Middle convolution of Fuchsian systems and the construction of rigid differential systems

In [M. Dettweiler, S. Reiter, An algorithm of Katz and its application to the inverse Galois problem, J. Symbolic Comput. 30 (2000) 761–798], a purely algebraic analogon of Katzʹ middle convolution functor (see [N.M. Katz, Rigid Local Systems, Ann. of Math. Stud., vol. 139, Princeton University Press, 1997]) is given. In this paper, we find an explicit Riemann–Hilbert correspondence for this functor. This leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the Riemann–Hilbert correspondence.