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# 1 Introduction to the GAP Character Table Library

### Sections

1. History of the GAP Character Table Library
2. Installing the GAP Character Table Library
3. Loading the GAP Character Table Library
4. Acknowledgements

The usefulness of GAP for character theoretic tasks depends on the availability of many known character tables, and there is a lot of character tables in the GAP table library. Of course, this library is ``open'' in the sense that it shall be extended. So we would be grateful for any further tables of interest sent to us for inclusion into our library. Please send interesting new character tables via e-mail to sam@math.rwth-aachen.de.

It depends on your GAP installation whether the character table library is available. You can check this as follows.

    gap> IsBound( LOADED_PACKAGES.ctbllib );
    true

If the result is `false` then the library is not installed, and you may ask your system administrator for installing it, or install the library in your home directory (see Installing the GAP Character Table Library).

For general information about character tables in GAP, see Chapter Character Tables in the GAP Reference Manual.

The `doc` directory of the GAP Character Table Library contains several files with demonstrations of computations with character tables. Currently these are `ctbldeco.dvi`, `ctblpope.dvi`, and `multfree.dvi`.

If you use the GAP Character Table Library to solve a problem then please send a short email to sam@math.rwth-aachen.de about it. The GAP Character Table Library database should be referenced with the entry Bre01 in the bibliography of this manual.

For referencing the GAP system in general, use the entry GAP4 in the bibliography of this manual.

## 1.1 History of the GAP Character Table Library

The first version of the GAP Character Table Library was released with GAP 3.1 in March 1992.

It was the first aim of this library to continue the character table library of the CAS system (see NPP84) in GAP, as a part of the process of reimplementing the algorithms of CAS in GAP (see History of Character Theory Stuff in GAP in the GAP Reference Manual). GAP 3.1 provided only very restricted methods for computing character tables from groups, so its character theory part was concerned mainly with library tables.

A second aspect of the character table library was to make all character tables shown in the ATLAS of Finite Groups (CCN85) available in GAP. In fact GAP turned out to provide a very good environment for systematic checks of these character tables.

To some extent, the access to the (ordinary) character tables in CCN85 was a prerequisite for storing also the corresponding Brauer character tables in the GAP character table library. Already GAP 3.1 contained many of these tables. They have been computed mainly ``outside of GAP'', using the methods described in HJLP92, and part of the library has been published in the ATLAS of Brauer Characters (JLPW95). One of the roles of GAP was again to perform systematic checks.

Besides these projects, many individual character tables have been added to the GAP Character Table Library since the times of GAP 3.1. They were computed from groups or with character theoretic methods or using a combination of these two possibilities (see, e.g., NPP84 and LP91). Section Contents of the GAP Character Table Library lists some of the sources. The changes in the GAP Character Table Library since the release of GAP 4.1 (in July 1999) are individually documented in the file `ctbldiff.dvi` which is part of the distribution of the library.

In the meantime, a rudimentary interface between groups in GAP and the tables in the GAP Character Table Library has been provided (see The Interface between Character Tables and Groups in the GAP Reference Manual). Similarly, there is an interface to the GAP Libary of Tables of Marks (see The Interface between Tables of Marks and Character Tables in the GAP Reference Manual).

Currently the main focus in the development of the GAP Character Table Library is --besides the addition of tables that appear to be interesting-- the better interaction with other databases, such as the ATLAS of Group Representations (see the GAP 4 package AtlasRep), and an improvement of the ``database'' aspect of the character table library itself, for example by providing a ``WWW table of contents''.

Until the release of GAP 4.3 in spring 2002, the GAP Character Table Library had been a part of the main GAP library. With GAP 4.3, it is ``split off'' as a GAP package.

## 1.2 Installing the GAP Character Table Library

To install the GAP package `ctbllib`, unpack the archive file in a directory in the `pkg` directory of your local copy of GAP 4. This might be the `pkg` directory of the GAP 4 home directory, see Section Installing GAP Packages of the GAP 4 Reference Manual for details. It is however also possible to keep an additional `pkg` directory in your private directories, see Installing a GAP Package in your home directory of the GAP 4 Reference Manual. The latter possibility must be chosen if you do not have write access to the GAP root directory.

The package consists entirely of GAP code, no external binaries need to be compiled.

Both dvi and PostScript versions of the manual for the GAP Character Table Library are available (as `manual.dvi` and `manual.ps` respectively) in the `doc` directory of the `ctbllib` package, and an HTML version can be found in the `htm` directory.

## 1.3 Loading the GAP Character Table Library

The GAP Character Table Library may be loaded automatically when GAP is started, or it has to be loaded within GAP as follows.

    gap> RequirePackage( "ctbllib" );

See Loading a GAP Package in the GAP Reference Manual for details about these alternatives; also the possibility to disable automatic loading of the library is described in this manual section. The default is that the GAP Character Table Library is loaded automatically.

## 1.4 Acknowledgements

The functions for the conversion of CAS tables to GAP format have been written by Götz Pfeiffer. The functions for converting the ``Cambridge format'' (in which the original data of the ATLAS of Finite Groups had been stored) to GAP format have been written by Christoph Jansen. The functions for checking library tables have been written by Thomas Breuer.

The development of the GAP Character Table Library has been supported by several DFG grants, in particular the project ``Representation Theory of Finite Groups and Finite Dimensional Algebras'' (until 1991), and the Schwerpunkt ``Algorithmische Zahlentheorie und Algebra'' (from 1991 until 1997).

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