  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  
  [1X1.1 [33X[0;0YGeneral aims of [5XWedderga[105X[101X[1X package[133X[101X
  
  [33X[0;0YThe  title  ``[5XWedderga[105X''  stands  for  ``[12XWedder[112Xburn  decomposition  of [12Xg[112Xroup
  [12Xa[112Xlgebras''.  This  is  a [5XGAP[105X 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.[133X
  
  [33X[0;0YThe  method implemented by the package produces the Wedderburn decomposition
  of  a group algebra [22XFG[122X provided [22XG[122X is a finite group and [22XF[122X is either a finite
  field  of  characteristic  coprime  to  the order of [22XG[122X, or an abelian number
  field (i.e. a subfield of a finite cyclotomic extension of the rationals).[133X
  
  [33X[0;0YOther  functions  of  [5XWedderga[105X  compute the primitive central idempotents of
  semisimple  group  algebras  and  a  complete  set  of  orthogonal primitive
  idempotents.[133X
  
  [33X[0;0YThe  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.[133X
  
  [33X[0;0YFurhermore, the package provides functions to create code words from a group
  ring element.[133X
  
  
  [1X1.2 [33X[0;0YInstallation and system requirements[133X[101X
  
  [33X[0;0Y[5XWedderga[105X  does  not  use  external  binaries  and,  therefore, works without
  restrictions  on the type of the operating system. It is designed for [5XGAP[105X4.4
  and no compatibility with previous releases of [5XGAP[105X4 is guaranteed.[133X
  
  [33X[0;0YTo  use the [5XWedderga[105X online help it is necessary to install the [5XGAP[105X4 package
  [5XGAPDoc[105X  by  Frank Lübeck and Max Neunhöffer, which is available from the [5XGAP[105X
  site or from [7Xhttp://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/[107X.[133X
  
  [33X[0;0Y[5XWedderga[105X  is distributed in standard formats ([11Xtar.gz[111X, [11Xtar.bz2[111X, [11X-win.zip[111X) and
  can  be  obtained  from  [7Xhttp://www.um.es/adelrio/wedderga.htm[107X,  its  mirror
  [7Xhttp://www.cs.st-andrews.ac.uk/~alexk/wedderga/[107X       or       the      page
  [7Xhttp://www.gap-system.org/Packages/wedderga.html[107X  at  the  [5XGAP[105X  web site. To
  install  [5XWedderga[105X,  unpack its archive into the [11Xpkg[111X subdirectory of your [5XGAP[105X
  installation.[133X
  
  [33X[0;0YWhen  you  don't have access to the directory of your main [5XGAP[105X installation,
  you can also install the package [13Xoutside the [5XGAP[105X main directory[113X by unpacking
  it  inside  a  directory  [11XMYGAPDIR/pkg[111X. Then to be able to load Wedderga you
  need to call GAP with the [10X-l ";MYGAPDIR"[110X option.[133X
  
  [33X[0;0YInstallation using other archive formats is performed in a similar way.[133X
  
  [33X[0;0YIf the package is installed correctly, it should be loaded as follows:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage("wedderga");[127X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28XLoading  Wedderga 4.4.4 (Wedderga)[128X[104X
    [4X[28Xby Osnel Broche Cristo (osnel@ufla.br),[128X[104X
    [4X[28X   Alexander Konovalov (http://www.cs.st-andrews.ac.uk/~alexk/),[128X[104X
    [4X[28X   Gabriela Olteanu (golteanu@um.es, olteanu@math.ubbcluj.ro),[128X[104X
    [4X[28X   Aurora Olivieri (olivieri@usb.ve), [128X[104X
    [4X[28X   Angel del Rio (http://www.um.es/adelrio) and[128X[104X
    [4X[28X   Inneke Van Gelder (ivgelder@vub.ac.be).[128X[104X
    [4X[28X-----------------------------------------------------------------------------[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X1.3 [33X[0;0YMain functions of [5XWedderga[105X[101X[1X package[133X[101X
  
  [33X[0;0YThe  main  functions  of  [5XWedderga[105X  are  [2XWedderburnDecomposition[102X ([14X2.1-1[114X) and
  [2XWedderburnDecompositionInfo[102X ([14X2.1-2[114X).[133X
  
  [33X[0;0Y[2XWedderburnDecomposition[102X ([14X2.1-1[114X) computes a list of simple algebras such that
  their  direct product is isomorphic to the group algebra [22XFG[122X, given as input.
  Thus,  the  direct  product  of  the entries of the output is the [13XWedderburn
  decomposition[113X ([14X9.3[114X) of [22XFG[122X.[133X
  
  [33X[0;0YIf  [22XF[122X is an abelian number field then the entries of the output are given as
  matrix  algebras  over  cyclotomic algebras (see [14X9.11[114X), thus, the entries of
  the  output  of  [2XWedderburnDecomposition[102X  ([14X2.1-1[114X)  are  realizations  of the
  [13XWedderburn  components[113X  ([14X9.3[114X)  of [22XFG[122X as algebras which are [13XBrauer equivalent[113X
  ([14X9.5[114X)  to  [13Xcyclotomic  algebras[113X  ([14X9.11[114X). Recall that the Brauer-Witt Theorem
  ensures  that  every  simple  factor of a semisimple group ring [22XFG[122X is Brauer
  equivalent  (that  is  represents  the same class in the Brauer group of its
  centre)  to  a  cyclotomic  algebra  ([Yam74]. In this case the algorithm is
  based  in  a  computational oriented proof of the Brauer-Witt Theorem due to
  Olteanu  [Olt07]  which  uses  previous  work by Olivieri, del Río and Simón
  [ORS04] for rational group algebras of [13Xstrongly monomial groups[113X ([14X9.16[114X).[133X
  
  [33X[0;0YThe Wedderburn components of [22XFG[122X are also matrix algebras over division rings
  which  are  finite  extensions  of  the  field [22XF[122X. If [22XF[122X is finite then by the
  Wedderburn  theorem these division rings are finite fields. In this case the
  output  of  [2XWedderburnDecomposition[102X  ([14X2.1-1[114X) represents the factors of [22XFG[122X as
  matrix algebras over finite extensions of the field [22XF[122X.[133X
  
  [33X[0;0YIn   theory   [5XWedderga[105X  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.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XQG := GroupRing( Rationals, SymmetricGroup(4) );[127X[104X
    [4X[28X<algebra-with-one over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecomposition(QG);[127X[104X
    [4X[28X[ Rationals, Rationals, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ),[128X[104X
    [4X[28X  <crossed product with center Rationals over CF(3) of a group of size 2> ][128X[104X
    [4X[25Xgap>[125X [27XFG := GroupRing( CF(5), SymmetricGroup(4) );[127X[104X
    [4X[28X<algebra-with-one over CF(5), with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecomposition( FG );[127X[104X
    [4X[28X[ CF(5), CF(5), ( CF(5)^[ 3, 3 ] ), ( CF(5)^[ 3, 3 ] ),[128X[104X
    [4X[28X  <crossed product with center CF(5) over AsField( CF(5), CF([128X[104X
    [4X[28X    15) ) of a group of size 2> ][128X[104X
    [4X[25Xgap>[125X [27XFG := GroupRing( GF(5), SymmetricGroup(4) ); [127X[104X
    [4X[28X<algebra-with-one over GF(5), with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecomposition( FG );[127X[104X
    [4X[28X[ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), [128X[104X
    [4X[28X  ( GF(5)^[ 3, 3 ] ), ( GF(5)^[ 3, 3 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XFG := GroupRing( GF(5), SmallGroup(24,3) );[127X[104X
    [4X[28X<algebra-with-one over GF(5), with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecomposition( FG );[127X[104X
    [4X[28X[ ( GF(5)^[ 1, 1 ] ), ( GF(5^2)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), [128X[104X
    [4X[28X  ( GF(5^2)^[ 2, 2 ] ), ( GF(5)^[ 3, 3 ] ) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YInstead  of  [2XWedderburnDecomposition[102X  ([14X2.1-1[114X),  that  returns  a list of [5XGAP[105X
  objects,   [2XWedderburnDecompositionInfo[102X   ([14X2.1-2[114X)   returns   the   numerical
  description of these objects. See Section [14X9.12[114X for theoretical background.[133X
  
