GAP (Wedderga) - Chapter 1: Introduction

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1 Introduction
1.1 General aims of Wedderga package
1.2 Installation and system requirements
1.3 Main functions of Wedderga package

### 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 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 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.

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.

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

#### 1.2 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 Lübeck and Max Neunhöffer, which is available from the GAP site or from http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/.

Wedderga is distributed in standard formats (`tar.gz`, `tar.bz2`, `-win.zip`) and can be obtained from https://gap-packages.github.io/wedderga/. To install Wedderga, unpack its archive into the `pkg` subdirectory of your GAP installation.

When you don't have access to the directory of your main GAP installation, you can also install the package outside the GAP main directory by unpacking it inside a directory `MYGAPDIR/pkg`. Then to be able to load Wedderga you need to call GAP with the `-l ";MYGAPDIR"` option.

Installation using other archive formats is performed in a similar way.

If the package is installed correctly, it should be loaded as follows:

    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

#### 1.3 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 (9.3) of FG.

If F is an abelian number field then the entries of the output are given as matrix algebras over cyclotomic algebras (see 9.11), thus, the entries of the output of `WedderburnDecomposition` (2.1-1) are realizations of the Wedderburn components (9.3) of FG as algebras which are Brauer equivalent (9.5) to cyclotomic algebras (9.11). 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 ([Yam74]. In this case the algorithm is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu [Olt07] which uses previous work by Olivieri, del Río and Simón [OdRS04] (see also [OdR03] ) for rational group algebras of strongly monomial groups (9.17). The algorithms are also based upon the work of Bakshi and Maheshwary [BM14] (see also [BM16]) on the rational group algebras of normally monomial groups (9.18).

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.

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.

    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 ] ) ]

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 9.12 for theoretical background.

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