In this paper we present a new and elementary approach for proving the main results of Katz (1996) using the Jordan-Pochhammer matrices of Takano and Bannai (1976) and Haraoka (1994). We find an explicit version of the middle convolution of Katz (1996) that connects certain tuples of matrices in linear groups. From this, Katz' existence algorithm for rigid tuples in linear groups can easily be deduced. It can further be shown that the convolution operation on tuples commutes with the braid group action.This yields a new approach in inverse Galois theory for realizing subgroups of linear groups regularly as Galois groups over Q. This approach is then applied to realize numerous series of classical groups regularly as Galois groups over Q.In the Appendix we treat an additive version of the convolution.

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 Katz' 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.

We classify all Sp 4 (C)-rigid, quasi-unipotent local systems and show that all of them have geometric origin. Furthermore, we investigate which of those having a maximal unipotent element are induced by fourth order Calabi-Yau operators. Via this approach, we reconstruct all known Calabi-Yau operators inducing a Sp 4 (C)-rigid monodromy tuple and obtain closed formulae for special solutions of them. (4, 8, 4) := Λ 2 L a+ 1 4 ⋆ H L 1 4 −a z 1 2 ⋆ H L b+ 3 4 ⋆ H L 3 4 −b z − 3 2 ⋆ H L 3 2 = 64 ϑ 4 + z −128 ϑ 4 − 256 ϑ 3 + ϑ 2 (128(a 2 + b 2 ) − 304) + z ϑ (128(a 2 + b 2 ) − 176) + 48(a 2 + b 2 ) + 256 a 2 b 2 − 39 + 64z 2 (a + 1 + ϑ − b) (a + 1 + ϑ + b) (a − 1 − ϑ − b) (−1 + a − ϑ + b) .

We use the middle convolution to obtain some old and new algebraic solutions of the Painlevé VI equations.

In a previous work of the authors, a middle convolution operation on the category of Fuchsian differential systems was introduced. In this note we show that the middle convolution of Fuchsian systems preserves the property of global nilpotence. This leads to a globally nilpotent Fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems.