  
  
                                    [1X[5XWedderga[105X[101X
  
  
                   [1XWedderburn Decomposition of Group Algebras[101X
  
  
                                 Version 4.9.1
  
  
                                11 December 2017
  
  
                              Gurmeet Kaur Bakshi
  
                              Osnel Broche Cristo
  
                                  Allen Herman
  
                              Alexander Konovalov
  
                              Sugandha Maheshwary
  
                                Aurora Olivieri
  
                                Gabriela Olteanu
  
                                 Ángel del Río
  
                               Inneke Van Gelder
  
  
  
  Gurmeet Kaur Bakshi
      Email:    [7Xmailto:gkbakshi@pu.ac.in[107X
      Address:  [33X[0;14YCenter for Advanced Study in Mathematics,[133X
                [33X[0;14YPanjab University, Chandigarh, PIN-160014, India[133X
  
  
  Osnel Broche Cristo
      Email:    [7Xmailto:osnel@ufla.br[107X
      Address:  [33X[0;14YDepartamento  de  Ciências  Exatas,  Universidade  Federal  de
                Lavras  -  UFLA,  Campus  Universitário  -  Caixa Postal 3037,
                37200-000, Lavras - MG, Brazil[133X
  
  
  Allen Herman
      Email:    [7Xmailto:aherman@math.uregina.ca[107X
      Homepage: [7Xhttp://www.math.uregina.ca/~aherman/[107X
      Address:  [33X[0;14YDepartment of Mathematics and Statistics,[133X
                [33X[0;14YUniversity of Regina,[133X
                [33X[0;14Y3737 Wascana Parkway,[133X
                [33X[0;14YRegina, SK, S0G 0E0, Canada[133X
  
  
  Alexander Konovalov
      Email:    [7Xmailto:alexk@mcs.st-andrews.ac.uk[107X
      Homepage: [7Xhttp://www.cs.st-andrews.ac.uk/~alexk/[107X
      Address:  [33X[0;14YSchool of Computer Science, University of St Andrews[133X
                [33X[0;14YJack Cole Building, North Haugh,[133X
                [33X[0;14YSt Andrews, Fife, KY16 9SX, Scotland[133X
  
  
  Sugandha Maheshwary
      Email:    [7Xmailto:sugandha@iisermohali.ac.in[107X
      Address:  [33X[0;14YDepartment of Mathematical Sciences[133X
                [33X[0;14YIndian Institute of Science Education and Research[133X
                [33X[0;14YMohali, Knowledge city, Sector 81, SAS Nagar,[133X
                [33X[0;14YManauli, PO 140306, India[133X
  
  
  Aurora Olivieri
      Email:    [7Xmailto:olivieri@usb.ve[107X
      Address:  [33X[0;14YDepartamento de Matemáticas[133X
                [33X[0;14YUniversidad Simón Bolívar[133X
                [33X[0;14YApartado Postal 89000, Caracas 1080-A, Venezuela[133X
  
  
  Gabriela Olteanu
      Email:    [7Xmailto:gabriela.olteanu@econ.ubbcluj.ro[107X
      Homepage: [7Xhttp://math.ubbcluj.ro/~olteanu[107X
      Address:  [33X[0;14YDepartment of Statistics-Forecasts-Mathematics[133X
                [33X[0;14YFaculty of Economics and Business Administration[133X
                [33X[0;14YBabes-Bolyai University[133X
                [33X[0;14YStr. T. Mihali 58-60, 400591 Cluj-Napoca, Romania[133X
  
  
  Ángel del Río
      Email:    [7Xmailto:adelrio@um.es[107X
      Homepage: [7Xhttp://www.um.es/adelrio[107X
      Address:  [33X[0;14YDepartamento de Matemáticas, Universidad de Murcia[133X
                [33X[0;14Y30100 Murcia, Spain[133X
  
  
  Inneke Van Gelder
      Email:    [7Xmailto:ivgelder@vub.ac.be[107X
      Homepage: [7Xhttp://homepages.vub.ac.be/~ivgelder[107X
      Address:  [33X[0;14YVrije Universiteit Brussel, Departement Wiskunde[133X
                [33X[0;14YPleinlaan 2[133X
                [33X[0;14Y1050 Brussels, Belgium[133X
  
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0Y``[5XWedderga[105X''  stands for ``[12XWEDDER[112Xburn decomposition of [12XG[112Xroup [12XA[112Xlgebras". This
  is  a  [5XGAP[105X  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 [5XWedderga[105X 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.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y©  2006-2017  by  Gurmeet  Kaur  Bakshi,  Osnel Broche Cristo, Allen Herman,
  Alexander Konovalov, Sugandha Maheshwary, Aurora Olivieri, Gabriela Olteanu,
  Ángel del Río and Inneke Van Gelder.[133X
  
  [33X[0;0Y[5XWedderga[105X  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
  [7Xhttp://www.gnu.org/licenses/gpl.html[107X.[133X
  
  [33X[0;0YIf you obtained [5XWedderga[105X, 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 [5XWedderga[105X, please cite it in the following form:[133X
  
  [33X[0;0YG.  K.  Bakshi, O. Broche Cristo, A. Herman, A. Konovalov, S. Maheshwary, A.
  Olivieri,  G. Olteanu, Á. del Río and I. Van Gelder. [13XWedderga --- Wedderburn
  Decomposition     of     Group     Algebras,     Version     4.9.1;[113X     2017
  ([7Xhttp://www.cs.st-andrews.ac.uk/~alexk/wedderga[107X).[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YWe  all  are very grateful to Steve Linton for communicating the package and
  to  the referee for careful testing [5XWedderga[105X and useful suggestions. Also we
  acknowledge  very much the members of the [5XGAP[105X 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
  [10XPrimitiveCentralIdempotentsByCharacterTable[110X for rational group algebras.[133X
  
  [33X[0;0YWe  gratefully  acknowledge  the  support  of  Wedderga  development  by the
  following institutions:[133X
  
  [30X    [33X[0;6YUniversity of Murcia;[133X
  
  [30X    [33X[0;6YFrancqui Stichting grant ADSI107;[133X
  
  [30X    [33X[0;6YM.E.C. of Romania (CEEX-ET 47/2006);[133X
  
  [30X    [33X[0;6YD.G.I. of Spain;[133X
  
  [30X    [33X[0;6YFundación Séneca of Murcia;[133X
  
  [30X    [33X[0;6YCAPES and FAPESP of Brazil;[133X
  
  [30X    [33X[0;6YResearch Foundation Flanders (FWO - Vlaanderen);[133X
  
  [30X    [33X[0;6YCCP CoDiMa (EP/M022641/1);[133X
  
  [30X    [33X[0;6YDepartment of Science and Technology (DST), India.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (Wedderga)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YGeneral aims of [5XWedderga[105X package[133X
    1.2 [33X[0;0YInstallation and system requirements[133X
    1.3 [33X[0;0YMain functions of [5XWedderga[105X package[133X
  2 [33X[0;0YWedderburn decomposition[133X
    2.1 [33X[0;0YWedderburn decomposition of a group algebra[133X
      2.1-1 WedderburnDecomposition
      2.1-2 WedderburnDecompositionInfo
    2.2 [33X[0;0YSimple quotients[133X
      2.2-1 SimpleAlgebraByCharacter
      2.2-2 SimpleAlgebraByCharacterInfo
      2.2-3 SimpleAlgebraByStrongSP
      2.2-4 SimpleAlgebraByStrongSPInfo
  3 [33X[0;0YShoda pairs[133X
    3.1 [33X[0;0YComputing extremely strong Shoda pairs[133X
      3.1-1 ExtremelyStrongShodaPairs
    3.2 [33X[0;0YComputing strong Shoda pairs[133X
      3.2-1 StrongShodaPairs
    3.3 [33X[0;0YProperties related with Shoda pairs[133X
      3.3-1 IsExtremelyStrongShodaPair
      3.3-2 IsStrongShodaPair
      3.3-3 IsShodaPair
      3.3-4 IsStronglyMonomial
      3.3-5 IsNormallyMonomial
  4 [33X[0;0YIdempotents[133X
    4.1 [33X[0;0YComputing idempotents from character table[133X
      4.1-1 PrimitiveCentralIdempotentsByCharacterTable
    4.2 [33X[0;0YTesting lists of idempotents for completeness[133X
      4.2-1 IsCompleteSetOfOrthogonalIdempotents
    4.3 [33X[0;0YIdempotents from Shoda pairs[133X
      4.3-1 PrimitiveCentralIdempotentsByESSP
      4.3-2 PrimitiveCentralIdempotentsByStrongSP
      4.3-3 PrimitiveCentralIdempotentsBySP
    4.4 [33X[0;0YComplete set of orthogonal primitive idempotents from Shoda pairs and
    cyclotomic classes[133X
      4.4-1 PrimitiveIdempotentsNilpotent
      4.4-2 PrimitiveIdempotentsTrivialTwisting
  5 [33X[0;0YCrossed products and their elements[133X
    5.1 [33X[0;0YConstruction of crossed products[133X
      5.1-1 CrossedProduct
    5.2 [33X[0;0YCrossed product elements and their properties[133X
      5.2-1 ElementOfCrossedProduct
  6 [33X[0;0YUseful properties and functions[133X
    6.1 [33X[0;0YSemisimple group algebras of finite groups[133X
      6.1-1 IsSemisimpleZeroCharacteristicGroupAlgebra
      6.1-2 IsSemisimpleRationalGroupAlgebra
      6.1-3 IsSemisimpleANFGroupAlgebra
      6.1-4 IsSemisimpleFiniteGroupAlgebra
      6.1-5 IsTwistingTrivial
    6.2 [33X[0;0YOperations with group rings elements[133X
      6.2-1 Centralizer
      6.2-2 OnPoints
      6.2-3 AverageSum
    6.3 [33X[0;0YCyclotomic classes[133X
      6.3-1 CyclotomicClasses
      6.3-2 IsCyclotomicClass
    6.4 [33X[0;0YOther commands[133X
      6.4-1 InfoWedderga
  7 [33X[0;0YFunctions for calculating Schur indices and identifying division algebras[133X
    7.1 [33X[0;0YMain Schur Index and Division Algebra Functions[133X
      7.1-1 WedderburnDecompositionWithDivAlgParts
      7.1-2 CyclotomicAlgebraWithDivAlgPart
      7.1-3 SchurIndex
      7.1-4 WedderburnDecompositionAsSCAlgebras
    7.2 [33X[0;0YCyclotomic Reciprocity Functions[133X
      7.2-1 PPartOfN
      7.2-2 PSplitSubextension
      7.2-3 SplittingDegreeAtP
    7.3 [33X[0;0YLocal index functions for Cyclic Cyclotomic Algebras[133X
      7.3-1 LocalIndicesOfCyclicCyclotomicAlgebra
      7.3-2 LocalIndexAtInfty
    7.4 [33X[0;0YLocal index functions for Non-Cyclic Cyclotomic Algebras[133X
      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 [33X[0;0YLocal index functions for Rational Quaternion Algebras[133X
      7.5-1 LocalIndicesOfRationalQuaternionAlgebra
      7.5-2 IsRationalQuaternionAlgebraADivisionRing
    7.6 [33X[0;0YFunctions involving Cyclic Algebras[133X
      7.6-1 DecomposeCyclotomicAlgebra
      7.6-2 ConvertCyclicAlgToCyclicCyclotomicAlg
      7.6-3 ConvertQuaternionAlgToQuadraticAlg
  8 [33X[0;0YApplications of the Wedderga package[133X
    8.1 [33X[0;0YCoding theory applications[133X
      8.1-1 CodeWordByGroupRingElement
      8.1-2 CodeByLeftIdeal
  9 [33X[0;0YThe basic theory behind [5XWedderga[105X[133X
    9.1 [33X[0;0YGroup rings and group algebras[133X
    9.2 [33X[0;0YSemisimple group algebras[133X
    9.3 [33X[0;0YWedderburn components[133X
    9.4 [33X[0;0YCharacters and primitive central idempotents[133X
    9.5 [33X[0;0YCentral simple algebras and Brauer equivalence[133X
    9.6 [33X[0;0YCrossed Products[133X
    9.7 [33X[0;0YCyclic Crossed Products[133X
    9.8 [33X[0;0YAbelian Crossed Products[133X
    9.9 [33X[0;0YClassical crossed products[133X
    9.10 [33X[0;0YCyclic Algebras[133X
    9.11 [33X[0;0YCyclotomic algebras[133X
    9.12 [33X[0;0YNumerical description of cyclotomic algebras[133X
    9.13 [33X[0;0YIdempotents given by subgroups[133X
    9.14 [33X[0;0YShoda pairs of a group[133X
    9.15 [33X[0;0YStrong Shoda pairs of a group[133X
    9.16 [33X[0;0YExtremely strong Shoda pairs of a group[133X
    9.17 [33X[0;0YStrongly monomial characters and strongly monomial groups[133X
    9.18 [33X[0;0YNormally monomial characters and normally monomial groups[133X
    9.19 [33X[0;0YCyclotomic Classes and Strong Shoda Pairs[133X
    9.20 [33X[0;0YTheory for Local Schur Index and Division Algebra Part Calculations[133X
    9.21 [33X[0;0YObtaining Algebras with structure constants as terms of the
    Wedderburn decomposition[133X
    9.22 [33X[0;0YA complete set of orthogonal primitive idempotents[133X
    9.23 [33X[0;0YApplications to coding theory[133X
  
  
  [32X
