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# Wedderga

## Wedderburn Decomposition of Group Algebras

Version 4.7.3

18 September 2015

Osnel Broche Cristo
Email: osnel@ufla.br
Address:
Departamento de Ciências Exatas, Universidade Federal de Lavras - UFLA, Campus Universitário - Caixa Postal 3037, 37200-000, Lavras - MG, Brazil

Allen Herman
Email: aherman@math.uregina.ca
Homepage: http://www.math.uregina.ca/~aherman/
Address:
Department of Mathematics and Statistics,
University of Regina,
3737 Wascana Parkway,
Regina, SK, S0G 0E0, Canada

Alexander Konovalov
Email: alexk@mcs.st-andrews.ac.uk
Homepage: 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
Email: olivieri@usb.ve
Address:
Departamento de Matemáticas
Universidad Simón Bolívar
Apartado Postal 89000, Caracas 1080-A, Venezuela

Gabriela Olteanu
Email: gabriela.olteanu@econ.ubbcluj.ro
Homepage: http://math.ubbcluj.ro/~olteanu
Address:
Department of Statistics-Forecasts-Mathematics
Faculty of Economics and Business Administration
Babes-Bolyai University
Str. T. Mihali 58-60, 400591 Cluj-Napoca, Romania

Ángel del Río
Email: adelrio@um.es
Homepage: http://www.um.es/adelrio
Address:
Departamento de Matemáticas, Universidad de Murcia
30100 Murcia, Spain

Inneke Van Gelder
Email: ivgelder@vub.ac.be
Homepage: http://homepages.vub.ac.be/~ivgelder
Address:
Vrije Universiteit Brussel, Departement Wiskunde
Pleinlaan 2
1050 Brussels, Belgium

### 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 rationals. It also contains functions that produce the primitive central idempotents of semisimple group algebras and a complete set of orthogonal primitive idempotents. Other functions of Wedderga allow 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

© 2006-2015 by Osnel Broche Cristo, Allen Herman, Alexander Konovalov, Aurora Olivieri, Gabriela Olteanu, Ángel del Río and Inneke Van Gelder.

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. Herman, A. Konovalov, A. Olivieri, G. Olteanu, Á. del Río and I. Van Gelder. Wedderga --- Wedderburn Decomposition of Group Algebras, Version 4.7.3; 2015 (http://www.cs.st-andrews.ac.uk/~alexk/wedderga).

### 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;

- CAPES and FAPESP of Brazil;

- Research Foundation Flanders (FWO - Vlaanderen).

We acknowledge with gratitude this support.

### Contents

1 Introduction
1.1 General aims of Wedderga package
1.2 Installation and system requirements
1.3 Main functions of Wedderga package
2 Wedderburn decomposition
2.1 Wedderburn decomposition of a group algebra
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
4.4 Complete set of orthogonal primitive idempotents from Shoda pairs and cyclotomic classes
4.4-1 PrimitiveIdempotentsNilpotent
4.4-2 PrimitiveIdempotentsTrivialTwisting
5 Crossed products and their elements
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
6.1-5 IsTwistingTrivial
6.2 Operations with 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
7 Functions for calculating Schur indices and identifying division algebras
7.1 Main Schur Index and Division Algebra Functions
7.1-1 WedderburnDecompositionWithDivAlgParts
7.1-2 CyclotomicAlgebraWithDivAlgPart
7.1-3 SchurIndex
7.1-4 WedderburnDecompositionAsSCAlgebras
7.2 Cyclotomic Reciprocity Functions
7.2-1 PPartOfN
7.2-2 PSplitSubextension
7.2-3 SplittingDegreeAtP
7.3 Local index functions for Cyclic Cyclotomic Algebras
7.3-1 LocalIndicesOfCyclicCyclotomicAlgebra
7.3-2 LocalIndexAtInfty
7.4 Local index functions for Non-Cyclic Cyclotomic Algebras
7.4-1 LocalIndicesOfCyclotomicAlgebra
7.4-2 RootOfDimensionOfCyclotomicAlgebra
7.4-3 DefiningGroupOfCyclotomicAlgebra
7.4-4 LocalIndexAtInftyByCharacter
7.4-5 DefectGroupOfConjugacyClassAtP
7.4-6 LocalIndexAtPByBrauerCharacter
7.4-7 LocalIndexAtOddPByCharacter
7.5 Local index functions for Rational Quaternion Algebras
7.5-1 LocalIndicesOfRationalQuaternionAlgebra
7.5-2 IsRationalQuaternionAlgebraADivisionRing
7.6 Functions involving Cyclic Algebras
7.6-1 DecomposeCyclotomicAlgebra
7.6-2 ConvertCyclicAlgToCyclicCyclotomicAlg
7.6-3 ConvertQuaternionAlgToQuadraticAlg
8 Applications of the Wedderga package
8.1 Coding theory applications
8.1-1 CodeWordByGroupRingElement
8.1-2 CodeByLeftIdeal
9 The basic theory behind Wedderga
9.1 Group rings and group algebras
9.2 Semisimple group algebras
9.3 Wedderburn components
9.4 Characters and primitive central idempotents
9.5 Central simple algebras and Brauer equivalence
9.6 Crossed Products
9.7 Cyclic Crossed Products
9.8 Abelian Crossed Products
9.9 Classical crossed products
9.10 Cyclic Algebras
9.11 Cyclotomic algebras
9.12 Numerical description of cyclotomic algebras
9.13 Idempotents given by subgroups
9.14 Shoda pairs of a group
9.15 Strong Shoda pairs of a group
9.16 Strongly monomial characters and strongly monomial groups
9.17 Cyclotomic Classes and Strong Shoda Pairs
9.18 Theory for Local Schur Index and Division Algebra Part Calculations
9.19 Obtaining Algebras with structure constants as terms of the Wedderburn decomposition
9.20 A complete set of orthogonal primitive idempotents
9.21 Applications to coding theory
References
Index
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