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<p><a id="X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5"></a></p>
<div class="pcenter">
<h1><strong class="pkg">Wedderga</strong></h1>
<h2>Wedderburn Decomposition of Group Algebras</h2>
<p>Version 4.3.2</p>
<p>January 2008</p>
</div>
<p><b>Osnel Broche Cristo
</b>
<br />Email: <span class="URL"><a href="mailto:osnel@ufla.br">osnel@ufla.br</a></span>
<br />Address: <br />Departamento de Ciências Exatas, Universidade Federal de Lavras - UFLA, Campus Universitário - Caixa Postal 3037, 37200-000, Lavras - MG, Brazil
</p><p><b>Alexander Konovalov
</b>
<br />Email: <span class="URL"><a href="mailto:konovalov@member.ams.org">konovalov@member.ams.org</a></span>
<br />Homepage: <span class="URL"><a href="http://www.cs.st-andrews.ac.uk/~alexk/">http://www.cs.st-andrews.ac.uk/~alexk/</a></span>
<br />Address: <br />School of Computer Science, University of St Andrews<br /> Jack Cole Building, North Haugh,<br /> St Andrews, Fife, KY16 9SX, Scotland
</p><p><b>Aurora Olivieri
</b>
<br />Email: <span class="URL"><a href="mailto:olivieri@usb.ve">olivieri@usb.ve</a></span>
<br />Address: <br />Departamento de Matemáticas<br /> Universidad Simón Bolívar<br /> Apartado Postal 89000, Caracas 1080-A, Venezuela
</p><p><b>Gabriela Olteanu
</b>
<br />Email: <span class="URL"><a href="mailto:golteanu@um.es, olteanu@math.ubbcluj.ro">golteanu@um.es, olteanu@math.ubbcluj.ro</a></span>
<br />Address: <br />Department of Mathematics and Computer Science<br /> North University of Baia Mare<br /> Victoriei 76, 430122 Baia Mare, Romania
</p><p><b>Ángel del Río
</b>
<br />Email: <span class="URL"><a href="mailto:adelrio@um.es">adelrio@um.es</a></span>
<br />Homepage: <span class="URL"><a href="http://www.um.es/adelrio">http://www.um.es/adelrio</a></span>
<br />Address: <br />Departamento de Matemáticas, Universidad de Murcia<br /> 30100 Murcia, Spain
</p>
<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p>The title ``<strong class="pkg">Wedderga</strong>'' stands for ``<strong class="button">WEDDER</strong>burn decomposition of <strong class="button">G</strong>roup <strong class="button">A</strong>lgebras. This is a <strong class="pkg">GAP</strong> 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 <strong class="pkg">Wedderga</strong> 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.</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2006-2008 by Osnel Broche Cristo, Alexander Konovalov, Aurora Olivieri, Gabriela Olteanu and Ángel del Río.</p>
<p><strong class="pkg">Wedderga</strong> 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 <span class="URL"><a href="http://www.gnu.org/licenses/gpl.html">http://www.gnu.org/licenses/gpl.html</a></span>.</p>
<p>If you obtained <strong class="pkg">Wedderga</strong>, we would be grateful for a short notification sent to one of the authors.</p>
<p>If you publish a result which was partially obtained with the usage of <strong class="pkg">Wedderga</strong>, please cite it in the following form:</p>
<p>O. Broche Cristo, A. Konovalov, A. Olivieri, G. Olteanu and Á. del Río. <em>Wedderga --- Wedderburn Decomposition of Group Algebras, Version 4.3.2;</em> 2008 (<span class="URL"><a href="http://www.um.es/adelrio/wedderga.htm">http://www.um.es/adelrio/wedderga.htm</a></span>).</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>We all are very grateful to Steve Linton for communicating the package and to the referee for careful testing <strong class="pkg">Wedderga</strong> and useful suggestions. Also we acknowledge very much the members of the <strong class="pkg">GAP</strong> 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 <code class="code">PrimitiveCentralIdempotentsByCharacterTable</code> for rational group algebras.</p>
<p>On various stages the development of the Wedderga package was supported by the following institutions:</p>
<ul>
<li><p>University of Murcia;</p>
</li>
<li><p>Francqui Stichting grant ADSI107;</p>
</li>
<li><p>M.E.C. of Romania (CEEX-ET 47/2006);</p>
</li>
<li><p>D.G.I. of Spain;</p>
</li>
<li><p>Fundación Séneca of Murcia;</p>
</li>
<li><p>CAPES and FAPESP of Brazil.</p>
</li>
</ul>
<p>We acknowledge with gratitude this support.</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents</h3>
<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap1.html#X7F8C3A087C875426">1.1 <span class="Heading">General aims of <strong class="pkg">Wedderga</strong> package</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap1.html#X7EC3E10184435AC0">1.2 <span class="Heading">Main functions of <strong class="pkg">Wedderga</strong> package</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap1.html#X7DB566D5785B7DBC">1.3 <span class="Heading">Installation and system requirements</span></a>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X87273420791F220E">2 <span class="Heading">Wedderburn decomposition</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X87273420791F220E">2.1 <span class="Heading">Wedderburn decomposition</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7F1779ED8777F3E7">2.1-1 WedderburnDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8710F98A85F0DD29">2.1-2 WedderburnDecompositionInfo</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7D06959F7D444C55">2.2 <span class="Heading">Simple quotients</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8349114C83161C2D">2.2-1 SimpleAlgebraByCharacter</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X876FD2367E64462D">2.2-2 SimpleAlgebraByCharacterInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X812D667D7D913EB5">2.2-3 SimpleAlgebraByStrongSP</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X858152C882129A0B">2.2-4 SimpleAlgebraByStrongSPInfo</a></span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X81DAF5267D30C83A">3 <span class="Heading">Strong Shoda pairs</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X807C74B07C4B99AF">3.1 <span class="Heading">Computing strong Shoda pairs</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X820A398687A79B9D">3.1-1 StrongShodaPairs</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7B49C1BC834E57E3">3.2 <span class="Heading">Properties related with Shoda pairs</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C17476F854F1E34">3.2-1 IsStrongShodaPair</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X823B8DEC7ECC3326">3.2-2 IsShodaPair</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X80C4ED17809FC547">3.2-3 IsStronglyMonomial</a></span>
</div>
</div>
<div class="ContChap"><a href="chap4.html#X7C651C9C78398FFF">4 <span class="Heading">Idempotents</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7DF49142844C278D">4.1 <span class="Heading">Computing idempotents from character table</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7BBEB4A084DBF0D6">4.1-1 PrimitiveCentralIdempotentsByCharacterTable</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X83F7CF1E87D02581">4.2 <span class="Heading">Testing lists of idempotents for completeness</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81FCD27E812078F0">4.2-1 IsCompleteSetOfOrthogonalIdempotents</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7C66102485AF5F80">4.3 <span class="Heading">Idempotents from Shoda pairs</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B48EE1A7ECAB151">4.3-1 PrimitiveCentralIdempotentsByStrongSP</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82460B1285A0A7D7">4.3-2 PrimitiveCentralIdempotentsBySP</a></span>
</div>
</div>
<div class="ContChap"><a href="chap5.html#X7FB21779832CE1CB">5 <span class="Heading">Crossed products</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X79122C7F877430A7">5.1 <span class="Heading">Construction of crossed products</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X797F31EF7B51A4DF">5.1-1 CrossedProduct</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X8560A2F37B608A9F">5.2 <span class="Heading">Crossed product elements and their properties</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7D2313AA82F1D5CC">5.2-1 ElementOfCrossedProduct</a></span>
</div>
</div>
<div class="ContChap"><a href="chap6.html#X7D3C0B1F7A66056F">6 <span class="Heading">Useful properties and functions</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap6.html#X7BA5D68A86B8C772">6.1 <span class="Heading">Semisimple group algebras of finite groups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7EF856E880722311">6.1-1 IsSemisimpleZeroCharacteristicGroupAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X85999B6A7C52E305">6.1-2 IsSemisimpleRationalGroupAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X79289F7F7FC04846">6.1-3 IsSemisimpleANFGroupAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B546E2D7FB561BA">6.1-4 IsSemisimpleFiniteGroupAlgebra</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap6.html#X86121BD77F7E5C7A">6.2 <span class="Heading">Operations with group rings elements</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7A2BF4527E08803C">6.2-1 Centralizer</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7FE417DD837987B4">6.2-2 OnPoints</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X798CEA1F80D355EE">6.2-3 AverageSum</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap6.html#X7AAB3882785C04E0">6.3 <span class="Heading">Cyclotomic classes</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7D7BDF5087C8F4C6">6.3-1 CyclotomicClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7FA101AE7BC33671">6.3-2 IsCyclotomicClass</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap6.html#X7B16423A7FBED034">6.4 <span class="Heading">Other commands</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X872510997A7AF31D">6.4-1 InfoWedderga</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8176F6DA7C8C057D">6.4-2 WEDDERGABuildManual</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B530E317C4CB2EC">6.4-3 WEDDERGABuildManualHTML</a></span>
</div>
</div>
<div class="ContChap"><a href="chap7.html#X840E625A81FDAEC6">7 <span class="Heading">The basic theory behind <strong class="pkg">Wedderga</strong></span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X815ECCD97B18314B">7.1 <span class="Heading">Group rings and group algebras</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X7FDD93FB79ADCC91">7.2 <span class="Heading">Semisimple group algebras</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X87273420791F220E">7.3 <span class="Heading">Wedderburn decomposition</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X87B6505C7C2EE054">7.4 <span class="Heading">Characters and primitive central idempotents</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X7A24D5407F72C633">7.5 <span class="Heading">Central simple algebras and Brauer equivalence</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X7FB21779832CE1CB">7.6 <span class="Heading">Crossed Products</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X828C42CD86AF605F">7.7 <span class="Heading">Cyclic Crossed Products</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X7869E2A48784C232">7.8 <span class="Heading">Abelian Crossed Products</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X80BABE5078A29793">7.9 <span class="Heading">Classical crossed products</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X84C98BB8859BBEE2">7.10 <span class="Heading">Cyclic Algebras</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X8099A8C784255672">7.11 <span class="Heading">Cyclotomic algebras</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X84A142407B7565E0">7.12 <span class="Heading">Numerical description of cyclotomic algebras</span></a>
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<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X8310E96086509397">7.13 <span class="Heading">Idempotents given by subgroups</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X80C058BE81824B23">7.14 <span class="Heading">Shoda pairs</span></a>
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<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X81DAF5267D30C83A">7.15 <span class="Heading">Strong Shoda pairs</span></a>
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<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X84C694978557EFE5">7.16 <span class="Heading">Strongly monomial characters and strongly monomial groups</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X800D8C5087D79DC8">7.17 <span class="Heading">Cyclotomic Classes and Strong Shoda Pairs</span></a>
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