71.1 Contents of the Character Table Library

This section gives a brief overview of the contents of the GAP character table library. For the details about, e.g., the structure of data files, see Organization of the Character Table Library.

The changes in the character table library are listed in a file that can be reached from http://www-gap.dcs.st-and.ac.uk/~gap/Info4/changes.html.

There are three different kinds of character tables in the GAP library, namely ordinary character tables, Brauer tables, and generic character tables. Note that the Brauer table and the corresponding ordinary table of a group determine the decomposition matrix of the group (or the decomposition matrices of its blocks). These decomposition matrices can be computed with GAP (see Operations Concerning Blocks for details). A collection of DVI and PostScript files of the known decomposition matrices of almost simple groups in the GAP table library can also be found at http://www.math.rwth-aachen.de/~MOC/decomposition/.

Ordinary Character Tables

Two different aspects are useful to list the ordinary character tables available in GAP, namely the aspect of the source of the tables and that of connections between the tables.

As for the source, there are two big sources, namely the ATLAS of Finite Groups (see ATLAS Tables) and the sf CAS library of character tables (see NPP84). Many ATLAS tables are contained in the sf CAS library, and difficulties may arise because the succession of characters and classes in sf CAS tables and ATLAS tables are in general different, so see CAS Tables for the relations between these two variants of character tables of the same group. A large subset of the sf CAS tables is the set of tables of Sylow normalizers of sporadic simple groups as published in Ost86 --this may be viewed as another source of character tables. The library also contains the character tables of factor groups of space groups (computed by W. Hanrath, see Han88) that are part of HP89 in form of two microfiches; these tables are given in sf CAS format (see CAS Tables) on the microfiches, but they had not been part of the ``official'' sf CAS library.

To avoid confusion about the ordering of classes and characters in a given table, authorship and so on, the InfoText (see InfoText) value of the table contains the information

origin: ATLAS of finite groups

for ATLAS tables (see ATLAS Tables),

origin: Ostermann

for tables contained in Ost86,

origin: CAS library

for any table of the sf CAS table library that is contained neither in the ATLAS nor in Ost86, and

origin: Hanrath library

for tables contained in the microfiches in HP89.

If one is interested in the aspect of connections between the tables, i.e., the internal structure of the library of ordinary tables, the contents can be listed up the following way.

We have

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all ATLAS tables (see ATLAS Tables), i.e., the tables of the simple groups which are contained in the ATLAS of Finite Groups, and the tables of cyclic and bicyclic extensions of these groups,

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most tables of maximal subgroups of sporadic simple groups (not all for the groups F₃₊, B, M),

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some tables of maximal subgroups of other ATLAS tables, where the list of maximal subgroups is complete if the Maxes value for the table is known (see Maxes),

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the tables of most nontrivial Sylow normalizers of sporadic simple groups as printed in Ost86, where nontrivial means that the Sylow p normalizer is not contained in p:(p-1) (not J₄N2, Co₁N2, Co₁N5, all of Fi₂₃, Fi₂₄^¢, B, M, HN, and Fi₂₂N2)

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some tables of element centralizers

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some tables of Sylow subgroups

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a few other tables, e.g. W(F4) (namely which?)

Note that class fusions stored on library tables are neither guaranteed to be consistent for any two subgroups of a group and their intersection, nor tested to be consistent with respect to composition of maps.

Brauer Tables

The library contains all tables of the ATLAS of Brauer Tables (JLPW95), and many other Brauer tables of bicyclic extensions of simple groups which are known yet. There is ongoing work in computing new tables, so this part of the library is growing.

The Brauer tables in the library contain the information

origin: modular ATLAS of finite groups

Generic Character Tables

See Generic Character Tables for an overview of generic tables available.

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