GAP (AtlasRep) - Chapter 1: Introduction to the AtlasRep Package

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1 Introduction to the AtlasRep Package
1.1 The ATLAS of Group Representations
1.2 The GAP Interface to the ATLAS of Group Representations
1.3 What's New in AtlasRep, Compared to Older Versions?
1.3-1 What's New in Version 1.5?
1.3-2 What's New in Version 1.4?
1.3-3 What's New in Version 1.3.1?
1.3-4 What's New in Version 1.3?
1.3-5 What's New in Version 1.2?
1.3-6 What's New in Version 1.1?
1.4 Acknowledgments

### 1 Introduction to the AtlasRep Package

The aim of the GAP 4 package AtlasRep is to provide a link between GAP and the "ATLAS of Group Representations", a database that comprises generating permutations and matrices for many almost simple groups, and information about their maximal subgroups. This database is available independent of GAP at

http://brauer.maths.qmul.ac.uk/Atlas

The AtlasRep package consists of this database (see Section 1.1) and a GAP interface (see Section 1.2); the latter is extended by further information available via the internet (see Section 4.4).

This package manual has the following parts.

A tutorial

  gives an overview how the functions of the package can be used, see Chapter 2.

User interface functions

  are described in Chapter 3.

Customizations of the package

  are described in Chapter 4.

Information how to extend the database

  can be found in Chapter 5.

More technical information

  can be found in the chapters 6 (concerning GAP objects that are introduced by the package) and 7 (concerning global variables and sanity checks).

#### 1.1 The ATLAS of Group Representations

The ATLAS of Group Representations consists of matrices over various rings, permutations, and shell scripts encoding so-called black box programs (see [Nic06] and Section 6.2). Many of these scripts are straight line programs (see [BSWW01], [SWW00], and Reference: Straight Line Programs) and straight line decisions (see Section 6.1). These programs can be used to compute certain elements in a group G from its standard generators (see [Wil96] and Section 3.3) for example generators of maximal subgroups of G or representatives of conjugacy classes of G.

The ATLAS of Group Representations has been prepared by Robert Wilson, Peter Walsh, Jonathan Tripp, Ibrahim Suleiman, Richard Parker, Simon Norton, Simon Nickerson, Steve Linton, John Bray, and Rachel Abbott (in reverse alphabetical order).

The information was computed and composed using computer algebra systems such as MeatAxe (see [Rin98]), Magma (see [CP96]), and GAP (in reverse alphabetical order). Part of the constructions have been documented in the literature on almost simple groups, or the results have been used in such publications, see for example the references in [CCNPW85] and [BN95].

If you use the ATLAS of Group Representations to solve a problem then please send a short email to R.A.Wilson@qmul.ac.uk about it. The ATLAS of Group Representations database should be referenced with the entry [ATLAS] in the bibliography of this manual.

If your work made use of functions of the GAP interface (see Section 1.2) then you should also reference this interface, as follows.

    @misc{ AtlasRep1.5,
      author =       {Wilson, R. A. and Parker, R. A. and Nickerson, S. and
                      Bray, J. N. and Breuer, T.},
      title =        {{AtlasRep}, A \textsf{GAP} Interface to the Atlas of
                      Group Representations,
                      {V}ersion 1.5},
      month =        {July},
      year =         {2011},
      note =         {Refereed \textsf{GAP} package},
      howpublished = {http://www.math.rwth-aachen.de/\~{}Thomas.Breuer/atlasrep}
    }

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

http://www.gap-system.org.

#### 1.2 The GAP Interface to the ATLAS of Group Representations

The GAP interface to the ATLAS of Group Representations consists of essentially two parts.

- First, there is the user interface which allows the user to get an overview of the contents of the database, and to access the data in GAP format; this is described in Chapter 3. Advanced users may add their own data to the database, this is described in Chapter 5.

- Second, there is administrational information, which covers also the declaration of GAP objects such as straight line decisions and black box programs. This is important mainly for users interested in the actual implementation (e. g., for modifying the package) or in using it together with the `C`-MeatAxe standalone (see [Rin98]); this is described in Chapter 7.

Information concerning the `C`-MeatAxe, including the manual [Rin98], can be found at

http://www.math.rwth-aachen.de/LDFM/homes/MTX

The interface and this manual have been provided by Thomas Breuer, except for the interpreter for black box programs (see Section 6.2), which is due to Simon Nickerson. Comments, bug reports, and hints for improving the interface can be sent to sam@math.rwth-aachen.de.

#### 1.3 What's New in AtlasRep, Compared to Older Versions?

##### 1.3-1 What's New in Version 1.5?

- The function `AtlasSubgroup` (3.5-8) now admits also the return value of `OneAtlasGeneratingSetInfo` (3.5-5) or the return value of `AtlasGroup` (3.5-7) as its first argument. The latter is implemented via the new attribute `AtlasRepInfoRecord` (3.5-9), which is set in the groups constructed by `AtlasGroup` (3.5-7).

- Information about transitivity, rank, primitivity, and point stabilizers of many permutation representations is now available. If applicable then this information appears in the records returned by `OneAtlasGeneratingSetInfo` (3.5-5), it is part of the overview shown by `DisplayAtlasInfo` (3.5-1), and it is shown also in the data overview in the web, see Section 4.4.

  Two new manual sections about point stabilizers have been added, see the sections 2.4-6 and 2.4-7.

- Information about the characters afforded by many matrix and permutation representations is now available. If applicable then this information appears in the records returned by `OneAtlasGeneratingSetInfo` (3.5-5), for matrix representations it is part of the overview shown by `DisplayAtlasInfo` (3.5-1), and it is shown also in the data overview in the web, see Section 4.4.

- The functions `Character` (Reference: Character (for a character table and a list)), `Identifier` (Reference: Identifier (for character tables)), `IsPrimitive` (Reference: IsPrimitive), `IsTransitive` (Reference: IsTransitive), `Transitivity` (Reference: Transitivity), and `RankAction` (Reference: RankAction) are now supported as input conditions in `DisplayAtlasInfo` (3.5-1), `OneAtlasGeneratingSetInfo` (3.5-5), etc..

- It is now possible to restrict the data shown by `DisplayAtlasInfo` (3.5-1) to private or non-private data. Also the data returned by `OneAtlasGeneratingSetInfo` (3.5-5) can be restricted to private or non-private data.

- A tutorial for beginners was added to the manual, see Chapter 2, and the manual was restructured.

- In the overview shown by `DisplayAtlasInfo` (3.5-1) and in the data overview in the web (see Section 4.4), the ordering of groups was improved such that, e.g., `"A9"` precedes `"A10"`.

- The function `AtlasClassNames` (3.4-2) now admits also a Brauer table as its argument, and works also for character tables of bicyclic extensions of simple groups.

- The group names that are entered in `DisplayAtlasInfo` (3.5-1), `OneAtlasGeneratingSetInfo` (3.5-5), etc., are now case insensitive, and if the package CTblLib is available then the admissible group names for the GAP character table of the group in question can be used in these functions.

- In order to reduce the number of global variables, several functions have been turned into components of the new global variable `AGR` (7.1-5). A few of these functions had been documented in the previous version, the old values are still available if the package file `gap/obsolete.g` has been read. This file is read automatically when the package gets loaded if `GAPInfo.UserPreferences.ReadObsolete` is `true`, see Reference: The gap.ini file.

- A few nicer characters are used by `DisplayAtlasInfo` (3.5-1) if `GAPInfo.TermEncoding` has the value `"UTF-8"` and if `AtlasOfGroupRepresentationsInfo.displayFunction` is different from `Print` (Reference: Print), see Section 4.3-5.

- A bug in the function `ReloadAtlasTableOfContents` (4.2-1) was fixed. Thanks to Jack Schmidt for reporting this bug.

##### 1.3-2 What's New in Version 1.4?

- In addition to the group orders that were added in version 1.3 (see Section 1.3-4), also many orders of maximal subgroups are now available. These values occur in the records returned by `AtlasProgram` (3.5-3) (for the case of `"maxes"` type programs) and of the three argument version of `AtlasGenerators` (3.5-2); now a `size` component may be bound. In these cases, the groups returned by `AtlasSubgroup` (3.5-8) have the `Size` (Reference: Size) attribute set.

- The information about the number of maximal subgroups, if available, is now used in `DisplayAtlasInfo` (3.5-1).

- In many cases, straight line programs for computing generators of maximal subgroups of a group G, say, can in fact be used to compute also generators of maximal subgroups of downward extensions of G; if not then it may suffice to extend the given straight line programs by additional generators.

  Currently this yields more than 200 new possibilities to compute maximal subgroups, this means a growth by about 25 percent. For example, all maximal subgroups of 12.M_22 and 2.Fi_22 can now be accessed via `AtlasGenerators` (3.5-2).

  (Of course this extension means only that one can access the straight line programs in question automatically via the GAP interface. In principle one could have used them already before, by explicitly applying a straight line program for a factor group to generators of a group, and perhaps adding some element in the kernel of the natural epimorphism.)

  For this feature, information about the compatibility of standard generators of groups and their factor groups was added.

- The bibliographies contained in the ATLAS of Finite Groups [CCNPW85] and in the ATLAS of Brauer Characters [JLPW95] are now available, see `BrowseBibliographySporadicSimple` (3.6-2).

- If the GAP package Browse (see [BL11]) is loaded then the new functions `BrowseMinimalDegrees` (3.6-1) and `BrowseBibliographySporadicSimple` (3.6-2) are available; these functions can be called also by choosing the corresponding menu entries of the Browse application `BrowseGapData` (Browse: BrowseGapData).

- The function `AtlasGroup` (3.5-7) now admits also the return value of `OneAtlasGeneratingSetInfo` (3.5-5) as its argument.

##### 1.3-3 What's New in Version 1.3.1?

This version was mainly released in order to fix a few problems. Now one does not get warnings about unbound variables when the package is loaded and the GAP package IO [Neu11] is not available, and pathological situations in `FFMatOrPermCMtxBinary` (7.3-5) (concerning extremely short corrupted data files and different byte orderings in binary files) are handled more carefully.

Besides this, the two functions `AtlasGroup` (3.5-7) and `AtlasSubgroup` (3.5-8) were introduced, and the extended function `QuaternionAlgebra` (Reference: QuaternionAlgebra) of GAP 4.4.10 can now be used for describing base rings in `OneAtlasGeneratingSetInfo` (3.5-5) and `AllAtlasGeneratingSetInfos` (3.5-6). (This is the reason why this version of the package requires at least version 4.4.10 of GAP.)

##### 1.3-4 What's New in Version 1.3?

- The database was extended, see Section 4.3-4 for the number and size of files.

- New data types and corresponding GAP objects have been introduced, for representing semi-presentations, presentations, and programs for finding standard generators. For details, see `AtlasProgram` (3.5-3), Chapter 6, and Section 7.6.

- The records returned by the functions `AtlasGenerators` (3.5-2), `OneAtlasGeneratingSetInfo` (3.5-5), and `AllAtlasGeneratingSetInfos` (3.5-6) now contain the name and (if known) the order of the group in question, and also components describing the degree in the case of permutation representations or the dimension and the base ring of the natural module in the case of matrix representations.

- For many of the groups, information about the minimal degree of faithful permutation representations and the minimal dimensions of faithful matrix representations in various characteristics is available for `DisplayAtlasInfo` (3.5-1), `OneAtlasGeneratingSetInfo` (3.5-5), and `AllAtlasGeneratingSetInfos` (3.5-6), see also Section 6.3. For these functions, also properties such as `IsPrimeInt` (Reference: IsPrimeInt) can be used to describe the intended restriction of the output.

- One can now use `Pager` (Reference: Pager) functionality in `DisplayAtlasInfo` (3.5-1), see Section 4.3-5.

  An interactive alternative to `DisplayAtlasInfo` (3.5-1) is provided by the function `BrowseAtlasInfo` (Browse: BrowseAtlasInfo) from the new (recommended) GAP package Browse [BL11].

- The functions `OneAtlasGeneratingSetInfo` (3.5-5) and `AllAtlasGeneratingSetInfos` (3.5-6) now admit also a list of group names as the first argument.

- The functions for actually accessing the data are more flexible now, see Section 4.3-6.

- For transferring remote data, the GAP package IO [Neu11] can now be used (and is recommended) as an alternative to `wget`, see Section 4.3-3.

- The address of the data server has changed. Since the access to the server is no longer possible via `ftp`, the mechanim used up to version 1.2, which was based on `ftp`, had to be rewritten.

  The main consequence of this change is that information about updates of the table of contents is now provided at the package's homepage. This means that on the one hand, now package users cannot compute the table of contents directly from the server data, but on the other hand the update information can be downloaded without the necessity to install `perl`.

  Another consequence is that the system program `ls` is no longer needed, see Section 1.3-6.

- The package manual has been restructured, extended and improved. It is now based on the package GAPDoc [LN11].

##### 1.3-5 What's New in Version 1.2?

Not much.

The release of Version 1.2 became necessary first of all in order to provide a package version that is compatible with GAP 4.4, since some cross-references into the GAP Reference Manual were broken due to changes of section names. Additionally, several web addresses concerning the package itself were changed and thus had to be adjusted.

This opportunity was used

- to upgrade the administrational part for loading the package to the mechanism that is recommended for GAP 4.4,

- to extend the test suite, which now covers more consistency checks using the GAP Character Table Library [Bre04],

- to make the function `ScanMeatAxeFile` (7.3-1) more robust, due to the fact that the GAP function `PermList` (Reference: PermList) now returns `fail` instead of raising an error,

- to change the way how representations with prescribed properties are accessed (the new function `OneAtlasGeneratingSetInfo` (3.5-5) is now preferred to the former `OneAtlasGeneratingSet`, and `AllAtlasGeneratingSetInfos` (3.5-6) has been added in order to provide programmatic access in parallel to the human readable descriptions printed by `DisplayAtlasInfo` (3.5-1)),

- and last but not least to include the current table of contents of the underlying database.

For AtlasRep users, the new feature of GAP 4.4 is particularly interesting that due to better kernel support, reading large matrices over finite fields is now faster than it was in GAP 4.3.

##### 1.3-6 What's New in Version 1.1?

The biggest change w.r.t. Version 1.1 is the addition of private extensions (see Chapter 5). It includes a new "free format" for straight line programs (see Section 5.2). Unfortunately, this feature requires the system program `ls`, so it may be not available for example under MS Windows operating systems. [But see Section 1.3-4.]

In order to admit the addition of other types of data, the implementation of several functions has been changed. Data types are described in Section 7.5. An example of a new data type are quaternionic representations (see Section 7.6). The user interface itself (see Chapter 3) remained the same.

As an alternative to `perl`, one can use `wget` now for transferring data files (see 4.3).

Data files can be read much more efficiently in GAP 4.3 than in GAP 4.2. In Version 1.1 of the AtlasRep package, this feature is used for reading matrices and permutations in MeatAxe text format with `ScanMeatAxeFile` (7.3-1). As a consequence, (at least) GAP 4.3 is required for AtlasRep Version 1.1.

The new `compress` component of the global variable `AtlasOfGroupRepresentationsInfo` (7.1-6) allows one to store data files automatically in `gzip`ped form.

For matrix representations in characteristic zero, invariant forms and generators for the centralizer algebra are now accessible in GAP if they are contained in the source files --this information had been ignored in Version 1.0 (see `AtlasOfGroupRepresentationsTestTableOfContentsRemoteUpdates` (4.2-4) for necessary updates).

Additional information is now available via the internet (see 4.4).

The update facilities have been extended (see 4.2).

The manual is now distributed also in pdf and HTML format; on the other hand, the PostScript format manual is no longer contained in the archives.

Apart from these changes, a few minor bugs in the handling of MeatAxe files have been fixed, typos in the documentation have been corrected, and the syntax checks for ATLAS straight line programs (see 7.4) have been improved.

#### 1.4 Acknowledgments

The perl script that had been used for fetching remote data until version 1.2 had been kindly provided by Frank Lübeck and Max Neunhöffer. Thanks also to Greg Gamble and Alexander Hulpke for technical hints concerning "standard" perl.

Ulrich Kaiser helped with preparing the package for MS Windows.

The idea to support private extensions of the package (see Chapter 5) is due to Klaus Lux. He used a preliminary version of AtlasRep Version 1.1, and helped to fix several bugs.

The functions `CMtxBinaryFFMatOrPerm` (7.3-4) and `FFMatOrPermCMtxBinary` (7.3-5) were contributed by Frank Lübeck.

The GAPDoc package [LN11], which is used for processing the manual of the AtlasRep package and for processing the bibliographies included in this package (see `BrowseBibliographySporadicSimple` (3.6-2)), was written by Frank Lübeck and Max Neunhöffer.

The GAP package IO [Neu11], which is recommended for transferring data, was written by Max Neunhöffer.

Max has also suggested the generalization of the data access described in Section 4.3-6, and the admissibility of the function `Character` (Reference: Character (for a character table and a list)) as a filter in `DisplayAtlasInfo` (3.5-1), `OneAtlasGeneratingSetInfo` (3.5-5), and `AllAtlasGeneratingSetInfos` (3.5-6).

Gunter Malle suggested to make the information about representations of minimal degree accessible, see Section 6.3.

Andries Brouwer suggested to add a tutorial, see Chapter 2. Klaus Lux suggested several improvements of this new chapter.

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