[1X1 Introduction[0X [1X1.1 General aims of [5XWedderga[1X package[0X The title ``[5XWedderga[0m'' stands for ``[12XWedder[0mburn decomposition of [12Xg[0mroup [12Xa[0mlgebras''. This is a [5XGAP[0m 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. 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). Other functions of [5XWedderga[0m compute the primitive central idempotents of semisimple group algebras. 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. [1X1.2 Main functions of [5XWedderga[1X package[0X The main functions of [5XWedderga[0m are [2XWedderburnDecomposition[0m ([14X2.1-1[0m) and [2XWedderburnDecompositionInfo[0m ([14X2.1-2[0m). [2XWedderburnDecomposition[0m ([14X2.1-1[0m) 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 [13XWedderburn decomposition[0m ([14X7.3[0m) of FG. If F is an abelian number field then the entries of the output are given as matrix algebras over cyclotomic algebras (see [14X7.11[0m), thus, the entries of the output of [2XWedderburnDecomposition[0m ([14X2.1-1[0m) are realizations of the [13XWedderburn components[0m ([14X7.3[0m) of FG as algebras which are [13XBrauer equivalent[0m ([14X7.5[0m) to [13Xcyclotomic algebras[0m ([14X7.11[0m). 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 ([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 RÃo and Simón [ORS04] for rational group algebras of [13Xstrongly monomial groups[0m ([14X7.16[0m). 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 [2XWedderburnDecomposition[0m ([14X2.1-1[0m) represents the factors of FG as matrix algebras over finite extensions of the field F. In theory [5XWedderga[0m 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. [4X--------------------------- Example ----------------------------[0X [4X[0X [4Xgap> QG := GroupRing( Rationals, SymmetricGroup(4) );[0X [4X<algebra-with-one over Rationals, with 2 generators>[0X [4Xgap> WedderburnDecomposition(QG);[0X [4X[ Rationals, Rationals, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ),[0X [4X <crossed product with center Rationals over CF(3) of a group of size 2> ][0X [4Xgap> FG := GroupRing( CF(5), SymmetricGroup(4) );[0X [4X<algebra-with-one over CF(5), with 2 generators>[0X [4Xgap> WedderburnDecomposition( FG );[0X [4X[ CF(5), CF(5), ( CF(5)^[ 3, 3 ] ), ( CF(5)^[ 3, 3 ] ),[0X [4X <crossed product with center CF(5) over AsField( CF(5), CF([0X [4X 15) ) of a group of size 2> ][0X [4Xgap> FG := GroupRing( GF(5), SymmetricGroup(4) ); [0X [4X<algebra-with-one over GF(5), with 2 generators>[0X [4Xgap> WedderburnDecomposition( FG );[0X [4X[ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), [0X [4X ( GF(5)^[ 3, 3 ] ), ( GF(5)^[ 3, 3 ] ) ][0X [4Xgap> FG := GroupRing( GF(5), SmallGroup(24,3) );[0X [4X<algebra-with-one over GF(5), with 4 generators>[0X [4Xgap> WedderburnDecomposition( FG );[0X [4X[ ( GF(5)^[ 1, 1 ] ), ( GF(5^2)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), [0X [4X ( GF(5^2)^[ 2, 2 ] ), ( GF(5)^[ 3, 3 ] ) ][0X [4X[0X [4X------------------------------------------------------------------[0X Instead of [2XWedderburnDecomposition[0m ([14X2.1-1[0m), that returns a list of [5XGAP[0m objects, [2XWedderburnDecompositionInfo[0m ([14X2.1-2[0m) returns the numerical description of these objects. See Section [14X7.12[0m for theoretical background. [1X1.3 Installation and system requirements[0X [5XWedderga[0m does not use external binaries and, therefore, works without restrictions on the type of the operating system. It is designed for [5XGAP[0m4.4 and no compatibility with previous releases of [5XGAP[0m4 is guaranteed. To use the [5XWedderga[0m online help it is necessary to install the [5XGAP[0m4 package [5XGAPDoc[0m by Frank Lübeck and Max Neunhöffer, which is available from the [5XGAP[0m site or from [7Xhttp://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/[0m. [5XWedderga[0m is distributed in standard formats ([11Xtar.gz[0m, [11Xtar.bz2[0m, [11X-win.zip[0m) and can be obtained from [7Xhttp://www.um.es/adelrio/wedderga.htm[0m, its mirror [7Xhttp://www.cs.st-andrews.ac.uk/~alexk/wedderga.htm[0m or the page [7Xhttp://www.gap-system.org/Packages/wedderga.html[0m at the [5XGAP[0m web site. The latter also offers [11Xzoo[0m-archive. To unpack the archive [11Xwedderga-4.3.2.zoo[0m you need the program [11Xunzoo[0m, which can be obtained from the [5XGAP[0m homepage [7Xhttp://www.gap-system.org/[0m (see section `Distribution'). To install [5XWedderga[0m, copy this archive into the [11Xpkg[0m subdirectory of your [5XGAP[0m4.4 installation. The subdirectory [11Xwedderga[0m will be created in the [11Xpkg[0m directory after the following command: [10Xunzoo -x wedderga-4.3.2.zoo[0m When you don't have access to the directory of your main [5XGAP[0m installation, you can also install the package [13Xoutside the [5XGAP[0m[13X main directory[0m by unpacking it inside a directory [11XMYGAPDIR/pkg[0m. Then to be able to load Wedderga you need to call GAP with the [10X-l ";MYGAPDIR"[0m option. Installation using other archive formats is performed in a similar way.