Given a monodromy representation of the m-times punctured projective line, the associated holomorphic vector bundle with connection extends canonically to P^1 with a logarithmic connection and, by Birkhoff-Grothendieck, splits into a direct sum of twisting sheaves O(n). This work computes that decomposition (the twisting parameters, or 'roots') explicitly from the representation via an algebraic, generic-representation approach: fully for all finite-dimensional representations when m=2 and for all representations of dimension under 3 when m=3. It introduces a 'monodromy derivative' built from an auxiliary connection and connects to Weil's Chern-class criterion and the Mehta-Seshadri correspondence with stable parabolic bundles.
A monodromy representation of the m-times punctured projective line (equivalently the punctured Riemann sphere), i.e. a representation of its fundamental group (a free group on m-1 generators), deter…