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1 Introduction to the GAP Character Table Library
 1.1 History of the GAP Character Table Library
 1.2 What's New in CTblLib, Compared to Older Versions?
 1.3 Acknowledgements

1 Introduction to the GAP Character Table Library

The usefulness of GAP for character theoretic tasks depends on the availability of many known character tables, and there are 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 offer interesting new character tables via e-mail to

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

gap> InstalledPackageVersion( "ctbllib" ) <> fail;

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 Section 4.5-1.

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

The doc and htm directories of the GAP Character Table Library contain several files with examples of character theoretic computations, in PDF and HTML format. Currently these are

If you use the GAP Character Table Library to solve a problem then please send a short e-mail to about it. The GAP Character Table Library database should be referenced as follows.

@misc{ CTblLib1.2.1,
  author =       {Breuer, T.},
  title =        {The \textsf{GAP} {C}haracter {T}able {L}ibrary,
                  {V}ersion 1.2.1},
  month =        {May},
  year =         {2012},
  note =         {\textsf{GAP} package},
  howpublished = {\~{}Thomas.Breuer/ctbllib}

For referencing the GAP system in general, use the entry [GAP] in the bibliography of this manual, see also

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 Reference: History of Character Theory Stuff in GAP. 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 [CCNPW85] 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 [CCNPW85] 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 [HJLP], 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 4.1 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 doc/ctbldiff.pdf of the package.

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 and the GAP Library of Tables of Marks (see the GAP packages AtlasRep and TomLib) and GAP's group libraries, and an improvement of the "database" aspect of the character table library itself, see the sections 3.1 and 3.5.

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 was "split off" as a GAP package.

1.2 What's New in CTblLib, Compared to Older Versions?

1.2-1 What's New in Version 1.2?

The following bugs were fixed in version 1.2.2.

Concerning character table data, we have the following.

Like in earlier versions, the PDF file doc/ctbldiff.pdf of the package lists the important changes, mainly related to the relevant simple groups. However, a probably more suitable overview is given by the GAP readable file data/ctbldiff.dat, which contains a complete overview of all changes, including the additions of class fusions. (Note that each added or changed fusion is mentioned twice in this list, once for the "from" table and once for the "to" table.) This list of changes can be shown using BrowseCTblLibDifferences (3.5-4) if the Browse package (see [BL12]) is available.

Besides these changes of the data, the following features are new.

1.2-2 What's New in Version 1.1?

First of all, of course several character tables were added; for an overview, see the file doc/ctbldiff.pdf in the home directory of the package. Also lots of class fusions were added. This includes factor fusions onto the tables of the factor groups modulo the largest normal p-subgroups whenever the tables of the factors are available; these maps admit the automatic construction of the p-modular Brauer tables if the corresponding tables of the factors are available. For example, the 2-modular Brauer table of the maximal subgroup of the type 2^10:M_22 in the group Fi_22 is available because of the known 2-modular table of M_22 and the stored factor fusion onto the table of M_22.

Second, more information has been made more explicit, in the following sense.

Third, several errors have been corrected (again see doc/ctbldiff.pdf). Most of them affect class fusions, and for most of those, the term "error" could be regarded as not really appropriate. See 4.9 for details.

Finally, the GAP functions for reading and writing other formats of character tables have been moved from the main GAP library to this package (see Chapter 6), because they are useful only for library tables. The GAP 3 format is now also supported, mainly for documentation purposes (see Section 6.3).

1.3 Acknowledgements

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).

The functions for the conversion of CAS tables to GAP format have been written by Götz Pfeiffer. The functions for converting the so-called "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 generic character tables of double covers of alternating and symmetric groups were contributed by Felix Noeske, see [Noe02].

The functions in Section 3.4 (DeligneLusztigName (3.4-3), DeligneLusztigNames (3.4-2), and UnipotentCharacter (3.4-1)) as well as the corresponding data for the finite groups of Lie type in the GAP Character Table Library have been contributed by Michael Claßen-Houben, see [Cla05].

Several character tables of maximal subgroups of covering groups of simple groups have been computed by Sebastian Dany, see [Dan06].

Thanks to Frank Lübeck and Max Neunhöffer for their help with solving technical problems concerning the HMTL part of the example files that belong to the package documentation, and to Ian Hutchinson whose TeX to HTML translator TtH was used to provide these HTML files.

Thanks to Frank Lübeck and Max Neunhöffer also for developing the GAPDoc package, on which the manual of the CTblLib package is based. The previously available documentation format had been completely inappropriate.

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