract

We work over an algebraically closed ground field of characteristic zero. A $G$-cover of ${\mathbb{P}}^1$ ramified at three points allows one to assign to each finite dimensional representation $V$ of $G$ a vector bundle $\oplus \mathcal{O}(s_i)$ on ${\mathbb{P}}^1$ with parabolic structure at the ramification points. This produces a tensor functor from representation of $G$ to vector bundles with parabolic structure that characterises the original cover. This work attempts to describe this tensor functor in terms of group theoretic data. More precisely, we construct a pullback functor on vector bundles with parabolic structure and describe the parabolic pullback of the previously described tensor functor.

25 pages