Wedderga
  
  
                   Wedderburn Decomposition of Group Algebras
  
  
                                 Version 4.3.2
  
  
                                  January 2008
  
  
                              Osnel Broche Cristo
  
                              Alexander Konovalov
  
                                Aurora Olivieri
  
                                Gabriela Olteanu
  
                                 Ángel del Río
  
  
  
  Osnel Broche Cristo
      Email:    mailto:osnel@ufla.br
      Address:  Departamento  de  Ciências  Exatas,  Universidade  Federal  de
                Lavras  -  UFLA,  Campus  Universitário  -  Caixa Postal 3037,
                37200-000, Lavras - MG, Brazil
  
  
  Alexander Konovalov
      Email:    mailto:konovalov@member.ams.org
      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:    mailto:olivieri@usb.ve
      Address:  Departamento de Matemáticas
                Universidad Simón Bolívar
                Apartado Postal 89000, Caracas 1080-A, Venezuela
  
  
  Gabriela Olteanu
      Email:    mailto:golteanu@um.es, olteanu@math.ubbcluj.ro
      Address:  Department of Mathematics and Computer Science
                North University of Baia Mare
                Victoriei 76, 430122 Baia Mare, Romania
  
  
  Ángel del Río
      Email:    mailto:adelrio@um.es
      Homepage: 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
  rationals.  It  also  contains  functions that produce the primitive central
  idempotents  of semisimple group algebras. 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.
  
  
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  Copyright
  ©  2006-2008  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 4.3.2; 2008
  (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;
  
  --    CAPES and FAPESP of Brazil.
  
  We acknowledge with gratitude this support.
  
  
  -------------------------------------------------------
  
  
  Contents (Wedderga)
  
  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 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
      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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