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<p><a id="X7DFB63A97E67C0A1" name="X7DFB63A97E67C0A1"></a></p>
<div class="ChapSects"><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>
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<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>
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<div class="ContSect"><span class="nocss"> </span><a href="chap1.html#X7DB566D5785B7DBC">1.3 <span class="Heading">Installation and system requirements</span></a>
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<h3>1 <span class="Heading">Introduction</span></h3>
<p><a id="X7F8C3A087C875426" name="X7F8C3A087C875426"></a></p>
<h4>1.1 <span class="Heading">General aims of <strong class="pkg">Wedderga</strong> package</span></h4>
<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. So the main functions of the package returns a list of simple algebras whose direct sum is isomorphic to the group algebra given as input.</p>
<p>The method implemented by the package produces the Wedderburn decomposition of a group algebra FG provided G is a finite group and F is either a finite field of characteristic coprime to the order of G, or an abelian number field (i.e. a subfield of a finite cyclotomic extension of the rationals).</p>
<p>Other functions of <strong class="pkg">Wedderga</strong> compute the primitive central idempotents of semisimple group algebras.</p>
<p>The package also provides functions 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="X7EC3E10184435AC0" name="X7EC3E10184435AC0"></a></p>
<h4>1.2 <span class="Heading">Main functions of <strong class="pkg">Wedderga</strong> package</span></h4>
<p>The main functions of <strong class="pkg">Wedderga</strong> are <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><b>2.1-1</b></a>) and <code class="func">WedderburnDecompositionInfo</code> (<a href="chap2.html#X8710F98A85F0DD29"><b>2.1-2</b></a>).</p>
<p><code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><b>2.1-1</b></a>) computes a list of simple algebras such that their direct product is isomorphic to the group algebra FG, given as input. Thus, the direct product of the entries of the output is the <em>Wedderburn decomposition</em> (<a href="chap7.html#X87273420791F220E"><b>7.3</b></a>) of FG.</p>
<p>If F is an abelian number field then the entries of the output are given as matrix algebras over cyclotomic algebras (see <a href="chap7.html#X8099A8C784255672"><b>7.11</b></a>), thus, the entries of the output of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><b>2.1-1</b></a>) are realizations of the <em>Wedderburn components</em> (<a href="chap7.html#X87273420791F220E"><b>7.3</b></a>) of FG as algebras which are <em>Brauer equivalent</em> (<a href="chap7.html#X7A24D5407F72C633"><b>7.5</b></a>) to <em>cyclotomic algebras</em> (<a href="chap7.html#X8099A8C784255672"><b>7.11</b></a>). Recall that the Brauer-Witt Theorem ensures that every simple factor of a semisimple group ring FG is Brauer equivalent (that is represents the same class in the Brauer group of its centre) to a cyclotomic algebra (<a href="chapBib.html#biBY">[Yam74]</a>. In this case the algorithm is based in a computational oriented proof of the Brauer-Witt Theorem due to Olteanu <a href="chapBib.html#biBO">[Olt07]</a> which uses previous work by Olivieri, del Río and Simón <a href="chapBib.html#biBORS">[ORS04]</a> for rational group algebras of <em>strongly monomial groups</em> (<a href="chap7.html#X84C694978557EFE5"><b>7.16</b></a>).</p>
<p>The Wedderburn components of FG are also matrix algebras over division rings which are finite extensions of the field F. If F is finite then by the Wedderburn theorem these division rings are finite fields. In this case the output of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><b>2.1-1</b></a>) represents the factors of FG as matrix algebras over finite extensions of the field F.</p>
<p>In theory <strong class="pkg">Wedderga</strong> could handle the calculation of the Wedderburn decomposition of group algebras of groups of arbitrary size but in practice if the order of the group is greater than 5000 then the program may crash. The way the group is given is relevant for the performance. Usually the program works better for groups given as permutation groups or pc groups.</p>
<table class="example">
<tr><td><pre>
gap> QG := GroupRing( Rationals, SymmetricGroup(4) );
<algebra-with-one over Rationals, with 2 generators>
gap> WedderburnDecomposition(QG);
[ Rationals, Rationals, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ),
<crossed product with center Rationals over CF(3) of a group of size 2> ]
gap> FG := GroupRing( CF(5), SymmetricGroup(4) );
<algebra-with-one over CF(5), with 2 generators>
gap> WedderburnDecomposition( FG );
[ CF(5), CF(5), ( CF(5)^[ 3, 3 ] ), ( CF(5)^[ 3, 3 ] ),
<crossed product with center CF(5) over AsField( CF(5), CF(
15) ) of a group of size 2> ]
gap> FG := GroupRing( GF(5), SymmetricGroup(4) );
<algebra-with-one over GF(5), with 2 generators>
gap> WedderburnDecomposition( FG );
[ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ),
( GF(5)^[ 3, 3 ] ), ( GF(5)^[ 3, 3 ] ) ]
gap> FG := GroupRing( GF(5), SmallGroup(24,3) );
<algebra-with-one over GF(5), with 4 generators>
gap> WedderburnDecomposition( FG );
[ ( GF(5)^[ 1, 1 ] ), ( GF(5^2)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ),
( GF(5^2)^[ 2, 2 ] ), ( GF(5)^[ 3, 3 ] ) ]
</pre></td></tr></table>
<p>Instead of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><b>2.1-1</b></a>), that returns a list of <strong class="pkg">GAP</strong> objects, <code class="func">WedderburnDecompositionInfo</code> (<a href="chap2.html#X8710F98A85F0DD29"><b>2.1-2</b></a>) returns the numerical description of these objects. See Section <a href="chap7.html#X84A142407B7565E0"><b>7.12</b></a> for theoretical background.</p>
<p><a id="X7DB566D5785B7DBC" name="X7DB566D5785B7DBC"></a></p>
<h4>1.3 <span class="Heading">Installation and system requirements</span></h4>
<p><strong class="pkg">Wedderga</strong> does not use external binaries and, therefore, works without restrictions on the type of the operating system. It is designed for <strong class="pkg">GAP</strong>4.4 and no compatibility with previous releases of <strong class="pkg">GAP</strong>4 is guaranteed.</p>
<p>To use the <strong class="pkg">Wedderga</strong> online help it is necessary to install the <strong class="pkg">GAP</strong>4 package <strong class="pkg">GAPDoc</strong> by Frank Lübeck and Max Neunhöffer, which is available from the <strong class="pkg">GAP</strong> site or from <span class="URL"><a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/">http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/</a></span>.</p>
<p><strong class="pkg">Wedderga</strong> is distributed in standard formats (<code class="file">tar.gz</code>, <code class="file">tar.bz2</code>, <code class="file">-win.zip</code>) and can be obtained from <span class="URL"><a href="http://www.um.es/adelrio/wedderga.htm">http://www.um.es/adelrio/wedderga.htm</a></span>, its mirror <span class="URL"><a href="http://www.cs.st-andrews.ac.uk/~alexk/wedderga.htm">http://www.cs.st-andrews.ac.uk/~alexk/wedderga.htm</a></span> or the page <span class="URL"><a href="http://www.gap-system.org/Packages/wedderga.html">http://www.gap-system.org/Packages/wedderga.html</a></span> at the <strong class="pkg">GAP</strong> web site. The latter also offers <code class="file">zoo</code>-archive. To unpack the archive <code class="file">wedderga-4.3.2.zoo</code> you need the program <code class="file">unzoo</code>, which can be obtained from the <strong class="pkg">GAP</strong> homepage <span class="URL"><a href="http://www.gap-system.org/">http://www.gap-system.org/</a></span> (see section `Distribution'). To install <strong class="pkg">Wedderga</strong>, copy this archive into the <code class="file">pkg</code> subdirectory of your <strong class="pkg">GAP</strong>4.4 installation. The subdirectory <code class="file">wedderga</code> will be created in the <code class="file">pkg</code> directory after the following command:</p>
<p><code class="code">unzoo -x wedderga-4.3.2.zoo</code></p>
<p>When you don't have access to the directory of your main <strong class="pkg">GAP</strong> installation, you can also install the package <em>outside the <strong class="pkg">GAP</strong> main directory</em> by unpacking it inside a directory <code class="file">MYGAPDIR/pkg</code>. Then to be able to load Wedderga you need to call GAP with the <code class="code">-l ";MYGAPDIR"</code> option.</p>
<p>Installation using other archive formats is performed in a similar way.</p>
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