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

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Abstract.

In this paper we solve an arbitrary matrix Riemann-Hilbert (inverse monodromy) problem with irreducible quasi-permutation monodromy representation 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 P1 . 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 tau-function 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 [13] and [14] where the 2× 2 case was solved.

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  1. Department of Mathematics and Statistics, Concordia University 7141 Sherbrooke West, Montreal H4B 1R6, Quebec, Canada

    D. Korotkin

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  1. D. Korotkin
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Correspondence to D. Korotkin.

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Mathematics Subject Classification (1991): 35Q15, 30F60, 32G81

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Korotkin, D. Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices. Math. Ann. 329, 335–364 (2004). https://doi.org/10.1007/s00208-004-0528-z

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