An Algorithm of Katz and its Application to the Inverse Galois Problem

Abstract

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.