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

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.

[1]  Martin W. Liebeck,et al.  The Subgroup Structure of the Finite Classical Groups , 1990 .

[2]  H. Völklein.  Rigid generators of classical groups , 1998 .

[3]  N. Spaltenstein.  Classes unipotentes et sous-groupes de Borel , 1982 .

[4]  Y. Haraoka.  Finite monodromy of Pochhammer equation , 1994 .

[5]  M. Dettweiler,et al.  On Rigid Tuples in Linear Groups of Odd Dimension , 1999 .

[6]  Frits Beukers,et al.  Monodromy for the hypergeometric functionnFn−1 , 1989 .

[7]  W. D. Gruyter,et al.  Zöpfe und Galoissche Gruppen. , 1991 .

[8]  H. Völklein.  The Braid Group and Linear Rigidity , 2001 .

[9]  A. Wagner.  Groups generated by elations , 1974 .

[10]  S. Reiter.  Galoisrealisierungen klassischer Gruppen , 1999 .

[11]  A. Wagner.  Collineation groups generated by homologies of order greater than 2 , 1978 .

[12]  H. Völklein.  GLn(q) as Galois group over the rationals , 1992 .

[13]  M. Fried,et al.  The inverse Galois problem and rational points on moduli spaces , 1991 .

[14]  William M. Kantor,et al.  Subgroups of classical groups generated by long root elements , 1979 .

[15]  M. Fried.  Fields of definition of function fields and hurwitz families — groups as galois groups , 1977 .

[16]  W. Magnus,et al.  Combinatorial Group Theory: COMBINATORIAL GROUP THEORY , 1967 .

[17]  L. Pochhammer,et al.  Ueber hypergeometrische Functionen nter Ordnung. , 1870 .

[18]  L. Scott,et al.  Matrices and cohomology , 1977 .

[19]  Jean-Pierre Serre,et al.  Topics in Galois Theory , 1992 .

[20]  Gunter Malle,et al.  Inverse Galois Theory , 2002 .

[21]  P. Deligne,et al.  Equations differentielles à points singuliers reguliers , 1970 .

[22]  K. Strambach,et al.  On linearly rigid tuples , 1999 .

[23]  N. M. Katz.  Rigid Local Systems , 1995 .

[24]  K. Strambach,et al.  Finite quotients of the pure symplectic braid group , 1998 .

[25]  Charles F. Miller,et al.  Combinatorial Group Theory , 2002 .

[26]  On the Deligne-Simpson problem , 1999, math/0011013.

[27]  R. Ree,et al.  A theorem on permutations , 1971 .

[28]  Symplectic groups as Galois groups , 1998 .

[29]  R. D. Carmichael,et al.  Group Theory: 15 , 1906 .

[30]  Roger W. Carter,et al.  Finite groups of Lie type: Conjugacy classes and complex characters , 1985 .