[1X[5XWedderga[0m[1X[0m [1XWedderburn Decomposition of Group Algebras[0m Version 4.3.2 January 2008 Osnel Broche Cristo Alexander Konovalov Aurora Olivieri Gabriela Olteanu Ãngel del RÃo Osnel Broche Cristo Email: [7Xmailto:osnel@ufla.br[0m Address: Departamento de Ciências Exatas, Universidade Federal de Lavras - UFLA, Campus Universitário - Caixa Postal 3037, 37200-000, Lavras - MG, Brazil Alexander Konovalov Email: [7Xmailto:konovalov@member.ams.org[0m Homepage: [7Xhttp://www.cs.st-andrews.ac.uk/~alexk/[0m Address: School of Computer Science, University of St Andrews Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland Aurora Olivieri Email: [7Xmailto:olivieri@usb.ve[0m Address: Departamento de Matemáticas Universidad Simón BolÃvar Apartado Postal 89000, Caracas 1080-A, Venezuela Gabriela Olteanu Email: [7Xmailto:golteanu@um.es, olteanu@math.ubbcluj.ro[0m Address: Department of Mathematics and Computer Science North University of Baia Mare Victoriei 76, 430122 Baia Mare, Romania Ãngel del RÃo Email: [7Xmailto:adelrio@um.es[0m Homepage: [7Xhttp://www.um.es/adelrio[0m Address: Departamento de Matemáticas, Universidad de Murcia 30100 Murcia, Spain ------------------------------------------------------- [1XAbstract[0m The title ``[5XWedderga[0m'' stands for ``[12XWEDDER[0mburn decomposition of [12XG[0mroup [12XA[0mlgebras. This is a [5XGAP[0m 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 [5XWedderga[0m 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. ------------------------------------------------------- [1XCopyright[0m © 2006-2008 by Osnel Broche Cristo, Alexander Konovalov, Aurora Olivieri, Gabriela Olteanu and Ãngel del RÃo. [5XWedderga[0m 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[0m. If you obtained [5XWedderga[0m, 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[0m, please cite it in the following form: O. Broche Cristo, A. Konovalov, A. Olivieri, G. Olteanu and Ã. del RÃo. [13XWedderga --- Wedderburn Decomposition of Group Algebras, Version 4.3.2;[0m 2008 ([7Xhttp://www.um.es/adelrio/wedderga.htm[0m). ------------------------------------------------------- [1XAcknowledgements[0m We all are very grateful to Steve Linton for communicating the package and to the referee for careful testing [5XWedderga[0m and useful suggestions. Also we acknowledge very much the members of the [5XGAP[0m 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[0m 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. ------------------------------------------------------- [1XContents (Wedderga)[0X 1 Introduction 1.1 General aims of [5XWedderga[0m package 1.2 Main functions of [5XWedderga[0m 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 [5XWedderga[0m 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 -------------------------------------------------------