<!-- ********************Introduction******************** -->

<Chapter Label="Intro">
<Heading>Introduction</Heading>    

<Section Label="IntroAims">
<Heading>General aims of &Wedderga; package</Heading>
The title ``&Wedderga;'' stands for ``<B>Wedder</B>burn decomposition of 
<B>g</B>roup <B>a</B>lgebras''. This is a &GAP; 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/>


The method implemented by the package produces the Wedderburn decomposition 
of a group algebra <M>FG</M> provided <M>G</M> is a finite group and 
<M>F</M> is either a finite field of characteristic coprime to the 
order of <M>G</M>, or an abelian number field (i.e. a subfield of a finite 
cyclotomic extension of the rationals). 
<P/>

Other functions of &Wedderga; 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/>

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/>

Furhermore, the package  provides functions to create code words from a group 
ring element.<P/>

</Section>

<!-- ********************************************************* -->

<Section Label="IntroInstall">
<Heading>Installation and system requirements</Heading>

&Wedderga; does not use external binaries and, therefore, works without
restrictions on the type of the operating system. It is designed for &GAP;4.4  
and no compatibility with previous releases of &GAP;4 is guaranteed. 
<P/>

To use the &Wedderga; online help it is necessary to install the &GAP;4 package 
&GAPDoc; by Frank L&#252;beck  and  Max Neunhöffer, which is available from the 
&GAP; site or from 
<URL>http://www.math.rwth-aachen.de/&tilde;Frank.Luebeck/GAPDoc/</URL>.
<P/> 

&Wedderga; is distributed in standard formats
(<File>tar.gz</File>, <File>tar.bz2</File>, <File>-win.zip</File>) 
and can be obtained from <URL>https://gap-packages.github.io/wedderga/</URL>.
To install &Wedderga;, unpack its archive into the 
<File>pkg</File> subdirectory of your &GAP; installation.
<P/>
When you don't have access to the directory of your main &GAP; 
installation, you can also install the package 
<E>outside the &GAP; main directory</E>
by unpacking it inside a directory <File>MYGAPDIR/pkg</File>. 
Then to be able to load Wedderga you need to call GAP with the 
<C>-l ";MYGAPDIR"</C> option.
<P/>
Installation using other archive formats is performed in a similar way.
<P/>
If the package is installed correctly, it should be loaded as follows:
<Log>
<![CDATA[
gap> LoadPackage("wedderga");
-----------------------------------------------------------------------------
Loading  Wedderga 4.9.0 (Wedderga)
by Gurmeet Kaur Bakshi (gkbakshi@pu.ac.in),
   Osnel Broche Cristo (osnel@ufla.br),
   Allen Herman (aherman@math.uregina.ca),
   Alexander Konovalov (https://alexk.host.cs.st-andrews.ac.uk),
   Sugandha Maheshwary (sugandha@iisermohali.ac.in),
   Gabriela Olteanu (http://math.ubbcluj.ro/~olteanu),
   Aurora Olivieri (olivieri@usb.ve),
   Angel del Rio (http://www.um.es/adelrio), and
   Inneke Van Gelder (http://homepages.vub.ac.be/~ivgelder).
Homepage: https://gap-packages.github.io/wedderga
-----------------------------------------------------------------------------
true
]]>
</Log>

</Section>

<!-- ********************************************************* -->

<Section Label="IntroMain">
<Heading>Main functions of &Wedderga; package</Heading>

The main functions of &Wedderga; are 
<Ref Attr="WedderburnDecomposition" /> and
<Ref Attr="WedderburnDecompositionInfo" />.
<P/>

<Ref Attr="WedderburnDecomposition" /> 
computes a list of simple algebras such that their direct product is
isomorphic to the group algebra <M>FG</M>, given as input. 
Thus, the direct product of the entries of the output is
the <E>Wedderburn decomposition</E> (<Ref Sect="WedDec" />) of <M>FG</M>. 
<P/>

If <M>F</M> is an abelian number field then the entries of the output are given
as matrix algebras over cyclotomic algebras (see <Ref Sect="Cyclotomic" />),
thus, the entries of the output of <Ref Attr="WedderburnDecomposition" /> are
realizations of the <E>Wedderburn components</E> (<Ref Sect="WedDec" />) of
<M>FG</M> as algebras which are <E>Brauer equivalent</E> (<Ref Sect="Brauer"/>)
to <E>cyclotomic algebras</E> (<Ref Sect="Cyclotomic" />). Recall that the
Brauer-Witt Theorem ensures that every simple factor of a semisimple group ring
<M>FG</M> is Brauer equivalent (that is represents the same class in the Brauer
group of its centre) to a cyclotomic algebra (<Cite Key="Y"/>. In this case the
algorithm is based on a computational oriented proof of the Brauer-Witt Theorem
due to Olteanu <Cite Key="O" /> which uses previous work by Olivieri, del Río
and Simón <Cite Key="ORS" /> (see also <Cite Key="OR" /> ) for rational group 
algebras of <E>strongly monomial groups</E> (<Ref Sect="StMon" />). The algorithms
 are also based upon the work of Bakshi and Maheshwary <Cite Key="BM14" /> 
 (see also <Cite Key="BM16" />) on the rational group algebras of <E>normally
  monomial groups</E> (<Ref Sect="NorMon" />).<P/>

The Wedderburn components of <M>FG</M> are also matrix algebras over division
rings which are finite extensions of the field <M>F</M>. If <M>F</M> is finite
then by the Wedderburn theorem these division rings are finite fields. In this
case the output of <Ref Attr="WedderburnDecomposition" /> represents the
factors of <M>FG</M> as matrix algebras over finite extensions of the field
<M>F</M>. 
<P/>

In theory &Wedderga; 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.
<Alt Only="LaTeX">\newpage</Alt>
<Example>
<![CDATA[
gap> QG := GroupRing( Rationals, SymmetricGroup(4) );
<algebra-with-one over Rationals, with 2 generators>
gap> WedderburnDecomposition(QG);
[ Rationals, Rationals, <crossed product with center Rationals over CF(
    3) of a group of size 2>, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ) ]
gap> FG := GroupRing( CF(5), SymmetricGroup(4) );
<algebra-with-one over CF(5), with 2 generators>
gap> WedderburnDecomposition( FG );
[ 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 ] ) ]
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 ] ) ]
]]>
</Example>

Instead of <Ref Attr="WedderburnDecomposition" />, that returns a list of 
&GAP; objects, <Ref Attr="WedderburnDecompositionInfo" /> returns the numerical 
description of these objects. 
See Section <Ref Sect="NumDesc" /> for theoretical background.

</Section>



</Chapter>
