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

TL;DR

This paper explores the relationship between G-covers of the projective line ramified at three points and associated parabolic vector bundles, aiming to describe the tensor functor in group-theoretic terms.

Contribution

It constructs a pullback functor on parabolic vector bundles and characterizes the tensor functor from G-covers using group-theoretic data.

Findings

Constructed a parabolic pullback functor for vector bundles.

Characterized the tensor functor via group-theoretic data.

Provided a description of the pullback of parabolic bundles in the context of G-covers.

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 $\oplus \mathscr{O}(s_i)$ 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.