J Austral Math Soc Ser A 57 pp1--16, 1994.

Testing Modules for Irreducibility

Derek F. Holt and Sarah Rees

Abstract

A practical method is described for deciding whether or not a finite-dimensional module for a group over a finite field is reducible or not. In the reducible case, an explicit submodule is found. The method is a generalisation of the Parker-Norton 'Meataxe' algorithm, but it does not depend for its efficiency on the field being small. The principal tools involved are the calculation of the nullspace and the characteristic polynomial of a matrix over a finite field, and the factorisation of the latter. Related algorithms to determine absolute irreducibility and module isomorphism for irreducibles are also described. Details of an implementation in the GAP system, together with some performance analyses are included.

1991 AMS Subject Classification: 20C40, 20-04

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Authors

Derek F. Holt

Mathematics Institute, University of Warwick, Coventry CV4 7AL, Great Britain.

mailto:dfh@maths.warwick.ac.uk

Sarah Rees

Department of Mathematics and Statistics, University of Newcastle, Newcastle-upon-Tyne NE1 7RU, Great Britain.

mailto:sarah.rees@newcastle.ac.uk