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Testing modules for irreducibility

Part of: Group theory and generalizations Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

Derek F. Holt  and
Sarah Rees
Derek F. Holt
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, Great Britian, e-mail: dfh@maths.warwick.ac.uk
Sarah Rees
Affiliation:
Department of Mathematics and Statistics, University of Newcastle, Newscastle-upon-Tyne NE1 7RU, Great Britian, e-mail: sarah.rees@newcastle.ac.uk
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Abstract

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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 generalistaion 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.

MSC classification

Secondary: 20C40: Computational methods 20-04: Explicit machine computation and programs (not the theory of computation or programming)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 57 , Issue 1 , August 1994 , pp. 1 - 16
Copyright
Copyright © Australian Mathematical Society 1994

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