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5. Technicalities of the AtlasRep Package
5.1 Global Variables Used by the AtlasRep Package
5.1-1 InfoAtlasRep
5.1-2 InfoCMeatAxe
5.1-3 InfoBBox
5.1-4 CMeatAxe.FastRead
5.1-5 AtlasOfGroupRepresentationsInfo
5.2 How to Customize the Access to Data files
5.3 Reading and Writing MeatAxe Format Files
5.3-1 ScanMeatAxeFile
5.3-2 MeatAxeString
5.3-3 FFList
5.3-4 CMtxBinaryFFMatOrPerm
5.3-5 FFMatOrPermCMtxBinary
5.4 Reading and Writing ATLAS Straight Line Programs
5.4-1 ScanStraightLineProgram
5.4-2 AtlasStringOfProgram
5.5 Data Types Used in the ATLAS of Group Representations
5.5-1 AGRDeclareDataType
5.6 Filenames Used in the ATLAS of Group Representations
5.6-1 AGRParseFilenameFormat
5.6-2 AGRFileContents
5.7 The Tables of Contents of the ATLAS of Group Representations
5.7-1 AGRGNAN
5.7-2 AGRRNG
5.8 Sanity Checks for the ATLAS of Group Representations
5.8-1 AtlasOfGroupRepresentationsTestGroupOrders
5.8-2 AtlasOfGroupRepresentationsTestFileHeaders
5.8-3 AtlasOfGroupRepresentationsTestWords
5.8-4 AtlasOfGroupRepresentationsTestFiles
5.8-5 AtlasOfGroupRepresentationsTestClassScripts

### 5. 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 2.).

Besides global variables used for administrational purposes (see Section 5.1) and several sanity checks (see Section 5.8), they can be regarded as the interface between the data actually contained in the files and the corresponding GAP objects (see Section 5.2, 5.3, 5.4, and 5.5), and the interface between the remote and the local version of the database (see Section 5.6 and 5.7). 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 [Rin98]). 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.

#### 5.1 Global Variables Used by the AtlasRep 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` (5.1-1), `InfoCMeatAxe` (5.1-2), and `InfoBBox` (5.1-3) (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` (5.1-1) 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).

##### 5.1-1 InfoAtlasRep

`> 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.

##### 5.1-2 InfoCMeatAxe

`> 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.

##### 5.1-3 InfoBBox

`> 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 4.2) is printed. The default level is 0, no information is printed on this level.

##### 5.1-4 CMeatAxe.FastRead

`> CMeatAxe.FastRead` | ( global variable )

If this component is bound and has the value `true` then `ScanMeatAxeFile` (5.3-1) 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.

##### 5.1-5 AtlasOfGroupRepresentationsInfo

`> 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 1.7) 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 1.7-1),

`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`

  a boolean that controls whether the GAP package IO[Neu07] or the external program `wget` is used to fetch data files, see 1.7-3,

`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 1.7-4,

`displayFunction`

  the function that is used by `DisplayAtlasInfo` (2.5-1) for printing the formatted data, see Section 1.7-5,

`accessFunctions`

  a list of records, each describing how to access the data files, see Sections 1.7-6 and 5.2.

`markprivate`

  a string used in `DisplayAtlasInfo` (2.5-1) to mark private data, see Section 3.2, and

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 2.2,

`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 5.6,

`ringinfo`

  a list of triples, each containing at the first position the name of a file with the matrix generators, at the second position a string describing the ring generated by the matrix entries, and at the third position this ring itself; `DisplayAtlasInfo` (2.5-1) displays this information for example for representations over proper extensions of the rational number field only if the representation is mentioned in the `ringinfo` list,

`private`

  a list of pairs of strings used for administrating private data (see Chapter 3.); the value is changed by `AtlasOfGroupRepresentationsNotifyPrivateDirectory` (3.1-1) and `AtlasOfGroupRepresentationsForgetPrivateDirectory` (3.1-2),

`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` (1.6-1), the value of the component `types` is set in `AGRDeclareDataType` (5.5-1), and the values of the components for local data directories are created by `AtlasOfGroupRepresentationsNotifyPrivateDirectory` (3.1-1).

#### 5.2 How to Customize the Access to Data files

We discuss the three steps listed in Section 1.7-6.

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` (5.1-5), the components of which 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` (3.1-1)), and type be the data type (see `AGRDeclareDataType` (5.5-1)). 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, it may be necessary to interpret the contents.

`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 shown when `AtlasOfGroupRepresentationsShowUserParameters` (1.7-8) is called.

`active`

  The current `accessFunctions` record is ignored by `AGRFileContents` (5.6-2) if the value is not `true`.

In `AGRFileContents` (5.6-2), the records in the `accessFunctions` component of `AtlasOfGroupRepresentationsInfo` (5.1-5) 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.

#### 5.3 Reading and Writing MeatAxe Format Files

##### 5.3-1 ScanMeatAxeFile

`> 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 MeatAxe manual [Rin98]), 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` (5.1-4). 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.

##### 5.3-2 MeatAxeString

`> 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(q)`, in 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 MeatAxe text format (see [Rin98]).

In the third form, for a permutation perm with largest moved point n, say, a prime power q, and a list dims of length 2 containing two positive integers larger than or equal to n, `MeatAxeString` returns a string that encodes perm as a matrix over `GF(q)`, of dimensions 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
    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> MeatAxeString( mat, 3 ) = str;
    true

##### 5.3-3 FFList

`> 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` (5.3-1) and `MeatAxeString` (5.3-2).

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

##### 5.3-4 CMtxBinaryFFMatOrPerm

`> CMtxBinaryFFMatOrPerm`( elm, def, outfile ) | ( 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 (pi, n) where pi 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 pi, viewed as a permutation on the points up to n, to the file with name outfile.

(The binary format is described in the `C`-MeatAxe manual [Rin98].)

    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" ) );

##### 5.3-5 FFMatOrPermCMtxBinary

`> 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 [Rin98]. `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

#### 5.4 Reading and Writing ATLAS Straight Line Programs

##### 5.4-1 ScanStraightLineProgram

`> 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.

`#anything`

  lines starting with a hash sign `#` are ignored,

`echo anything`

  lines 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 n`

  means that there are n inputs, referred to via the labels `1`, `2`, ..., n,

`inp k a1 a2 ... ak`

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

`cjr a b`

  means that a is replaced by `b^(-1) * a * b`,

`cj a b c`

  means that c is defined as `b^(-1) * a * b`,

`com a b c`

  means that c is defined as `a^(-1) * b^(-1) * a * b`,

`iv a b`

  means that b is defined as `a^(-1)`,

`mu a b c`

  means that c is defined as `a * b`,

`pwr a b c`

  means that c is defined as `b^a`,

`cp a b`

  means that b is defined as a copy of a,

`oup l`

  means that there are l outputs, stored in the labels `1`, `2`, ..., l, and

`oup l b1 b2 ... bl`

  means 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` (5.1-2) 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 2.4 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 names `3`, `1`, `2`, `4`, and `5`.

##### 5.4-2 AtlasStringOfProgram

`> 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` (4.1-1)), 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` (5.4-1).

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 [Rin98]) rather than the format described in Section 5.4. (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` (4.2-1)).

    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"

#### 5.5 Data Types Used in the ATLAS of Group Representations

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` (2.5-1) 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` (2.5-1) when called with first argument a group name,

- an input format of its own for `AtlasProgram` (2.5-3),

- an input format of its own for `OneAtlasGeneratingSetInfo` (2.5-4), and

- specific tests for the data of this data type; these functions are used by the global tests described in Section 5.8.

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

##### 5.5-1 AGRDeclareDataType

`> AGRDeclareDataType`( kind, name, record ) | ( function )

Let kind be one of the strings `"rep"` or `"prg"`, and record be a record. `AGRDeclareDataType` 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 with 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 `AGRParseFilenameFormat` (5.6-1), such that only filenames of the type in question match. (It is not checked whether this "detection function" matches exactly one type, so one must be very careful here when declaring a new type.)

`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 `AGRParseFilenameFormat` (5.6-1) 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` (5.1-5), see Section 5.2. 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` (2.5-2), `OneAtlasGeneratingSetInfo` (2.5-4), or `AllAtlasGeneratingSetInfos` (2.5-5)) 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` (2.5-1) 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` (2.5-1) 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 to print no column for the data type.)

`DisplayPRG` (for `prg` only)

  This is used in `DisplayAtlasInfo` (2.5-1) 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` (2.5-1) and `OneAtlasGeneratingSetInfo` (2.5-4). The value must be a function that takes two arguments (a list as returned by `AGRParseFilenameFormat` (5.6-1), 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` (2.5-1).

`AccessPRG` (for `prg` only)

  This is used in `AtlasProgram` (2.5-3). 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` (2.5-3)), 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` (2.5-3) 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.)

`TOCEntryString`

  This is used in `StoreAtlasTableOfContents` (1.6-2). The value must be a function that takes two arguments (the name name of the type and a list as returned by `AGRParseFilenameFormat` (5.6-1) 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` (1.6-1), 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` (1.6-1)), 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 `AGRParseFilenameFormat` (5.6-1), 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 `AtlasOfGroupRepresentationsTestFileHeaders` (5.8-2). The value must be a function that takes the same four arguments as `AGRFileContents` (5.6-2), except that the first argument `"datagens"` can be replaced by `"local"` and that the third argument is a list as returned by `AGRParseFilenameFormat` (5.6-1). (The default value is `ReturnTrue` (Reference: ReturnTrue).)

`TestFiles` (for `rep` only)

  This is used in the function `AtlasOfGroupRepresentationsTestFiles` (5.8-4). 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 `AtlasOfGroupRepresentationsTestWords` (5.8-3). The value must be a function that takes five arguments where the first four are the same arguments as for `AGRFileContents` (5.6-2), except that the first argument `"dataword"` can be replaced by `"local"`, and the fifth argument is `true` or `false`, indicating verbose mode or not.

#### 5.6 Filenames Used in the ATLAS of Group Representations

The data of each local GAP version of the ATLAS of Group Representations is either private (see Chapter 3.) or is 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 2.3).

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 2.2). The translations of the name groupname to the name(s) used within GAP is given by the component `GAPnames` of `AtlasOfGroupRepresentationsInfo` (5.1-5).

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.

`groupnameGi-cycWn`

  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`.

`groupnameGi-cclsWn`

  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`.

`groupnameGicycWn-cclsWm`

  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`.

`groupnameGi-maxkWn`

  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`.

`groupnameGimaxkWn-subgroupnameGjWm`

  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.

`groupnameGi-aoutnameWn`

  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 alpha of G given by the name outname; if this name is empty then alpha is the unique nontrivial outer automorphism of G; if it is a positive integer k then alpha 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 alpha 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^prime, and O_8^+(3).2_2^primeprimeprimeprimeprime, respectively.

`groupnameGi-GjWn`

  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`.

`groupnameGi-checkn`

  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.

`groupnameGi-Pn`

  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.

`groupnameGi-findn`

  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.

`groupnameGi-XdescrWn`

  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 3.), such a script can be accessed with descr as the third argument of `AtlasProgram` (2.5-3).

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).

`groupnameGi-fqrdimidBm.mnr`

  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`.

`groupnameGi-pnidBm.mnr`

  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`.

`groupnameGi-ArdimidBm.g`

  a 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`.

`groupnameGi-ZrdimidBm.g`

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

`groupnameGi-HrdimidBm.g`

  a 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`.

`groupnameGi-ZnrdimidBm.g`

  a 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`.

##### 5.6-1 AGRParseFilenameFormat

`> AGRParseFilenameFormat`( 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`.

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

##### 5.6-2 AGRFileContents

`> AGRFileContents`( 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 `AGRDeclareDataType` (5.5-1)). dirname must be one of `"datagens"`, `"dataword"`, or the dirid value of a private directory, see `AtlasOfGroupRepresentationsNotifyPrivateDirectory` (3.1-1). If groupname is the ATLAS-file name of a group G (see Section 2.2), 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 `AGRFileContents` returns the contents of the corresponding file(s), in the sense that the file(s) (or equivalent ones, see Section 1.7-6) 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 `AGRFileContents` does not check whether the table of contents of the ATLAS of Group Representations actually contains filename.

#### 5.7 The Tables of Contents of the ATLAS of Group Representations

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` (1.6-1).

It is assumed that the local data directories contain only files that are also available on servers. Private extensions to the database (cf. Section 1.8 and Chapter 3.) 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 2.2) 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 5.5) for which data are available.

Note that the name mapping between the ATLAS-file and GAP names of the groups is provided by the `groupnames` component of `AtlasOfGroupRepresentationsInfo` (5.1-5), and information about the base rings of matrix representations is provided by the `ringinfo` component. Group names are notified with `AGRGNAN` (5.7-1), and base ring information can be notified with `AGRRNG` (5.7-2); these two administrational functions may be useful for private extensions of the package (see Chapter 3.).

##### 5.7-1 AGRGNAN

`> AGRGNAN`( gapname, atlasname[, size] ) | ( function )

Let gapname be a string denoting a GAP group name, and atlasname be a string denoting the corresponding ATLAS-file name used in filenames of the ATLAS of Group Representations. If size is given then it must be the order of the corresponding group. `AGRGNAN` adds the pair `[ gapname, atlasname ]` or the triple `[ gapname, atlasname, size ]`, respectively, to the list stored in the `GAPnames` component of `AtlasOfGroupRepresentationsInfo` (5.1-5), making the ATLAS data involving atlasname accessible for the group with name gapname.

An example of a valid call is `AGRGNAN("A6.2_2","PGL29",360)`, see also Section 3.3.

##### 5.7-2 AGRRNG

`> AGRRNG`( filename, descr ) | ( function )

Let filename be a string denoting the name of a file containing the generators of a matrix representation over a ring that is not determined by the filename, and let descr be a string describing this ring R, say. `AGRRNG` adds the triple `[ filename, descr, R ]` to the list stored in the `ringinfo` component of `AtlasOfGroupRepresentationsInfo` (5.1-5).

An example of a valid call is `AGRRNG("A5G1-Ar3aB0","Field([Sqrt(5)])")`.

#### 5.8 Sanity Checks for the ATLAS of Group Representations

The fact that the ATLAS of Group Representations is designed as an open database (see Section 1.7-1) 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 the following can also be used for private extensions of the package (see Chapter 3.).

The file `tst/testall.g` of the package contains `ReadTest` (Reference: ReadTest) statements for executing a collection of such sanity checks; one can run them by starting GAP in the `tst` directory, and then calling `Read( "testall.g" )`. If no problem occurs then GAP prints only lines starting with one of the following.

    + $Id:
    + GAP4stones:

The required space and time for running these tests depends on the amount of locally available data.

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

The file `tst/atlasrep.tst` contains calls to the functions `AtlasOfGroupRepresentationsTestGroupOrders` (5.8-1), which checks the consistency of the stored group orders and the actual data, `AtlasOfGroupRepresentationsTestFileHeaders` (5.8-2), which checks the consistency of the names of MeatAxe text files and the first lines of the files, and `AtlasOfGroupRepresentationsTestWords` (5.8-3), which checks whether the available programs do what they promise.

The calls to `AtlasOfGroupRepresentationsTestFiles` (5.8-4), `AtlasOfGroupRepresentationsTestClassScripts` (5.8-5), and `AGR_TestMinimalDegrees` (4.3-5) are not part of the tests that are run by reading `tst/testall.g`.

All these tests apply only to the local table of contents (see Section 5.7), that is, only those data files are checked that are actually available in the local GAP installation. No files are fetched from servers during these tests.

Further tests, such as the consistency of different versions of server data, exist but are not part of the distributed package.

##### 5.8-1 AtlasOfGroupRepresentationsTestGroupOrders

`> AtlasOfGroupRepresentationsTestGroupOrders`( ) | ( function )

Returns: `false` if a contradiction was found, `true` otherwise.

This function checks whether the group orders stored in the `GAPnames` component of `AtlasOfGroupRepresentationsInfo` (5.1-5) coincide with the orders computed from an ATLAS permutation representation of degree up to 10^4, from the character table or the table of marks with the given name, from the inner 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). A message is printed for each group name for which no order is stored (and perhaps now can be stored), for which the stored group order cannot be verified, for which a contradiction was found.

##### 5.8-2 AtlasOfGroupRepresentationsTestFileHeaders

`> AtlasOfGroupRepresentationsTestFileHeaders`( [tocid[, groupname]] ) | ( function )

Returns: `false` if an error occurs, otherwise `true`.

First suppose that this function is called with two arguments tocid, the identifier of a directory (see Section `sect:Adding a Private Data Directory` (3.1)), and groupname, an ATLAS-file name that occurs as a component name in the table of contents of the directory. The function checks for those data files for groupname in the tocid directory that are in MeatAxe text format whether the filename and the header line are consistent; it checks the data file in GAP format whether the file name is consistent with the contents of the file.

If only one argument tocid is given then all representations available for groupname are checked with the three argument version.

If only one argument tocid is given then all available groups in the directory with identifier tocid are checked; the contents of the local `dataword` directory can be checked by entering `"local"`, which is also the default for tocid.

##### 5.8-3 AtlasOfGroupRepresentationsTestWords

`> AtlasOfGroupRepresentationsTestWords`( [tocid[, groupname]] ) | ( function )

Returns: `false` if an error occurs, otherwise `true`.

Called with one argument tocid, a string, `AtlasOfGroupRepresentationsTestWords` processes all programs that are stored in the directory with identifier tocid (see Section `sect:Adding a Private Data Directory` (3.1)), using the function stored in the `TestWords` component of the data type in question. The contents of the local `dataword` directory can be checked by entering `"local"`, which is also the default.

If the string groupname, an ATLAS-file name that occurs as a component name in the table of contents of the directory, is given as the second argument then only the data files for this group are tested.

##### 5.8-4 AtlasOfGroupRepresentationsTestFiles

`> AtlasOfGroupRepresentationsTestFiles`( [tocid[, groupname]] ) | ( function )

Returns: `false` if an error occurs, otherwise `true`.

This function is an analogue of `AtlasOfGroupRepresentationsTestFileHeaders` (5.8-2). It checks whether reading MeatAxe text files with `ScanMeatAxeFile` (5.3-1) returns non-`fail` results. It does not check whether the first line of a MeatAxe text file is consistent with the filename, since this is tested by `AtlasOfGroupRepresentationsTestFileHeaders` (5.8-2).

##### 5.8-5 AtlasOfGroupRepresentationsTestClassScripts

`> AtlasOfGroupRepresentationsTestClassScripts`( [groupname] ) | ( function )

Returns: `false` if an error occurs, otherwise `true`.

First suppose that `AtlasOfGroupRepresentationsTestClassScripts` is called with one argument groupname, the name of a component in `AtlasOfGroupRepresentationsInfo.TableOfContents.( "local" )`. If the GAP table library contains an ordinary character table with `Identifier` (Reference: Identifier!for character tables) value the GAP name corresponding to groupname then it is checked whether all those straight line programs for this group that return class representatives are consistent with the character table in the sense that the class names used occur for the table, and that the element orders and centralizer orders for the classes are correct.

If no argument is given then all available groups are checked with the one argument version.

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