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# Wedderga

## Wedderburn Decomposition of Group Algebras

Version 4.2

March 2007

Osnel Broche Cristo
e-mail: osnelier@ime.usp.br
Address:
Departamento de Matemática
Instituto de Ciências Exatas
Universidade Federal de Juiz de Fora
Campus-Cidade Universitária, 36036-900, Juiz de Fora

Alexander Konovalov
e-mail: konovalov@member.ams.org
WWW: http://www.cs.st-andrews.ac.uk/~alexk/
Address:
School of Computer Science, University of St Andrews
Jack Cole Building, North Haugh,
St Andrews, Fife, KY16 9SX, Scotland

Aurora Olivieri
e-mail: olivieri@usb.ve
Address:
Departamento de Matemáticas
Universidad Simón Bolívar
Apartado Postal 89000, Caracas 1080-A, Venezuela

Gabriela Olteanu
e-mail: golteanu@um.es, olteanu@math.ubbcluj.ro
Address:
Departamento de Matemáticas, Universidad de Murcia
30100 Murcia, Spain

Ángel del Río
e-mail: adelrio@um.es
WWW: http://www.um.es/adelrio
Address:
Departamento de Matemáticas, Universidad de Murcia
30100 Murcia, Spain

### Abstract

The title ``Wedderga'' stands for ``WEDDERburn decomposition of Group Algebras. This is a GAP package to compute the simple components of the Wedderburn decomposition of semisimple group algebras of finite groups over finite fields and over subfields of finite cyclotomic extensions of the rational. It also contains functions that produce the primitive central idempotents of semisimple group algebras. Other functions of Wedderga allows to construct crossed products over a group with coefficients in an associative ring with identity and the multiplication determined by a given action and twisting.

### Copyright

(C) 2006-2007 by Osnel Broche Cristo, Alexander Konovalov, Aurora Olivieri, Gabriela Olteanu and Ángel del Río.

Wedderga is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the FSF's own site http://www.gnu.org/licenses/gpl.html.

If you obtained Wedderga, we would be grateful for a short notification sent to one of the authors.

If you publish a result which was partially obtained with the usage of Wedderga, please cite it in the following form:

O. Broche Cristo, A. Konovalov, A. Olivieri, G. Olteanu and Á. del Río. Wedderga --- Wedderburn Decomposition of Group Algebras, Version 4.2; 2007 (http://www.um.es/adelrio/wedderga.htm).

### Acknowledgements

We all are very grateful to Steve Linton for communicating the package and to the referee for careful testing Wedderga and useful suggestions. Also we acknowledge very much the members of the GAP team: Thomas Breuer, Alexander Hulpke, Frank Lübeck and many other colleagues for helpful comments and advise. We would like also to thank Thomas Breuer for the code of `PrimitiveCentralIdempotentsByCharacterTable` for rational group algebras.

On various stages the development of the Wedderga package was supported by the following institutions:

- University of Murcia;

- Francqui Stichting grant ADSI107;

- M.E.C. of Romania (CEEX-ET 47/2006);

- D.G.I. of Spain;

- Fundación Séneca of Murcia.

We acknowledge with gratitude this support.

### Contents

1. Introduction
1.1 General aims of Wedderga package
1.2 Main functions of Wedderga package
1.3 Installation and system requirements
2. Wedderburn decomposition
2.1 Wedderburn decomposition
2.1-1 WedderburnDecomposition
2.1-2 WedderburnDecompositionInfo
2.2 Simple quotients
2.2-1 SimpleAlgebraByCharacter
2.2-2 SimpleAlgebraByCharacterInfo
2.2-3 SimpleAlgebraByStrongSP
2.2-4 SimpleAlgebraByStrongSPInfo
3. Strong Shoda pairs
3.1 Computing strong Shoda pairs
3.1-1 StrongShodaPairs
3.2 Properties related with Shoda pairs
3.2-1 IsStrongShodaPair
3.2-2 IsShodaPair
3.2-3 IsStronglyMonomial
4. Idempotents
4.1 Computing idempotents from character table
4.1-1 PrimitiveCentralIdempotentsByCharacterTable
4.2 Testing lists of idempotents for completeness
4.2-1 IsCompleteSetOfOrthogonalIdempotents
4.3 Idempotents from Shoda pairs
4.3-1 PrimitiveCentralIdempotentsByStrongSP
4.3-2 PrimitiveCentralIdempotentsBySP
5. Crossed products
5.1 Construction of crossed products
5.1-1 CrossedProduct
5.2 Crossed product elements and their properties
5.2-1 ElementOfCrossedProduct
6. Useful properties and functions
6.1 Semisimple group algebras of finite groups
6.1-1 IsSemisimpleZeroCharacteristicGroupAlgebra
6.1-2 IsSemisimpleRationalGroupAlgebra
6.1-3 IsSemisimpleANFGroupAlgebra
6.1-4 IsSemisimpleFiniteGroupAlgebra
6.2 Operations over group rings elements
6.2-1 Centralizer
6.2-2 OnPoints
6.2-3 AverageSum
6.3 Cyclotomic classes
6.3-1 CyclotomicClasses
6.3-2 IsCyclotomicClass
6.4 Other commands
6.4-1 InfoWedderga
6.4-2 WEDDERGABuildManual
6.4-3 WEDDERGABuildManualHTML
7. The basic theory behind Wedderga
7.1 Group rings and group algebras
7.2 Semisimple group algebras
7.3 Wedderburn decomposition
7.4 Characters and primitive central idempotents
7.5 Central simple algebras and Brauer equivalence
7.6 Crossed Products
7.7 Cyclic Crossed Products
7.8 Abelian Crossed Products
7.9 Classical crossed products
7.10 Cyclic Algebras
7.11 Cyclotomic algebras
7.12 Numerical description of cyclotomic algebras
7.13 Idempotents given by subgroups
7.14 Shoda pairs
7.15 Strong Shoda pairs
7.16 Strongly monomial characters and strongly monomial groups
7.17 Cyclotomic Classes and Strong Shoda Pairs

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