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### 1. Introduction

#### 1.1 General aims of Wedderga package

The title ``Wedderga'' stands for ``Wedderburn decomposition of group algebras''. 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.

The methods 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 with the order of G, or an abelian number field (i.e. a subfield of a finite cyclotomic extension of the rationals).

Other functions of Wedderga 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.

#### 1.2 Main functions of Wedderga package

The main functions of Wedderga are `WedderburnDecomposition` (2.1-1) and `WedderburnDecompositionInfo` (2.1-2).

`WedderburnDecomposition` (2.1-1) 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 Wedderburn decomposition (7.3) of FG.

If F is an abelian number field then the entries of the output are given as matrix algebras over cyclotomic algebras (see 7.11), thus, the entries of the output of `WedderburnDecomposition` (2.1-1) are realizations of the Wedderburn components (7.3) of FG as algebras which are Brauer equivalent (7.5) to cyclotomic algebras (7.11). 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 ([Y74], [O07]).

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 `WedderburnDecomposition` (2.1-1) represents the factors of FG as matrix algebras over finite extensions of the field F.

.

    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 ] ) ]

Instead of `WedderburnDecomposition` (2.1-1), that returns a list of GAP objects, `WedderburnDecompositionInfo` (2.1-2) returns the numerical description of these objects. See Section 7.12 for theoretical background.

#### 1.3 Installation and system requirements

Wedderga does not use external binaries and, therefore, works without restrictions on the type of the operating system. It is designed for GAP4.4 and no compatibility with previous releases of GAP4 is guaranteed.

To use the Wedderga online help it is necessary to install the GAP4 package GAPDoc by Frank L\"ubeck and Max Neunh\"offer, which is available from the GAP site or from http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/.

Wedderga is distributed in standard formats (`zoo`, `tar.gz`, `tar.bz2`, `-win.zip`) and can be obtained from http://www.um.es/adelrio/wedderga.htm. To unpack the archive `wedderga-4.2.zoo` you need the program `unzoo`, which can be obtained from the GAP homepage http://www.gap-system.org/ (see section `Distribution'). To install Wedderga, copy this archive into the `pkg` subdirectory of your GAP4.4 installation. The subdirectory `wedderga` will be created in the `pkg` directory after the following command:

`unzoo -x wedderga-4.2.zoo`

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