2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68107We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n by n matrices for which one can find matrices in their closures whose product is equal to the identity matrix. Both answers depend on the root system of a Kac-Moody Lie algebra. Our proofs use Ringel's theory of tubular algebras, work of Mihai on the existence of logarithmic connections, the Riemann-Hilbert correspondence and an algebraic version, due to Dettweiler and Reiter, of Katz's middle convolution operation.30 pages, various corrections and improvementsAlgebraic GeometryRings and AlgebrasPrimary 14H60, 15A24; Secondary 16G99, 34M50text