<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 <span class="SimpleMath">FG</span> provided <span class="SimpleMath">G</span> is a finite group and <span class="SimpleMath">F</span> is either a finite field of characteristic coprime to the order of <span class="SimpleMath">G</span>, 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 and a complete set of orthogonal primitive idempotents, and calculate Schur indices of simple 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>Furhermore, the package provides functions to create code words from a group ring element.</p>
<h4>1.2 <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="https://gap-packages.github.io/wedderga/">https://gap-packages.github.io/wedderga/</a></span>. To install <strong class="pkg">Wedderga</strong>, unpack its archive into the <code class="file">pkg</code> subdirectory of your <strong class="pkg">GAP</strong> installation.</p>
<p>When you don't have access to the directory of your main GAP installation, you can also install the package outside the GAP main directory by unpacking it inside a directory MYGAPDIR/pkg. Then to be able to load Wedderga you need to call GAP with the -l ";MYGAPDIR" option.
<p>Installation using other archive formats is performed in a similar way.</p>
<p>If the package is installed correctly, it should be loaded as follows:</p>
<h4>1.3 <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"><span class="RefLink">2.1-1</span></a>) and <code class="func">WedderburnDecompositionInfo</code> (<a href="chap2.html#X8710F98A85F0DD29"><span class="RefLink">2.1-2</span></a>).</p>
<p><code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) computes a list of simple algebras such that their direct product is isomorphic to the group algebra <span class="SimpleMath">FG</span>, given as input. Thus, the direct product of the entries of the output is the <em>Wedderburn decomposition</em> (<a href="chap9.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">FG</span>.</p>
<p>If <span class="SimpleMath">F</span> is an abelian number field then the entries of the output are given as matrix algebras over cyclotomic algebras (see <a href="chap9.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>), thus, the entries of the output of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) are realizations of the <em>Wedderburn components</em> (<a href="chap9.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">FG</span> as algebras which are <em>Brauer equivalent</em> (<a href="chap9.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) to <em>cyclotomic algebras</em> (<a href="chap9.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>). Recall that the Brauer-Witt Theorem ensures that every simple factor of a semisimple group ring <span class="SimpleMath">FG</span> 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 on 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">[OdRS04]</a> (see also <a href="chapBib.html#biBOR">[OdR03]</a> ) for rational group algebras of <em>strongly monomial groups</em> (<a href="chap9.html#X84C694978557EFE5"><span class="RefLink">9.17</span></a>). The algorithms are also based upon the work of Bakshi and Maheshwary <a href="chapBib.html#biBBM14">[BM14]</a> (see also <a href="chapBib.html#biBBM16">[BM16]</a>) on the rational group algebras of <em>normally monomial groups</em> (<a href="chap9.html#X7C8D47C180E0ACAD"><span class="RefLink">9.18</span></a>).</p>
<p>The Wedderburn components of <span class="SimpleMath">FG</span> are also matrix algebras over division rings which are finite extensions of the field <span class="SimpleMath">F</span>. If <span class="SimpleMath">F</span> 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"><span class="RefLink">2.1-1</span></a>) represents the factors of <span class="SimpleMath">FG</span> as matrix algebras over finite extensions of the field <span class="SimpleMath">F</span>.</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>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QG := GroupRing( Rationals, SymmetricGroup(4) );</span>
<algebra-with-one over Rationals, with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnDecomposition(QG);</span>
[ Rationals, Rationals, <crossed product with center Rationals over CF(
3) of a group of size 2>, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FG := GroupRing( CF(5), SymmetricGroup(4) );</span>
<algebra-with-one over CF(5), with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnDecomposition( FG );</span>
[ CF(5), CF(5), <crossed product with center CF(5) over AsField( CF(5), CF(
15) ) of a group of size 2>, ( CF(5)^[ 3, 3 ] ), ( CF(5)^[ 3, 3 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FG := GroupRing( GF(5), SymmetricGroup(4) ); </span>
<algebra-with-one over GF(5), with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnDecomposition( FG );</span>
[ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ),
( GF(5)^[ 3, 3 ] ), ( GF(5)^[ 3, 3 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FG := GroupRing( GF(5), SmallGroup(24,3) );</span>
<algebra-with-one over GF(5), with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnDecomposition( FG );</span>
[ ( GF(5)^[ 1, 1 ] ), ( GF(5^2)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ),
( GF(5^2)^[ 2, 2 ] ), ( GF(5)^[ 3, 3 ] ) ]
</pre></div>
<p>Instead of <code class="func">WedderburnDecomposition</code> (<a href="chap2.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>), that returns a list of <strong class="pkg">GAP</strong> objects, <code class="func">WedderburnDecompositionInfo</code> (<a href="chap2.html#X8710F98A85F0DD29"><span class="RefLink">2.1-2</span></a>) returns the numerical description of these objects. See Section <a href="chap9.html#X84A142407B7565E0"><span class="RefLink">9.12</span></a> for theoretical background.</p>
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