7 Technicalities of the **AtlasRep** Package

This chapter describes those parts of the **GAP** interface to the **ATLAS** of Group Representations that do not belong to the user interface (cf. Chapter 3).

Besides global variables used for administrational purposes (see Section 7.1) and several sanity checks (see Section 7.8), they can be regarded as the interface between the data actually contained in the files and the corresponding **GAP** objects (see Section 7.2, 7.3, 7.4, and 7.5), and the interface between the remote and the local version of the database (see Section 7.6 and 7.7). The former interface contains functions to read and write files in **MeatAxe** format, which may be interesting for users familiar with **MeatAxe** standalones (see for example [Rin]). Other low level functions may be undocumented in the sense that they are not described in this manual. Users interested in them may look at the actual implementation in the `gap`

directory of the package, but it may happen that this will be changed in future versions of the package.

For debugging purposes, the functions from the **GAP** interface to the **ATLAS** of Group Representations print information depending on the info level of the info classes `InfoAtlasRep`

(7.1-1), `InfoCMeatAxe`

(7.1-2), and `InfoBBox`

(7.1-3) (cf. Reference: Info Functions).

The info level of an info class can be changed using `SetInfoLevel`

(Reference: SetInfoLevel). For example, the info level of `InfoAtlasRep`

(7.1-1) can be set to the nonnegative integer `n` using `SetInfoLevel( InfoAtlasRep, `

.`n` )

Information about files being read can be obtained by setting the value of the global variable `InfoRead1`

to `Print`

(Reference: Print).

`‣ InfoAtlasRep` | ( info class ) |

If the info level of `InfoAtlasRep`

is at least 1 then information about `fail`

results of functions in the **AtlasRep** package is printed. If the info level is at least 2 then information about calls to external programs is printed. The default level is 0, no information is printed on this level.

`‣ InfoCMeatAxe` | ( info class ) |

If the info level of `InfoCMeatAxe`

is at least 1 then information about `fail`

results of `C`

-**MeatAxe** functions is printed. The default level is zero, no information is printed on this level.

`‣ InfoBBox` | ( info class ) |

If the info level of `InfoBBox`

is at least 1 then information about `fail`

results of functions dealing with black box programs (see Section 6.2) is printed. The default level is 0, no information is printed on this level.

`‣ CMeatAxe.FastRead` | ( global variable ) |

If this component is bound and has the value `true`

then `ScanMeatAxeFile`

(7.3-1) reads text files via `StringFile`

(GAPDoc: StringFile). Otherwise each file containing a matrix over a finite field is read line by line via `ReadLine`

(Reference: ReadLine), and the **GAP** matrix is constructed line by line, in a compressed representation (see Reference: Row Vectors over Finite Fields and Reference: Matrices over Finite Fields), which makes it possible to read large matrices in a reasonable amount of space. The `StringFile`

(GAPDoc: StringFile) approach is faster but needs more intermediate space when text files containing matrices over finite fields are read.

`‣ AGR` | ( global variable ) |

is a record whose components are functions and data that are used by the higher level interface functions.

`‣ AtlasOfGroupRepresentationsInfo` | ( global variable ) |

This is a record that is defined in the file `gap/types.g`

of the package, with the following components.

Components corresponding to *user parameters* (see Section 4.3) are

`remote`

a boolean that controls what files are available; if the value is

`true`

then**GAP**is allowed to try remotely accessing any**ATLAS**file from the servers (see below) and thus all files listed in the global table of contents are available, if the value is`false`

then**GAP**may access only those files that are stored in the database directories of the local**GAP**installation (see Section 4.3-1),`servers`

a list of pairs

`[`

`server``,`

`path``]`

, where`server`is a string denoting the`http`

address of a server where files can be fetched that are not stored in the local database, and`path`is a string describing the path where the data directories on the server reside,`wget`

controls whether the

**GAP**package**IO**[Neu14] or the external program`wget`

is used to fetch data files, see 4.3-3,`compress`

a boolean that controls whether

**MeatAxe**format text files are stored in compressed form; if the value is`true`

then these files are compressed with`gzip`

after they have been fetched from a server, see Section 4.3-4,`displayFunction`

the function that is used by

`DisplayAtlasInfo`

(3.5-1) for printing the formatted data, see Section 4.3-5,`accessFunctions`

a list of records, each describing how to access the data files, see Sections 4.3-6 and 7.2, and

`markprivate`

a string used in

`DisplayAtlasInfo`

(3.5-1) to mark private data, see Section 5.2.

*System components* (which are computed automatically) are

`GAPnames`

a list of pairs, each containing the

**GAP**name and the**ATLAS**-file name of a group, see Section 3.2,`groupnames`

a list of triples, each containing at the first position the name of the directory on each server that contains data about the group G in question, at the second position the name of the (usually simple) group for which a subdirectory exists that contains the data about G, and at the third position the

**ATLAS**-file name used for G, see Section 7.6,`private`

a list of pairs of strings used for administrating private data (see Chapter 5); the value is changed by

`AtlasOfGroupRepresentationsNotifyPrivateDirectory`

(5.1-1) and`AtlasOfGroupRepresentationsForgetPrivateDirectory`

(5.1-2),`characterinfo`

,`permrepinfo`

,`ringinfo`

additional information about representations, concerning the characters afforded, the point stabilizers of permutation representations, and the ring of definition of matrix representations; this information is used by

`DisplayAtlasInfo`

(3.5-1),`TableOfContents`

a record with at most the components

`local`

,`remote`

,`types`

, and the names of private data directories. The values of the components`local`

and`remote`

can be computed automatically by`ReloadAtlasTableOfContents`

(4.2-1), the value of the component`types`

is set in`AGR.DeclareDataType`

(7.5-1), and the values of the components for local data directories are created by`AtlasOfGroupRepresentationsNotifyPrivateDirectory`

(5.1-1).

We discuss the three steps listed in Section 4.3-6.

For creating an overview of the locally available data, the first of these steps must be available independent of actually accessing the file in question. For updating the local copy of the server data, the second of the above steps must be available independent of the third one. Therefore, the package provides the possibility to extend the default behaviour by adding new records to the `accessFunctions`

component of `AtlasOfGroupRepresentationsInfo`

(7.1-6). Its components are as follows.

`location(`

`filename`,`groupname`,`dirname`,`type`)Let

`filename`be the default filename (without path) of the required file, or a list of such filenames. Let`groupname`be the**ATLAS**name of the group to which the data in these files belong,`dirname`be the default directory name (one of`"datagens"`

,`"dataword"`

, or the`dirid`value of a private directory, see`AtlasOfGroupRepresentationsNotifyPrivateDirectory`

(5.1-1)), and`type`be the data type (see`AGR.DeclareDataType`

(7.5-1)). This function must return either the absolute path(s) where the mechanism implemented by the current record expects the local version of the given file(s), or`fail`

if this function does not feel responsible for these file(s). In the latter case, the`location`

function in another record will know a path.The file(s) is/are regarded as not locally available if all installed

`location`

functions return either`fail`

or paths of nonexisting files, in the sense of`IsExistingFile`

(Reference: IsExistingFile).`fetch(`

`filepath`,`filename`,`groupname`,`dirname`,`type`)This function is called when a file is not locally available and if the

`location`

function in the current record has returned a path or a list of paths. The arguments`dirname`and`type`must be the same as for the`location`

function, and`filepath`and`filename`must be strings (*not*lists of strings).The return value must be

`true`

if the function succeeded with making the file locally available (including postprocessing if applicable), and`false`

otherwise.`contents(`

`filepath`,`type`)This function is called when the

`location`

function in the current record has returned the path(s)`filepath`, and if either these are paths of existing files or the`fetch`

function in the current record has been called for these paths, and the return value was`true`

. The argument`type`must be the same as for the`location`

and the`fetch`

functions.The return value must be the contents of the file(s), in the sense that the

**GAP**matrix, matrix list, permutation, permutation list, or program described by the file(s) is returned. This means that besides reading the file(s) via the appropriate function, interpreting the contents may be necessary.`description`

This must be a short string that describes for which kinds of files the functions in the current record are intended, which file formats are supported etc. The value is used by

`AtlasOfGroupRepresentationsUserParameters`

(4.3-8).`active`

The current

`accessFunctions`

record is ignored by`AGR.FileContents`

(7.6-2) if the value is not`true`

.

In `AGR.FileContents`

(7.6-2), the records in the `accessFunctions`

component of `AtlasOfGroupRepresentationsInfo`

(7.1-6) are considered in reversed order.

By default, the `accessFunctions`

list contains three records. Only for one of them, the `active`

component has the value `true`

. One of the other two records can be used to change the access to permutation representations and to matrix representations over finite fields such that **MeatAxe** binary files are transferred and read instead of **MeatAxe** text files. The fourth record makes sense only if a local server is accessible, i. e., if the server files can be read directly, without being transferred into the data directories of the package.

`‣ ScanMeatAxeFile` ( filename[, q][, "string"] ) | ( function ) |

Returns: the matrix or list of permutations stored in the file or encoded by the string.

Let `filename` be the name of a **GAP** readable file (see Reference: Filename) that contains a matrix or a permutation or a list of permutations in **MeatAxe** text format (see the section about the program `zcv`

in the `C`

-**MeatAxe** documentation [Rin]), and let `q` be a prime power. `ScanMeatAxeFile`

returns the corresponding **GAP** matrix or list of permutations, respectively.

If the file contains a matrix then the way how it is read by `ScanMeatAxeFile`

depends on the value of the global variable `CMeatAxe.FastRead`

(7.1-4).

If the parameter `q` is given then the result matrix is represented over the field with `q` elements, the default for `q` is the field size stored in the file.

If the file contains a list of permutations then it is read with `StringFile`

(GAPDoc: StringFile); the parameter `q`, if given, is ignored in this case.

If the string `"string"`

is entered as the third argument then the first argument must be a string as obtained by reading a file in **MeatAxe** text format as a text stream (see `InputTextFile`

(Reference: InputTextFile)). Also in this case, `ScanMeatAxeFile`

returns the corresponding **GAP** matrix or list of permutations, respectively.

`‣ MeatAxeString` ( mat, q ) | ( operation ) |

`‣ MeatAxeString` ( perms, degree ) | ( operation ) |

`‣ MeatAxeString` ( perm, q, dims ) | ( operation ) |

Returns: a string encoding the **GAP** objects given as input in **MeatAxe** format.

In the first form, for a matrix `mat` whose entries lie in the finite field with `q` elements, `MeatAxeString`

returns a string that encodes `mat` as a matrix over `GF(`

, in `q`)**MeatAxe** text format.

In the second form, for a nonempty list `perms` of permutations that move only points up to the positive integer `degree`, `MeatAxeString`

returns a string that encodes `perms` as permutations of degree `degree`, in `C`

-**MeatAxe** text format (see [Rin]).

In the third form, for a permutation `perm` with largest moved point n, say, a prime power `q`, and a list `dims` of length two containing two positive integers larger than or equal to n, `MeatAxeString`

returns a string that encodes `perm` as a matrix over `GF(`

, of dimensions `q`)`dims`, whose first n rows and columns describe the permutation matrix corresponding to `perm`, and the remaining rows and columns are zero.

When strings are printed to files using `PrintTo`

(Reference: PrintTo) or `AppendTo`

(Reference: AppendTo) then line breaks are inserted whenever lines exceed the number of characters given by the second entry of the list returned by `SizeScreen`

(Reference: SizeScreen), see Reference: Operations for Output Streams. This behaviour is not desirable for creating data files. So the recommended functions for printing the result of `MeatAxeString`

to a file are `FileString`

(GAPDoc: FileString) and `WriteAll`

(Reference: WriteAll).

gap> mat:= [ [ 1, -1 ], [ 0, 1 ] ] * Z(3)^0;; gap> str:= MeatAxeString( mat, 3 ); "1 3 2 2\n12\n01\n" gap> mat = ScanMeatAxeFile( str, "string" ); true gap> str:= MeatAxeString( mat, 9 ); "1 9 2 2\n12\n01\n" gap> mat = ScanMeatAxeFile( str, "string" ); true gap> perms:= [ (1,2,3)(5,6) ];; gap> str:= MeatAxeString( perms, 6 ); "12 1 6 1\n2\n3\n1\n4\n6\n5\n" gap> perms = ScanMeatAxeFile( str, "string" ); true gap> str:= MeatAxeString( perms, 8 ); "12 1 8 1\n2\n3\n1\n4\n6\n5\n7\n8\n" gap> perms = ScanMeatAxeFile( str, "string" ); true

Note that the output of `MeatAxeString`

in the case of permutation matrices depends on the user preference `WriteMeatAxeFilesOfMode2`

.

gap> perm:= (1,2,4);; gap> str:= MeatAxeString( perm, 3, [ 5, 6 ] ); "2 3 5 6\n2\n4\n3\n1\n5\n" gap> mat:= ScanMeatAxeFile( str, "string" );; Print( mat, "\n" ); [ [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ] ] gap> pref:= UserPreference( "AtlasRep", "WriteMeatAxeFilesOfMode2" );; gap> SetUserPreference( "AtlasRep", "WriteMeatAxeFilesOfMode2", true ); gap> MeatAxeString( mat, 3 ) = str; true gap> SetUserPreference( "AtlasRep", "WriteMeatAxeFilesOfMode2", false ); gap> MeatAxeString( mat, 3 ); "1 3 5 6\n010000\n000100\n001000\n100000\n000010\n" gap> SetUserPreference( "AtlasRep", "WriteMeatAxeFilesOfMode2", pref );

`‣ FFList` ( F ) | ( function ) |

Returns: a list of elements in the given finite field.

`‣ FFLists` | ( global variable ) |

`FFList`

is a utility program for the conversion of vectors and matrices from **MeatAxe** format to **GAP** format and vice versa. It is used by `ScanMeatAxeFile`

(7.3-1) and `MeatAxeString`

(7.3-2).

For a finite field `F`, `FFList`

returns a list l giving the correspondence between the **MeatAxe** numbering and the **GAP** numbering of the elements in `F`.

The element of `F` corresponding to **MeatAxe** number n is l[ n+1 ], and the **MeatAxe** number of the field element z is `Position( `

l, z` ) - 1`

.

The global variable `FFLists`

is used to store the information about `F` once it has been computed.

gap> FFList( GF(4) ); [ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2 ] gap> IsBound( FFLists[4] ); true

`‣ CMtxBinaryFFMatOrPerm` ( elm, def, outfile[, base] ) | ( function ) |

Let the pair (`elm`, `def`) be either of the form (M, q) where M is a matrix over a finite field F, say, with q ≤ 256 elements, or of the form (π, n) where π is a permutation with largest moved point at most n. Let `outfile` be a string. `CMtxBinaryFFMatOrPerm`

writes the `C`

-**MeatAxe** binary format of M, viewed as a matrix over F, or of π, viewed as a permutation on the points up to n, to the file with name `outfile`.

In the case of a permutation π, the optional argument `base` prescribes whether the binary file contains the points from 0 to `deg`- 1 (`base`= 0, supported by version 2.4 of the `C`

-**MeatAxe**) or the points from 1 to `deg` (`base`= 1, supported by older versions of the `C`

-**MeatAxe**). The default for `base` is given by the value of the user preference `BaseOfMeatAxePermutation`

, see Section 4.3-11.

(The binary format is described in the `C`

-**MeatAxe** manual [Rin].)

gap> tmpdir:= DirectoryTemporary();; gap> mat:= Filename( tmpdir, "mat" );; gap> q:= 4;; gap> mats:= GeneratorsOfGroup( GL(10,q) );; gap> CMtxBinaryFFMatOrPerm( mats[1], q, Concatenation( mat, "1" ) ); gap> CMtxBinaryFFMatOrPerm( mats[2], q, Concatenation( mat, "2" ) ); gap> prm:= Filename( tmpdir, "prm" );; gap> n:= 200;; gap> perms:= GeneratorsOfGroup( SymmetricGroup( n ) );; gap> CMtxBinaryFFMatOrPerm( perms[1], n, Concatenation( prm, "1" ) ); gap> CMtxBinaryFFMatOrPerm( perms[2], n, Concatenation( prm, "2" ) ); gap> CMtxBinaryFFMatOrPerm( perms[1], n, Concatenation( prm, "1a" ), 0 ); gap> CMtxBinaryFFMatOrPerm( perms[2], n, Concatenation( prm, "2b" ), 1 );

`‣ FFMatOrPermCMtxBinary` ( fname ) | ( function ) |

Returns: the matrix or permutation stored in the file.

Let `fname` be the name of a file that contains the `C`

-**MeatAxe** binary format of a matrix over a finite field or of a permutation, as is described in [Rin]. `FFMatOrPermCMtxBinary`

returns the corresponding **GAP** matrix or permutation.

gap> FFMatOrPermCMtxBinary( Concatenation( mat, "1" ) ) = mats[1]; true gap> FFMatOrPermCMtxBinary( Concatenation( mat, "2" ) ) = mats[2]; true gap> FFMatOrPermCMtxBinary( Concatenation( prm, "1" ) ) = perms[1]; true gap> FFMatOrPermCMtxBinary( Concatenation( prm, "2" ) ) = perms[2]; true gap> FFMatOrPermCMtxBinary( Concatenation( prm, "1a" ) ) = perms[1]; true gap> FFMatOrPermCMtxBinary( Concatenation( prm, "2b" ) ) = perms[2]; true

`‣ ScanStraightLineProgram` ( filename[, "string"] ) | ( function ) |

Returns: a record containing the straight line program.

Let `filename` be the name of a file that contains a straight line program in the sense that it consists only of lines in the following form.

`#`

anythinglines starting with a hash sign

`#`

are ignored,`echo`

anythinglines starting with

`echo`

are ignored for the`program`

component of the result record (see below), they are used to set up the bijection between the labels used in the program and conjugacy class names in the case that the program computes dedicated class representatives,`inp`

nmeans that there are n inputs, referred to via the labels

`1`

,`2`

, ..., n,`inp`

k a1 a2 ... akmeans that the next k inputs are referred to via the labels a1, a2, ..., ak,

`cjr`

a bmeans that a is replaced by b

`^(-1) *`

a`*`

b,`cj`

a b cmeans that c is defined as b

`^(-1) *`

a`*`

b,`com`

a b cmeans that c is defined as a

`^(-1) *`

b^(-1)`*`

a`*`

b,`iv`

a bmeans that b is defined as a

`^(-1)`

,`mu`

a b cmeans that c is defined as a

`*`

b,`pwr`

a b cmeans that c is defined as b

`^`

a,`cp`

a bmeans that b is defined as a copy of a,

`oup`

lmeans that there are l outputs, stored in the labels

`1`

,`2`

, ..., l, and`oup`

l b1 b2 ... blmeans that the next l outputs are stored in the labels b1, b2, ... bl.

Each of the labels a, b, c can be any nonempty sequence of digits and alphabet characters, except that the first argument of `pwr`

must denote an integer.

If the `inp`

or `oup`

statements are missing then the input or output, respectively, is assumed to be given by the labels `1`

and `2`

. There can be multiple `inp`

lines at the beginning of the program and multiple `oup`

lines at the end of the program. Only the first `inp`

or `oup`

line may omit the names of the elements. For example, an empty file `filename` or an empty string `string` represent a straight line program with two inputs that are returned as outputs.

No command except `cjr`

may overwrite its own input. For example, the line `mu a b a`

is not legal. (This is not checked.)

`ScanStraightLineProgram`

returns a record containing as the value of its component `program`

the corresponding **GAP** straight line program (see `IsStraightLineProgram`

(Reference: IsStraightLineProgram)) if the input string satisfies the syntax rules stated above, and returns `fail`

otherwise. In the latter case, information about the first corrupted line of the program is printed if the info level of `InfoCMeatAxe`

(7.1-2) is at least 1.

If the string `"string"`

is entered as the second argument then the first argument must be a string as obtained by reading a file in **MeatAxe** text format as a text stream (see `InputTextFile`

(Reference: InputTextFile)). Also in this case, `ScanStraightLineProgram`

returns either a record with the corresponding **GAP** straight line program or `fail`

.

If the input describes a straight line program that computes certain class representatives of the group in question then the result record also contains the component `outputs`

. Its value is a list of strings, the entry at position i denoting the name of the class in which the i output of the straight line program lies; see Section 3.4 for the definition of the class names that occur.

Such straight line programs must end with a sequence of output specifications of the following form.

echo "Classes 1A 2A 3A 5A 5B" oup 5 3 1 2 4 5

This example means that the list of outputs of the program contains elements of the classes `1A`

, `2A`

, `3A`

, `5A`

, and `5B`

(in this order), and that inside the program, these elements are referred to by the five names `3`

, `1`

, `2`

, `4`

, and `5`

.

`‣ AtlasStringOfProgram` ( prog[, outputnames] ) | ( function ) |

`‣ AtlasStringOfProgram` ( prog[, "mtx"] ) | ( function ) |

Returns: a string encoding the straight line program/decision in the format used in **ATLAS** files.

For a straight line program or straight line decision `prog` (see `IsStraightLineProgram`

(Reference: IsStraightLineProgram) and `IsStraightLineDecision`

(6.1-1)), this function returns a string describing the input format of an equivalent straight line program or straight line decision as used in the **ATLAS** of Group Representations, that is, the lines are of the form described in `ScanStraightLineProgram`

(7.4-1).

A list of strings that is given as the optional second argument `outputnames` is interpreted as the class names corresponding to the outputs; this argument has the effect that appropriate `echo`

statements appear in the result string.

If the string `"mtx"`

is given as the second argument then the result has the format used in the `C`

-**MeatAxe** (see [Rin]) rather than the format described in Section 7.4. (Note that the `C`

-**MeatAxe** format does not make sense if the argument `outputnames` is given, and that this format does not support `inp`

and `oup`

statements.)

The argument `prog` must not be a black box program (see `IsBBoxProgram`

(6.2-1)).

gap> str:= "inp 2\nmu 1 2 3\nmu 3 1 2\niv 2 1\noup 2 1 2";; gap> prg:= ScanStraightLineProgram( str, "string" ); rec( program := <straight line program> ) gap> prg:= prg.program;; gap> Display( prg ); # input: r:= [ g1, g2 ]; # program: r[3]:= r[1]*r[2]; r[2]:= r[3]*r[1]; r[1]:= r[2]^-1; # return values: [ r[1], r[2] ] gap> StringOfResultOfStraightLineProgram( prg, [ "a", "b" ] ); "[ (aba)^-1, aba ]" gap> AtlasStringOfProgram( prg ); "inp 2\nmu 1 2 3\nmu 3 1 2\niv 2 1\noup 2\n" gap> prg:= StraightLineProgram( "(a^2b^3)^-1", [ "a", "b" ] ); <straight line program> gap> Print( AtlasStringOfProgram( prg ) ); inp 2 pwr 2 1 4 pwr 3 2 5 mu 4 5 3 iv 3 4 oup 1 4 gap> prg:= StraightLineProgram( [ [2,3], [ [3,1,1,4], [1,2,3,1] ] ], 2 ); <straight line program> gap> Print( AtlasStringOfProgram( prg ) ); inp 2 pwr 3 2 3 pwr 4 1 5 mu 3 5 4 pwr 2 1 6 mu 6 3 5 oup 2 4 5 gap> Print( AtlasStringOfProgram( prg, "mtx" ) ); # inputs are expected in 1 2 zsm pwr3 2 3 zsm pwr4 1 5 zmu 3 5 4 zsm pwr2 1 6 zmu 6 3 5 echo "outputs are in 4 5" gap> str:= "inp 2\nchor 1 2\nchor 2 3\nmu 1 2 3\nchor 3 5";; gap> prg:= ScanStraightLineDecision( str );; gap> AtlasStringOfProgram( prg.program ); "inp 2\nchor 1 2\nchor 2 3\nmu 1 2 3\nchor 3 5\n"

Each representation or program that is administrated by the **AtlasRep** package belongs to a unique *data type*. Informally, examples of data types are "permutation representation", "matrix representation over the integers", or "straight line program for computing class representatives".

The idea is that for each data type, there can be

a column of its own in the output produced by

`DisplayAtlasInfo`

(3.5-1) when called without arguments or with only argument a list of group names,a line format of its own for the output produced by

`DisplayAtlasInfo`

(3.5-1) when called with first argument a group name,an input format of its own for

`AtlasProgram`

(3.5-3),an input format of its own for

`OneAtlasGeneratingSetInfo`

(3.5-5), andspecific tests for the data of this data type; these functions are used by the global tests described in Section 7.8.

Formally, a data type is defined by a record whose components are used by the interface functions. The details are described in the following.

`‣ AGR.DeclareDataType` ( kind, name, record ) | ( function ) |

Let `kind` be one of the strings `"rep"`

or `"prg"`

, and `record` be a record. `AGR.DeclareDataType`

declares a new data type of representations (if `kind` is `"rep"`

) or of programs (if `kind` is `"prg"`

). For each group used in the **AtlasRep** package, the record that contains the information about the data will have a component `name` whose value is a list containing the data about the new type. Examples of `name` are `"perm"`

, `"matff"`

, and `"classes"`

.

*Mandatory components* of `record` are

`FilenameFormat`

This defines the format of the filenames containing data of the type in question. The value must be a list that can be used as the second argument of

`AGR.ParseFilenameFormat`

(7.6-1), such that only filenames of the type in question match. (It is not checked whether this "detection function" matches exactly one type, so declaring a new type needs care.)`AddFileInfo`

This defines the information stored in the table of contents for the data of the type. The value must be a function that takes three arguments (the current list of data for the type and the given group, a list returned by

`AGR.ParseFilenameFormat`

(7.6-1) for the given type, and a filename). This function adds the necessary parts of the data entry to the list, and returns`true`

if the data belongs to the type, otherwise`false`

is returned; note that the latter case occurs if the filename matches the format description but additional conditions on the parts of the name are not satisfied (for example integer parts may be required to be positive or prime powers).`ReadAndInterpretDefault`

This is the function that does the work for the default

`contents`

value of the`accessFunctions`

component of`AtlasOfGroupRepresentationsInfo`

(7.1-6), see Section 7.2. This function must take a path and return the**GAP**object given by this file.`AddDescribingComponents`

(for`rep`

only)This function takes two arguments, a record (that will be returned by

`AtlasGenerators`

(3.5-2),`OneAtlasGeneratingSetInfo`

(3.5-5), or`AllAtlasGeneratingSetInfos`

(3.5-6)) and the type record`record`. It sets the components`p`

,`dim`

,`id`

, and`ring`

that are promised for return values of the abovementioned three functions.`DisplayGroup`

(for`rep`

only)This defines the format of the lines printed by

`DisplayAtlasInfo`

(3.5-1) for a given group. The value must be a function that takes a list as returned by the function given in the component`AddFileInfo`

, and returns the string to be printed for the representation in question.

*Optional components* of `record` are

`DisplayOverviewInfo`

This is used to introduce a new column in the output of

`DisplayAtlasInfo`

(3.5-1) when this is called without arguments or with a list of group names as its only argument. The value must be a list of length three, containing at its first position a string used as the header of the column, at its second position one of the strings`"r"`

or`"l"`

, denoting right or left aligned column entries, and at its third position a function that takes two arguments (a list of tables of contents of the**AtlasRep**package and a group name), and returns a list of length two, containing the string to be printed as the column value and`true`

or`false`

, depending on whether private data is involved or not. (The default is`fail`

, indicating that no new column shall be printed.)`DisplayPRG`

(for`prg`

only)This is used in

`DisplayAtlasInfo`

(3.5-1) for**ATLAS**programs. The value must be a function that takes four arguments (a list of tables of contents to examine, the name of the given group, a list of integers or`true`

for the required standardization, and a list of all available standardizations), and returns the list of lines (strings) to be printed as the information about the available programs of the current type and for the given group. (The default is to return an empty list.)`AccessGroupCondition`

(for`rep`

only)This is used in

`DisplayAtlasInfo`

(3.5-1) and`OneAtlasGeneratingSetInfo`

(3.5-5). The value must be a function that takes two arguments (a list as returned by`OneAtlasGeneratingSetInfo`

(3.5-5), and a list of conditions), and returns`true`

or`false`

, depending on whether the first argument satisfies the conditions. (The default value is`ReturnFalse`

(Reference: ReturnFalse).)The function must support conditions such as

`[ IsPermGroup, true ]`

and`[ NrMovedPoints, [ 5, 6 ] ]`

, in general a list of functions followed by a prescribed value, a list of prescribed values, another (unary) function, or the string`"minimal"`

. For an overview of the interesting functions, see`DisplayAtlasInfo`

(3.5-1).`AccessPRG`

(for`prg`

only)This is used in

`AtlasProgram`

(3.5-3). The value must be a function that takes three arguments (the record with the information about the given group in the current table of contents, an integer or a list of integers or`true`

for the required standardization, and a list of conditions given by the optional arguments of`AtlasProgram`

(3.5-3)), and returns either`fail`

or a list that together with the group name forms the identifier of a program that matches the conditions. (The default value is`ReturnFail`

(Reference: ReturnFail).)`AtlasProgram`

(for`prg`

only)This is used in

`AtlasProgram`

(3.5-3) to create the result value from the identifier. (The default value is`AtlasProgramDefault`

, which works whenever the second entry of the identifier is the filename; this is not the case for example if the program is the composition of several programs.)`AtlasProgramInfo`

(for`prg`

only)This is used in

`AtlasProgramInfo`

(3.5-4) to create the result value from the identifier. (The default value is`AtlasProgramDefault`

.)`TOCEntryString`

This is used in

`StoreAtlasTableOfContents`

(4.2-2). The value must be a function that takes two arguments (the name`name`of the type and a list as returned by`AGR.ParseFilenameFormat`

(7.6-1) and returns a string that describes the appropriate function call. (The default value is`TOCEntryStringDefault`

.)`PostprocessFileInfo`

This is used in the construction of a table of contents via

`ReloadAtlasTableOfContents`

(4.2-1), for testing or rearranging the data of the current table of contents. The value must be a function that takes two arguments, the table of contents record and the record in it that belongs to one fixed group. (The default function does nothing.)`SortTOCEntries`

This is used in the construction of a table of contents (see

`ReloadAtlasTableOfContents`

(4.2-1)), for sorting the entries after they have been added and after the value of the component`PostprocessFileInfo`

has been called. The value must be a function that takes a list as returned by`AGR.ParseFilenameFormat`

(7.6-1), and returns the sorting key. (There is no default value, which means that no sorting is needed.)`TestFileHeaders`

(for`rep`

only)This is used in the function

`AGR.Test.FileHeaders`

. The value must be a function that takes the same four arguments as`AGR.FileContents`

(7.6-2), except that the first argument`"datagens"`

can be replaced by`"local"`

and that the third argument is a list as returned by`AGR.ParseFilenameFormat`

(7.6-1). (The default value is`ReturnTrue`

(Reference: ReturnTrue).)`TestFiles`

(for`rep`

only)This is used in the function

`AGR.Test.Files`

. The format of the value and the default are the same as for the value of the component`TestFileHeaders`

.`TestWords`

(for`prg`

only)This is used in the function

`AGR.Test.Words`

. The value must be a function that takes five arguments where the first four are the same arguments as for`AGR.FileContents`

(7.6-2), except that the first argument`"dataword"`

can be replaced by`"local"`

, and the fifth argument is`true`

or`false`

, indicating verbose mode or not.

The data of each local **GAP** version of the **ATLAS** of Group Representations are either private (see Chapter 5) or are stored in the two directories `datagens`

and `dataword`

. In the following, we describe the format of filenames in the latter two directories, as a reference of the "official" part of the **ATLAS**.

In the directory `datagens`

, the generators for the *representations* available are stored, the directory `dataword`

contains the *programs* to compute conjugacy class representatives, generators of maximal subgroups, images of generators under automorphisms of a given group G from standard generators of G, and to check and compute standard generators (see Section 3.3).

The name of each data file in the **ATLAS** of Group Representations describes the contents of the file. This section lists the definitions of the filenames used.

Each filename consists of two parts, separated by a minus sign `-`

. The first part is always of the form `groupname``G`

`i`, where the integer `i` denotes the `i`-th set of standard generators for the group G, say, with **ATLAS**-file name `groupname` (see 3.2). The translations of the name `groupname` to the name(s) used within **GAP** is given by the component `GAPnames`

of `AtlasOfGroupRepresentationsInfo`

(7.1-6).

The filenames in the directory `dataword`

have one of the following forms. In each of these cases, the suffix `W`

`n` means that `n` is the version number of the program.

`groupname`G`i`-cycW`n`In this case, the file contains a straight line program that returns a list of representatives of generators of maximally cyclic subgroups of G. An example is

`Co1G1-cycW1`

.`groupname`G`i`-cclsW`n`In this case, the file contains a straight line program that returns a list of conjugacy class representatives of G. An example is

`RuG1-cclsW1`

.`groupname`G`i`cycW`n`-cclsW`m`In this case, the file contains a straight line program that takes the return value of the program in the file

`groupname``G`

`i``-cycW`

`n`(see above), and returns a list of conjugacy class representatives of G. An example is`M11G1cycW1-cclsW1`

.`groupname`G`i`-max`k`W`n`In this case, the file contains a straight line program that takes generators of G w.r.t. the

`i`-th set of standard generators, and returns a list of generators (in general*not*standard generators) for a subgroup U in the`k`-th class of maximal subgroups of G. An example is`J1G1-max7W1`

.`groupname`G`i`max`k`W`n`-`subgroupname`G`j`W`m`In this case, the file contains a straight line program that takes the return value of the program in the file

`groupname``G`

`i``-max`

`k``W`

`n`(see above), which are generators for a group U, say;`subgroupname`is a name for U, and the return value is a list of standard generators for U, w.r.t. the`j`-th set of standard generators. (Of course this implies that the groups in the`k`-th class of maximal subgroups of G are isomorphic to the group with name`subgroupname`.) An example is`J1G1max1W1-L211G1W1`

; the first class of maximal subgroups of the Janko group J_1 consists of groups isomorphic to the linear group L_2(11), for which standard generators are defined.`groupname`G`i`-a`outname`W`n`In this case, the file contains a straight line program that takes generators of G w.r.t. the

`i`-th set of standard generators, and returns the list of their images under the outer automorphism α of G given by the name`outname`; if this name is empty then α is the unique nontrivial outer automorphism of G; if it is a positive integer k then α is a generator of the unique cyclic order k subgroup of the outer automorphism group of G; if it is of the form`2_1`

or`2a`

,`4_2`

or`4b`

,`3_3`

or`3c`

... then α generates the cyclic group of automorphisms induced on G by G.2_1, G.4_2, G.3_3 ...; finally, if it is of the form`k``p`

`d`, with`k`one of the above forms and`d`an integer then`d`denotes the number of dashes appended to the automorphism described by`k`; if`d`= 1 then`d`can be omitted. Examples are`A5G1-aW1`

,`L34G1-a2_1W1`

,`U43G1-a2_3pW1`

, and`O8p3G1-a2_2p5W1`

; these file names describe the outer order 2 automorphism of A_5 (induced by the action of S_5) and the order 2 automorphisms of L_3(4), U_4(3), and O_8^+(3) induced by the actions of L_3(4).2_1, U_4(3).2_2^', and O_8^+(3).2_2^{'''''}, respectively.`groupname`G`i`-G`j`W`n`In this case, the file contains a straight line program that takes generators of G w.r.t. the

`i`-th set of standard generators, and returns standard generators of G w.r.t. the`j`-th set of standard generators. An example is`L35G1-G2W1`

.`groupname`G`i`-check`n`In this case, the file contains a straight line decision that takes generators of G, and returns

`true`

if these generators are standard generators w.r.t. the`i`-th standardization, and`false`

otherwise.`groupname`G`i`-P`n`In this case, the file contains a straight line decision that takes some group elements, and returns

`true`

if these elements are standard generators for`G`, w.r.t. the`i`-th standardization, and`false`

otherwise.`groupname`G`i`-find`n`In this case, the file contains a black box program that takes a group, and returns (if it is successful) a set of standard generators for

`G`, w.r.t. the`i`-th standardization.`groupname`G`i`-X`descr`W`n`In this case, the file contains a straight line program that takes generators of G w.r.t. the

`i`-th set of standard generators, and whose return value corresponds to`descr`. This format is used only in private extensions (see Chapter 5), such a script can be accessed with`descr`as the third argument of`AtlasProgram`

(3.5-3).

The filenames in the directory `datagens`

have one of the following forms. In each of these cases, `id` is a (possibly empty) string that starts with a lowercase alphabet letter (see `IsLowerAlphaChar`

(Reference: IsLowerAlphaChar)), and `m` is a nonnegative integer, meaning that the generators are written w.r.t. the `m`-th basis (the meaning is defined by the **ATLAS** developers).

`groupname`G`i`-f`q`r`dim``id`B`m`.m`nr`a file in

**MeatAxe**text file format containing the`nr`-th generator of a matrix representation over the field with`q`elements, of dimension`dim`. An example is`S5G1-f2r4aB0.m1`

.`groupname`G`i`-p`n``id`B`m`.m`nr`a file in

**MeatAxe**text file format containing the`nr`-th generator of a permutation representation on`n`points. An example is`M11G1-p11B0.m1`

.`groupname`G`i`-Ar`dim``id`B`m`.ga

**GAP**readable file containing all generators of a matrix representation of dimension`dim`over an algebraic number field not specified further. An example is`A5G1-Ar3aB0.g`

.`groupname`G`i`-Zr`dim``id`B`m`.ga

**GAP**readable file containing all generators of a matrix representation over the integers, of dimension`dim`. An example is`A5G1-Zr4B0.g`

.`groupname`G`i`-Hr`dim``id`B`m`.ga

**GAP**readable file containing all generators of a matrix representation over a quaternion algebra over an algebraic number field, of dimension`dim`. An example is`2A6G1-Hr2aB0.g`

.`groupname`G`i`-Z`n`r`dim``id`B`m`.ga

**GAP**readable file containing all generators of a matrix representation of dimension`dim`over the ring of integers mod`n`. An example is`2A8G1-Z4r4aB0.g`

.

`‣ AGR.ParseFilenameFormat` ( string, format ) | ( function ) |

Returns: a list of strings and integers if `string` matches `format`, and `fail`

otherwise.

Let `string` be a filename, and `format` be a list [ [ c_1, c_2, ..., c_n ], [ f_1, f_2, ..., f_n ] ] such that each entry c_i is a list of strings and of functions that take a character as their argument and return `true`

or `false`

, and such that each entry f_i is a function for parsing a filename, such as the currently undocumented functions `ParseForwards`

and `ParseBackwards`

.

`AGR.ParseFilenameFormat`

returns a list of strings and integers such that the concatenation of their `String`

(Reference: String) values yields `string` if `string` matches `format`, and `fail`

otherwise. Matching is defined as follows. Splitting `string` at each minus character (`-`

) yields m parts s_1, s_2, ..., s_m. The string `string` matches `format` if s_i matches the conditions in c_i, for 1 ≤ i ≤ n, in the sense that applying f_i to s_i and c_i yields a non-`fail`

result.

gap> format:= [ [ [ IsChar, "G", IsDigitChar ], > [ "p", IsDigitChar, AGR.IsLowerAlphaOrDigitChar, > "B", IsDigitChar, ".m", IsDigitChar ] ], > [ ParseBackwards, ParseForwards ] ];; gap> AGR.ParseFilenameFormat( "A6G1-p10B0.m1", format ); [ "A6", "G", 1, "p", 10, "", "B", 0, ".m", 1 ] gap> AGR.ParseFilenameFormat( "A6G1-p15aB0.m1", format ); [ "A6", "G", 1, "p", 15, "a", "B", 0, ".m", 1 ] gap> AGR.ParseFilenameFormat( "A6G1-f2r16B0.m1", format ); fail

`‣ AGR.FileContents` ( dirname, groupname, filename, type ) | ( function ) |

Returns: the **GAP** object obtained from reading and interpreting the file(s) with name(s) `filename`.

Let `dirname` and `groupname` be strings, `filename` be a string or a list of strings, and `type` be a data type (see `AGR.DeclareDataType`

(7.5-1)). `dirname` must be one of `"datagens"`

, `"dataword"`

, or the `dirid` value of a private directory, see `AtlasOfGroupRepresentationsNotifyPrivateDirectory`

(5.1-1). If `groupname` is the **ATLAS**-file name of a group G (see Section 3.2), and if `filename` is either the name of an accessible file in the `dirname` directory of the **ATLAS**, or a list of such filenames, with data concerning G and for the data type `type`

, then `AGR.FileContents`

returns the contents of the corresponding file(s), in the sense that the file(s) (or equivalent ones, see Section 4.3-6) is/are read, and the result is interpreted if necessary; otherwise `fail`

is returned.

Note that if `filename` refers to file(s) already stored in the `dirname` directory then `AGR.FileContents`

does *not* check whether the table of contents of the **ATLAS** of Group Representations actually contains `filename`.

The list of data currently available is stored in several *tables of contents*, one for the local **GAP** data, one for the data on remote servers, and one for each private data directory. These tables of contents are created by `ReloadAtlasTableOfContents`

(4.2-1).

It is assumed that the local data directories contain only files that are also available on servers. Private extensions to the database (cf. Section 4.5 and Chapter 5) cannot be handled by putting the data files into the local directories.

Each table of contents is represented by a record whose components are the **ATLAS**-file names of the groups (see Section 3.2) and `lastupdated`

, a string describing the date of the last update of this table of contents. The value for each group name is a record whose components are the names of those data types (see Section 7.5) for which data are available.

Here are the administrational functions that are used to build the tables of contents. Some of them may be useful also for private extensions of the package (see Chapter 5).

The following functions define group names, available representations, and straight line programs.

`AGR.GNAN(`

`gapname`,`atlasname`)Called with two strings

`gapname`(the**GAP**name of the group) and`atlasname`(the**ATLAS**name of the group),`AGR.GNAN`

stores the information in the list`AtlasOfGroupRepresentationsInfo.GAPnames`

, which defines the name mapping between the**ATLAS**names and**GAP**names of the groups.This function may be used also for private extensions of the database.

An example of a valid call is

`AGR.GNAN("A5.2","S5")`

.`AGR.GRP(`

`dirname`,`simpname`,`groupname`)Called with three strings,

`AGR.GRP`

stores in the`groupname`

component of`AtlasOfGroupRepresentationsInfo`

(7.1-6) in which path on the servers the data about the group with**ATLAS**name`groupname`can be found.This function is

*not*intended for private extensions of the database.An example of a valid call is

`AGR.GRP("alt","A5","S5")`

.`AGR.TOC(`

`typename`,`filename`,`crcfile`)Called with two strings

`typename`and`filename`, and a list`crc`of integers,`AGR.TOC`

notifies an entry to the`TableOfContents.remote`

component of`AtlasOfGroupRepresentationsInfo`

(7.1-6), where`typename`must be the name of the data type to which the entry belongs,`filename`must be the prefix of the data file(s), and`crc`must be the list of`CrcFile`

(Reference: CrcFile) values of the file(s).This function is

*not*intended for private extensions of the database.An example of a valid call is

`AGR.TOC("perm","S5G1-p5B0.m",[-3581724,115937465])`

.

The following functions add data about the groups and their standard generators. The function calls must be executed after the corresponding `AGR.GNAN`

calls.

`AGR.GRS(`

`gapname`,`size`)Called with the string

`gapname`(the**GAP**name of the group) and the integer`size`(the order of the group),`AGR.GRS`

stores this information in`AtlasOfGroupRepresentationsInfo.GAPnames`

.An example of a valid call is

`AGR.GRS("A5.2",120)`

.`AGR.MXN(`

`gapname`,`nrMaxes`)Called with the string

`gapname`(the**GAP**name of the group) and the integer`nrMaxes`(the number of classes of maximal subgroups of the group),`AGR.MXN`

stores the information in`AtlasOfGroupRepresentationsInfo.GAPnames`

.An example of a valid call is

`AGR.MXN("A5.2",4)`

.`AGR.MXO(`

`gapname`,`sizesMaxes`)Called with the string

`gapname`(the**GAP**name of the group) and the list`sizesMaxes`(of subgroup orders of the classes of maximal subgroups of the group, not necessarily dense, in non-increasing order),`AGR.MXO`

stores the information in`AtlasOfGroupRepresentationsInfo.GAPnames`

.An example of a valid call is

`AGR.MXO("A5.2",[60,24,20,12])`

.`AGR.MXS(`

`gapname`,`structureMaxes`)Called with the string

`gapname`(the**GAP**name of the group) and the list`structureMaxes`(of strings describing the structures of the maximal subgroups of the group, not necessarily dense),`AGR.MXS`

stores the information in`AtlasOfGroupRepresentationsInfo.GAPnames`

.An example of a valid call is

`AGR.MXS("A5.2",["A5","S4","5:4","S3x2"])`

.`AGR.KERPRG(`

`gapname`,`kernelProgram`)Called with the string

`gapname`(the**GAP**name of the group) and the list`kernelProgram`(with entries the standardization of the group, the**GAP**name of a factor group, and the list of lines of a straight line program that computes generators of the kernel of the epimorphism from the group to the factor group),`AGR.KERPRG`

stores the information in`AtlasOfGroupRepresentationsInfo.GAPnames`

.An example of a valid call is

`AGR.KERPRG("2.J2",[1,"J2",[[[1,2]]]])`

.`AGR.STDCOMP`

Called with the string

`gapname`(the**GAP**name of the group) and the list`factorCompatibility`(with entries the standardization of the group, the**GAP**name of a factor group, the standardization of this factor group, and`true`

or`false`

, indicating whether mapping the standard generators for`gapname`to those of`factgapname`defines an epimorphism),`AGR.STDCOMP`

stores the information in`AtlasOfGroupRepresentationsInfo.GAPnames`

.An example of a valid call is

`AGR.STDCOMP("2.A5.2",[1,"A5.2",1,true])`

.

The following functions add data about representations or straight line programs that are already known. The function calls must be executed after the corresponding `AGR.TOC`

calls.

`AGR.RNG(`

`repname`,`descr`)Called with two strings

`repname`(denoting the name of a file containing the generators of a matrix representation over a ring that is not determined by the filename) and`descr`(describing this ring R, say),`AGR.RNG`

adds the triple [`repname`,`descr`, R ] to the list stored in the`ringinfo`

component of`AtlasOfGroupRepresentationsInfo`

(7.1-6).An example of a valid call is

`AGR.RNG("A5G1-Ar3aB0","Field([Sqrt(5)])")`

.`AGR.TOCEXT(`

`atlasname`,`std`,`maxnr`,`files`)Called with the string

`atlasname`(the**ATLAS**name of the group), the positive integers`std`(the standardization) and`maxnr`(the number of the class of maximal subgroups), and the list`files`(of filenames of straight line programs for computing generators of the`maxnr`-th maximal subgroup, using a straight line program for a factor group plus perhaps some straight line program for computing kernel generators),`AGR.TOCEXT`

stores the information in the`maxext`

component of the`atlasname`component of the`"remote"`

table of contents.An example of a valid call is

`AGR.TOCEXT("2A5",1,3,["A5G1-max3W1"])`

.`AGR.API(`

`repname`,`info`)Called with the string

`repname`(denoting the name of a permutation representation) and the list`info`(describing the point stabilizer of this representation),`AGR.API`

binds the component`repname`of the record`AtlasOfGroupRepresentationsInfo.permrepinfo`

to`info`.`info`has the following entries.At position 1, the transitivity is stored.

If the transitivity is zero then the second entry is the list of orbit lengths.

If the transitivity is positive then the second entry is the rank of the action.

If the transitivity is positive then the third entry is one of the strings

`"prim"`

,`"imprim"`

, denoting primitivity or not.If the transitivity is positive then the fourth entry is a string describing the structure of the point stabilizer. If the third entry is

`"imprim"`

then this description consists of a subgroup part and a maximal subgroup part, separated by`" < "`

.If the third entry is

`"prim"`

then the fifth entry is either`"???"`

or it denotes the number of the class of maximal subgroups that are the point stabilizers.

An example of a valid call is

`AGR.API("A5G1-p5B0",[3,2,"prim","A4",1])`

.`AGR.CHAR(`

`groupname`,`repname`,`char`,`pos`[,`charname`] )Called with the strings

`groupname`(the**GAP**name of the group) and`repname`(denoting the name of the representation), the integer`char`(the characteristic of the representation), and`pos`(the position or list of positions of the irreducible constituent(s)),`AGR.CHAR`

stores the information in`AtlasOfGroupRepresentationsInfo.characterinfo`

. A string describing the character can be entered as`charname`.An example of a valid call is

`AGR.CHAR("M11","M11G1-p11B0",0,[1,2],"1a+10a")`

.

These functions are used to create the initial table of contents for the server data of the **AtlasRep** package when the file `gap/atlasprm.g`

of the package is read.

The fact that the **ATLAS** of Group Representations is designed as an open database (see Section 4.3-1) makes it especially desirable to have consistency checks available which can be run automatically whenever new data are added by the developers of the **ATLAS**. The tests described in Section 7.8-1 can be used also for data from private extensions of the package (see Chapter 5), Section 7.8-2 lists tests which do not have this property.

All these tests apply only to the *local* table of contents (see Section 7.7) or to private extensions. So only those data files are checked that are actually available in the local **GAP** installation. No files are fetched from servers during these tests. The required space and time for running these tests depend on the amount of locally available data.

The file `tst/testall.g`

of the package contains `Test`

(Reference: Test) statements for executing a collection of such sanity checks; one can run them by calling `ReadPackage( "AtlasRep", "tst/testall.g" )`

. If no problem occurs then **GAP** prints only lines starting with one of the following.

+ Input file: + GAP4stones:

Some of the checks compute and verify additional data, such as information about point stabilizers of permutation representations. In these cases, output lines starting with `#E`

are error messages that point to inconsistencies, whereas output lines starting with `#I`

inform about data that have been computed and were not yet stored, or about stored data that were not verified.

The examples in the package manual form a part of the tests, they are collected in the file `tst/docxpl.tst`

of the package.

The following tests can be used to check the data that belong to a given table of contents. Each of these tests is given by a function with optional argument `tocid`, the identifying string that had been entered as the second argument of `AtlasOfGroupRepresentationsNotifyPrivateDirectory`

(5.1-1). The contents of the local `dataword`

directory can be checked by entering `"local"`

, which is also the default for `tocid`. The function returns `false`

if an error occurs, otherwise `true`

. Currently the following tests of this kind are available.

`AGR.Test.Words( [`

`tocid`] )processes all straight line programs that are stored in the directory with identifier

`tocid`, using the function stored in the`TestWords`

component of the data type in question.`AGR.Test.FileHeaders( [`

`tocid`] )checks whether all

**MeatAxe**text format data files in the directory with identifier`tocid`have a header line that is consistent with the filename, and whether the contents of all**GAP**format data files in this directory is consistent with the contents of the file.`AGR.Test.Files( [`

`tocid`] )checks whether the

**MeatAxe**text files that are stored in the directory with identifier`tocid`can be read with`ScanMeatAxeFile`

(7.3-1) such that the result is not`fail`

. The function does not check whether the first line of a**MeatAxe**text file is consistent with the filename, since this can be tested with`AGR.Test.FileHeaders`

.`AGR.Test.BinaryFormat( [`

`tocid`] )checks whether all

**MeatAxe**text format data files in the directory with identifier`tocid`satisfy that applying first`CMtxBinaryFFMatOrPerm`

(7.3-4) and then`FFMatOrPermCMtxBinary`

(7.3-5) yields the same object.`AGR.Test.Primitivity( [`

`tocid`] )checks the stored primitivity information for the permutation representations that are stored in the directory with identifier

`tocid`.`AGR.Test.Characters( [`

`tocid`] )checks the stored character information for the matrix and permutation representations that are stored in the directory with identifier

`tocid`.

The tests described in this section are not intended for checking data from private extensions of the **AtlasRep** package. Each of the tests is given by a function without arguments that returns `false`

if a contradiction was found during the test, and `true`

otherwise. Additionally, certain messages are printed when contradictions between stored and computed data are found, when stored data cannot be verified computationally, or when the computations yield improvements of the stored data. Currently the following tests of this kind are available.

`AGR.Test.GroupOrders()`

checks whether the group orders stored in the

`GAPnames`

component of`AtlasOfGroupRepresentationsInfo`

(7.1-6) coincide with the group orders computed from an**ATLAS**permutation representation of degree up to`AGR.Test.MaxTestDegree`

, from the character table or the table of marks with the given name, or from the structure of the name. Supported is a splitting of the name at the first dot (`.`

), where the two parts of the name are examined with the same criteria in order to derive the group order.`AGR.Test.MaxesOrders()`

checks whether the orders of maximal subgroups stored in the component

`GAPnames`

of`AtlasOfGroupRepresentationsInfo`

(7.1-6) coincide with the orders computed from the restriction of an**ATLAS**permutation representation of degree up to`AGR.Test.MaxTestDegree`

, from the character table, or the table of marks with the given name, or from the information about maximal subgroups of a factor group modulo a normal subgroup that is contained in the Frattini subgroup.`AGR.Test.MaxesStructure()`

checks whether the names of maximal subgroups stored in the component

`GAPnames`

of`AtlasOfGroupRepresentationsInfo`

(7.1-6) coincide with the names computed from the**GAP**character table with the given name.`AGR.Test.StdCompatibility()`

checks whether the information about the compatibility of standard generators of a group and its factor groups that is stored in the

`GAPnames`

component of`AtlasOfGroupRepresentationsInfo`

(7.1-6) coincides with computed values.The following criterion is used for computing the value for a group G. Use the

**GAP**Character Table Library to determine factor groups F of G for which standard generators are defined and moreover a presentation in terms of these standard generators is known. Evaluate the relators of the presentation in the standard generators of G, and let N be the normal closure of these elements in G. Then mapping the standard generators of F to the N-cosets of the standard generators of G is an epimorphism. If |G/N| = |F| holds then G/N and F are isomorphic, and the standard generators of G and F are compatible in the sense that mapping the standard generators of G to their N-cosets yields standard generators of F.`AGR.Test.CompatibleMaxes()`

checks whether the information about deriving straight line programs for restricting to subgroups from straight line programs that belong to a factor group coincide with computed values.

The following criterion is used for computing the value for a group G. If F is a factor group of G such that the standard generators of G and F are compatible (see the test function

`AGR.Test.StdCompatibility`

) and if there are a presentation for F and a permutation representation of G then it is checked whether the`"maxes"`

type straight line programs for F can be used to compute generators for the maximal subgroups of G; if not then generators of the kernel of the natural epimorphism from G to F, must be added.`AGR.Test.ClassScripts()`

checks whether the straight line programs that compute representatives of certain conjugacy classes are consistent with information stored on the

**GAP**character table of the group in question, in the sense that the given class names really occur in the character table and that the element orders and centralizer orders for the classes are correct.`AGR.Test.CycToCcls()`

checks whether some straight line program that computes representatives of conjugacy classes of a group can be computed from the ordinary

**GAP**character table of that group and a straight line program that computes representatives of cyclic subgroups. In this case the missing scripts are printed if the level of`InfoAtlasRep`

(7.1-1) is at least 1.`AGR.Test.Standardization()`

checks whether all generating sets corresponding to the same set of standard generators have the same element orders; for the case that straight line programs for computing certain class representatives are available, also the orders of these representatives are checked w. r. t. all generating sets.

`AGR.Test.StdTomLib()`

checks whether the standard generators are compatible with those that occur in the

**TomLib**package.`AGR.Test.KernelGenerators()`

checks whether the information stored in the

`GAPnames`

component of`AtlasOfGroupRepresentationsInfo`

(7.1-6) about straight line programs for computing generators of the kernels of natural epimorphisms between**ATLAS**groups coincides with computed values.The following criterion is used for computing the value for a group G. Use the

**GAP**Character Table Library to determine factor groups F of G for which standard generators are defined such that mapping standard generators of G to those of F defines a homomorphism, and such that a presentation of F in terms of its standard generators is known. Evaluating the relators of the presentation in the standard generators of G yields normal subgroup generators for the kernel.A message is printed for each group name for which some straight line program for computing kernel generators was not stored but now was computed, or for which the stored info cannot be verified,

`AGR.Test.MinimalDegrees()`

checks that the (permutation and matrix) representations available in the

**ATLAS**of Group Representations do not have smaller degree than the claimed minimum.

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