Computing the Roots of Twisting Sheaves over the Projective Line arising from Monodromy Representations

Given a monodromy representation $\rho$ of the projective line minus $m$ points, one can extend the resulting vector bundle with connection map canonically to a vector bundle with logarithmic connection map over all of the projective line. Now, since vector bundles split as twisting sheaves over the projective line, the focus of this work regards knowing the exact decomposition; i.e. computing the roots. Particularly, we compute the roots for all finite-dimensional $\rho$ when $m = 2$ and for all $\rho$ of dimension less than $3$ when $m = 3$.