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<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 methods 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 with 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. 
<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/>

</Section>

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<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"/>, <Cite Key="O" />). 
<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/>

<Alt Only="LaTeX">\pagebreak</Alt>.

<Example>
<![CDATA[
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 ] ) ]
]]>
</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>

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<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\"ubeck and Max Neunh\"offer, 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>zoo</File>, <File>tar.gz</File>, <File>tar.bz2</File>, 
<File>-win.zip</File>) and can be obtained from 
<URL>http://www.um.es/adelrio/wedderga.htm</URL>.
To unpack the archive <File>wedderga-4.2.zoo</File> you need the program 
<File>unzoo</File>, which can be obtained from the &GAP; homepage 
<URL>http://www.gap-system.org/</URL> (see section `Distribution').
To install &Wedderga;, copy this archive into the <File>pkg</File> subdirectory of your 
&GAP;4.4 installation. The subdirectory <File>wedderga</File> will be created in 
the <File>pkg</File> directory after the following command:
<P/>
  <C>unzoo -x wedderga-4.2.zoo</C>
<P/>

</Section>

</Chapter>
