Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices

In this paper we solve an arbitrary matrix Riemann-Hilbert (inverse monodromy) problem with quasi-permutation monodromy representations outside of a divisor in the space of monodromy data. This divisor is characterized in terms of the theta-divisor on the Jacobi manifold of an auxiliary compact Riemann surface realized as an appropriate branched covering of CP 1 . The solution is given in terms of a generalization of Szegö kernel on the Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. The isomonodromy taufunction of these solutions is computed up to a nowhere vanishing factor independent of the elements of monodromy matrices. Results of this work generalize the results of papers [14] and [5] where the 2× 2 case was solved. subjclass: Primary 35Q15; Secondary 30F60, 32G81.