Monodromy Representations: Decomposing Rank-Three Bundles over the Projective Line with Three Marked Points

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 use the monodromy derivative to compute the roots for all three-dimensional $\rho$ when $m=3$.