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 7.9), 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 7.8). 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, **AtlasRep** functions 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: InfoLevel). For example, the info level of `InfoAtlasRep`

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

\(n\)` )`

.

`‣ InfoAtlasRep` | ( info class ) |

If the info level of `InfoAtlasRep`

is at least \(1\) then information about `fail`

results of **AtlasRep** functions is printed. If the info level is at least \(2\) then also 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 (see Section 7.3) 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.

`‣ AGR` | ( global variable ) |

is a record whose components are functions and data that are used by the high level interface functions. Some of the components are documented, see for example the index of the package manual.

`‣ AtlasOfGroupRepresentationsInfo` | ( global variable ) |

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

of the package, with the following components.

`GAPnames`

a list of pairs, each containing the

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

a list used for administrating extensions of the database (see Chapter 5); the value is changed by

`AtlasOfGroupRepresentationsNotifyData`

(5.1-1) and`AtlasOfGroupRepresentationsForgetData`

(5.1-2),`characterinfo`

,`permrepinfo`

,`ringinfo`

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

`DisplayAtlasInfo`

(3.5-1),`TableOfContents`

a record with at most the components

`core`

,`internal`

,`local`

,`merged`

,`types`

, and the identifiers of database extensions. The value of the component`types`

is set in`AGR.DeclareDataType`

(7.5-1), and the values of the other components are created by`AtlasOfGroupRepresentationsNotifyData`

(5.1-1).`accessFunctions`

a list of records, each describing how to access the data files, see Sections 4.2-5 and 7.2, and

By default, locally available data files are stored in prescribed directories, and the files are exactly the text files that have been downloaded from appropriate places in the internet. However, a more flexible approach may be useful.

First, one may want to use *different file formats*, for example **MeatAxe** binary files may be provided parallel to **MeatAxe** text files. Second, one may want to use *a different directory structure*, for example the same structure as used on some server –this makes sense for example if a local mirror of a server is available, because then one can read the server files directly, without transferring/copying them to another directory.

In order to achieve this (and perhaps more), we admit to customize the meaning of the following three access steps.

**Are the required data locally available?**There may be different file formats available, such as text or binary files, and it may happen that the data are available in one file or are distributed to several files.

**How can a file be made locally available?**A different remote file may be fetched, or some postprocessing may be required.

**How is the data of a file accessed by****GAP**?A different function may be needed to evaluate the file contents.

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`

(7.1-5). The relevant record components are as follows.

`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 as key in the user preference

`FileAccessFunctions`

, see Section 4.2-5.`location(`

\(files, type\)`)`

Let \(files\) be a list of pairs

`[ dirname, filename ]`

, and \(type\) be the data type (see`AGR.DeclareDataType`

(7.5-1)) to which the files belong. This function must return either the absolute paths where the mechanism implemented by the current record expects the local version of the given files, or`fail`

if this function does not feel responsible for these files.The files 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(`

\(filepath, filename, dirname, type\)`)`

This function is called if a file is not locally available and if the

`location`

function in the current record has returned a list of paths. The argument \(type\) 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), a string with the contents of the data file if the remote data were directly loaded into the**GAP**session (if no local caching is possible), and`false`

otherwise.`contents(`

\(files, type, filepaths\)`)`

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 first three arguments must be the same as for the`location`

function.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.

In `AGR.FileContents`

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

component of `AtlasOfGroupRepresentationsInfo`

(7.1-5) are considered –in reversed order– whose `description`

component occurs in the user preference `FileAccessFunctions`

, see Section 4.2-5.

`‣ 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 user preference `HowToReadMeatAxeTextFiles`

, see Section 4.2-7.

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 ) |

`‣ MeatAxeString` ( intmat ) | ( operation ) |

Returns: a string encoding the **GAP** objects given as input in `C`

-**MeatAxe** text format, see [Rin].

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 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 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.

In the fourth form, for a matrix `intmat` of integers, `MeatAxeString`

returns a string that encodes `intmat` as an integer matrix.

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

The **MeatAxe** defines the bijection between the elements in the field with \(q = p^d\) elements and the set \(\{ 0, 1, \ldots, q-1 \}\) of integers by assigning the field element \(\sum_{{i=0}}^{{d-1}} c_i z^i\) to the integer \(\sum_{{i=0}}^{{d-1}} c_i p^i\), where the \(c_i\) are in the set \(\{ 0, 1, \ldots, p-1 \}\) and \(z\) is the primitive root of the field with \(q\) elements that corresponds to the residue class of the indeterminate, modulo the ideal spanned by the Conway polynomial of degree \(d\) over the field with \(p\) elements.

The finite fields introduced by the **StandardFF** package [Lüb21] are supported by `FFList`

and `FFLists`

, in the sense that the bijection defined by `StandardIsomorphismGF`

(StandardFF: StandardIsomorphismGF) is applied automatically when `F` is a field in the filter `IsStandardFiniteField`

(StandardFF: IsStandardFiniteField).

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

Let the pair \((\textit{elm}, \textit{def})\) be either of the form \((M, q)\) where \(M\) is a matrix over a finite field \(F\), say, with \(q \leq 256\) 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`.

In the case of a permutation \(\pi\), 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 4.2-10.

(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, or `fail`

.

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`

, \(\ldots\), \(n\),`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`

, \(\ldots\), \(l\), 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`

(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 data files, 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 for `ScanStraightLineProgram`

(7.4-1). (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`

(3.5-4),an input format of its own for

`OneAtlasGeneratingSetInfo`

(3.5-6), andspecific tests for the data of this data type; these functions are used by the global tests described in Section 7.9.

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. If `kind` is `"rep"`

then `AGR.DeclareDataType`

declares a new data type of representations, if `kind` is `"prg"`

then it declares a new data type of programs. The string `name` is the name of the type, for example `"perm"`

, `"matff"`

, or `"classes"`

. **AtlasRep** stores the data for each group internally in a record whose component `name` holds the list of the data about the type with this name.

*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`

(7.1-5), 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`

(3.5-3),`OneAtlasGeneratingSetInfo`

(3.5-6), or`AllAtlasGeneratingSetInfos`

(3.5-7)) 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, a list containing the**GAP**name and the**ATLAS**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`

(3.5-6). The value must be a function that takes two arguments (a list as returned by`OneAtlasGeneratingSetInfo`

(3.5-6), 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`

(3.5-4). The value must be a function that takes four arguments (the current table of contents, the group name, 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`

(3.5-4)), 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`

(3.5-4) 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`

(3.5-5) to create the result value from the identifier. (The default value is`AtlasProgramDefault`

.)`TOCEntryString`

This is used in

`StringOfAtlasTableOfContents`

(5.1-3). The value must be a function that takes two or three arguments (the name`name`of the type, a list as returned by`AGR.ParseFilenameFormat`

(7.6-1), and optionally a string that indicates the "remote" format) and returns a string that describes the appropriate data format. (The default value is`TOCEntryStringDefault`

.)`PostprocessFileInfo`

This is used in the construction of a table of contents 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 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 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 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 fifth argument is`true`

or`false`

, indicating verbose mode or not.

**AtlasRep** expects that the filename of each data file describes the contents of the file. This section lists the definitions of the supported structures of filenames.

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`

(7.1-5).

The names of files that contain straight line programs or straight line decisions 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 \(\alpha\) 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`

\(\ldots\) then \(\alpha\) generates the cyclic group of automorphisms induced on \(G\) by \(G.2_1\), \(G.4_2\), \(G.3_3\) \(\ldots\); 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 \(O_8^+(3).2_2^{{\prime\prime\prime\prime\prime}}\), respectively.**\(groupname\)**`G`

\(i\)`-ker`

\(factgroupname\)`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 generators of the kernel of an epimorphism that maps \(G\) to a group with

**ATLAS**-file name \(factgroupname\). An example is`2A5G1-kerA5W1`

.**\(groupname\)**`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`

(3.5-4).

The names of files that contain group generators 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`

\(m\)`.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`

.**\(groupname\)**`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`

\(m\)`.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`

.**\(groupname\)**`G`

\(i\)`-Z`

\(n\)`r`

\(dim\)\(id\)`B`

\(m\)`.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`

.

`‣ 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 \([ [ c_1, c_2, \ldots, c_n ], [ f_1, f_2, \ldots, f_n ] ]\) 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 \(s_1, s_2, \ldots, s_m\). The string `string` matches `format` if \(s_i\) matches the conditions in \(c_i\), for \(1 \leq i \leq n\), 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` ( files, type ) | ( function ) |

Returns: the **GAP** object obtained from reading and interpreting the file(s) given by `files`.

Let `files` be a list of pairs of the form `[ dirname, filename ]`

, where `dirname`

and `filename`

are strings, and let `type` be a data type (see `AGR.DeclareDataType`

(7.5-1)). Each `dirname`

must be one of `"datagens"`

, `"dataword"`

, or the `dirid`

value of a data extension (see `AtlasOfGroupRepresentationsNotifyData`

(5.1-1)). If the contents of each of the files in question is accessible and their data belong to the data type `type`

then `AGR.FileContents`

returns the contents of the files; otherwise `fail`

is returned.

Note that if some file is already stored in the `dirname` directory then `AGR.FileContents`

does *not* check whether the relevant table of contents actually contains `filename`.

`identifier`

used by the The functions `AtlasGenerators`

(3.5-3), `AtlasProgram`

(3.5-4), `AtlasProgramInfo`

(3.5-5), `OneAtlasGeneratingSetInfo`

(3.5-6), and `AllAtlasGeneratingSetInfos`

(3.5-7) return records which have a component `identifier`

. The value of this component describes the record in the sense that one can reconstruct the whole record from it, and the `identifier`

value can be used as an input for `AtlasGenerators`

(3.5-3), `AtlasProgram`

(3.5-4), `AtlasProgramInfo`

(3.5-5), `AtlasGroup`

(3.5-8), and `AtlasSubgroup`

(3.5-9).

The `identifier`

component has the following format.

For records describing representations, it is a list of the form

`[ gapname, files, std, info ]`

.For records describing straight line programs and straight line decisions, it is a list of the form

`[ gapname, files, std ]`

.

Here `gapname`

is the **GAP** name of the group in question, `files`

defines the data files, `std`

is the standardization of its generators, and `info`

is some information that depends on the type of the representation, for example the number of moved points in the case of a permutation representation.

The `files`

entry has one of the following formats:

a string, in the case that exactly one file is needed that does not belong to a private extension; an example of such an

`identifier`

value is`[ "J1", "J1G1-cycW1", 1 ]`

a list whose entries are strings (which refer to files from the core part of the database) and pairs of the form

`[ tocid, file ]`

(which refer to files from the extension given by`tocid`

); examples of`identifier`

values are`[ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ]`

,`[ "2.M12", [ [ "mfer", "2M12G1-cclsW1" ] ], 1 ]`

,`[ "2.M12", [ "M12G1-max1W1", [ "internal", "2M12G1-kerM12W1" ] ], 1 ]`

,`[ "2.M12", [ [ "mfer", "2M12G1-p24bB0.m1" ], [ "mfer", "2M12G1-p24bB0.m2" ] ], 1, 24 ]`

.

Up to version 1.5 of the **AtlasRep** package, a different `identifier`

format was used for files from extensions of the database. Namely, the first entry of the list was a pair `[ tocid, groupname ]`

, and the second entry was either a string or a list of strings. Note that with that old format, it was not possible to describe a combination of several files from different sources (core part and extension, or different extensions). The function `AtlasRepIdentifier`

(7.7-1) can be used to convert between the two formats.

`‣ AtlasRepIdentifier` ( oldid ) | ( function ) |

`‣ AtlasRepIdentifier` ( id, "old" ) | ( function ) |

This function converts between the "old format" (the one used up to version 1.5.1 of the package) and the "new format" (the one used since version 2.0) of the `identifier`

component of the records returned by **AtlasRep** functions. Note that the two formats differ only for `identifier`

components that describe data from non-core parts of the database.

If the only argument is a list `oldid` that is an `identifier`

in old format then the function returns the corresponding `identifier`

in new format. If there are two arguments, a list `id` that is an `identifier`

in new format and the string `"old"`, then the function returns the corresponding `identifier`

in old format if this is possible, and `fail`

otherwise.

gap> id:= [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ];; gap> AtlasRepIdentifier( id ) = id; true gap> id:= [ "L2(8)", "L28G1-check1", 1, 1 ];; gap> AtlasRepIdentifier( id ) = id; true gap> oldid:= [ [ "priv", "C4" ], [ "C4G1-p4B0.m1" ], 1, 4 ];; gap> newid:= AtlasRepIdentifier( oldid ); [ "C4", [ [ "priv", "C4G1-p4B0.m1" ] ], 1, 4 ] gap> oldid = AtlasRepIdentifier( newid, "old" ); true gap> oldid:= [ [ "priv", "C4" ], "C4G1-max1W1", 1 ];; gap> newid:= AtlasRepIdentifier( oldid ); [ "C4", [ [ "priv", "C4G1-max1W1" ] ], 1 ] gap> oldid = AtlasRepIdentifier( newid, "old" ); true gap> oldid:= [ [ "priv", "C4" ], "C4G1-Ar1aB0.g", 1, 1 ];; gap> newid:= AtlasRepIdentifier( oldid ); [ "C4", [ [ "priv", "C4G1-Ar1aB0.g" ] ], 1, 1 ] gap> oldid = AtlasRepIdentifier( newid, "old" ); true gap> oldid:= [ [ "priv", "C4" ], "C4G1-XtestW1", 1 ];; gap> newid:= AtlasRepIdentifier( oldid ); [ "C4", [ [ "priv", "C4G1-XtestW1" ] ], 1 ] gap> oldid = AtlasRepIdentifier( newid, "old" ); true gap> oldid:= [ [ "mfer", "2.M12" ], > [ "2M12G1-p264aB0.m1", "2M12G1-p264aB0.m2" ], 1, 264 ];; gap> newid:= AtlasRepIdentifier( oldid ); [ "2.M12", [ [ "mfer", "2M12G1-p264aB0.m1" ], [ "mfer", "2M12G1-p264aB0.m2" ] ] , 1, 264 ] gap> oldid = AtlasRepIdentifier( newid, "old" ); true

The list of **AtlasRep** data is stored in several *tables of contents*, which are given essentially by JSON documents, one for the core data and one for each data extension in the sense of Chapter 5. The only exception are data extensions by locally available files in a given directory, where the contents of this directory itself describes the data in question. One can create such a JSON document for the contents of a given local data directory with the function `StringOfAtlasTableOfContents`

(5.1-3).

Here are the administrational functions that are called when a data extension gets notified with `AtlasOfGroupRepresentationsNotifyData`

(5.1-1). In each case, \(gapname\) and \(atlasname\) denote the **GAP** and **ATLAS** name of the group in question (see Section 3.2), and \(dirid\) denotes the identifier of the data extension.

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

`AGR.GNAN(`

\(gapname, atlasname[, dirid]\)`)`

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.An example of a valid call is

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

.`AGR.TOC(`

\(typename, filename, crc[, dirid]\)`)`

`AGR.TOC`

notifies an entry to the`TableOfContents.(`

\(dirid\)`)`

component of`AtlasOfGroupRepresentationsInfo`

(7.1-5). The string \(typename\) must be the name of the data type to which the entry belongs, the string \(filename\) must be the prefix of the data file(s), and \(crc\) must be a list that contains the checksums of the data files, which are either integers (see`CrcFile`

(Reference: CrcFile)) or strings (see`HexSHA256`

). In particular, the number of files that are described by the entry equals the length of \(crc\).The optional argument \(dirid\) is equal to the argument with the same name in the corresponding call of

`AtlasOfGroupRepresentationsNotifyData`

(5.1-1). If no \(dirid\) argument is given then the current value of`AGR.DIRID`

is taken as the default; this value is set automatically before a`toc.json`

file gets evaluated by`AtlasOfGroupRepresentationsNotifyData`

(5.1-1), and is reset afterwards. If`AGR.DIRID`

is not bound and \(dirid\) is not given then this function has no effect.An example of a valid call is

`AGR.TOC("perm","alt/A5/mtx/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(`

\(gapname, size[, dirid]\)`)`

The integer \(size\) is stored as the order of the group with

**GAP**name \(gapname\), in`AtlasOfGroupRepresentationsInfo.GAPnames`

.An example of a valid call is

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

.`AGR.MXN(`

\(gapname, nrMaxes[, dirid]\)`)`

The integer \(nrMaxes\) is stored as the number of classes of maximal subgroups of the group with

**GAP**name \(gapname\), in`AtlasOfGroupRepresentationsInfo.GAPnames`

.An example of a valid call is

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

.`AGR.MXO(`

\(gapname, sizesMaxes[, dirid]\)`)`

The list \(sizesMaxes\) of subgroup orders of the classes of maximal subgroups of the group with

**GAP**name \(gapname\) (not necessarily dense, in non-increasing order) is stored in`AtlasOfGroupRepresentationsInfo.GAPnames`

.An example of a valid call is

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

.`AGR.MXS(`

\(gapname, structureMaxes[, dirid]\)`)`

The list \(structureMaxes\) of strings describing the structures of the maximal subgroups of the group with

**GAP**name \(gapname\) (not necessarily dense), is stored in`AtlasOfGroupRepresentationsInfo.GAPnames`

.An example of a valid call is

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

.`AGR.STDCOMP(`

\(gapname, factorCompatibility[, dirid]\)`)`

The list \(factorCompatibility\) (with entries the standardization of the group with

**GAP**name \(gapname\) , 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) is stored 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(`

\(repname, descr[, dirid]\)`)`

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`

(7.1-5).An example of a valid call is

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

.`AGR.TOCEXT(`

\(atlasname, std, maxnr, files[, dirid]\)`)`

Called with \(atlasname\), 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`AtlasOfGroupRepresentationsInfo.GAPnames`

.An example of a valid call is

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

.`AGR.API(`

\(repname, info[, dirid]\)`)`

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 a record that describes the contents of \(info\).\(info\) has the following entries.

At position \(1\), the transitivity is stored.

If the transitivity is zero then \(info\) has length two, and the second entry is the list of orbit lengths.

If the transitivity is positive then \(info\) has length four or five, and 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 either the string

`"???"`

or a string that describes 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 the string`"???"`

or 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(`

\(gapname, repname, char, pos[, charname[, dirid]]\)`)`

Called with the strings \(gapname\) 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 \(charname\).

If \(dirid\) is given but no \(charname\) is known then one can enter

`fail`

as the fifth argument.An example of a valid call is

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

.

The file `tst/testall.g`

of the package contains `Test`

(Reference: Test) statements for checking whether the **AtlasRep** functions behave as documented. One can run these tests by calling `ReadPackage( "AtlasRep", "tst/testall.g" )`

. 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 remainder of this section deals with consistency checks of the data. The tests described in Section 7.9-1 can be used for data from any extension of the database (see Chapter 5), Section 7.9-2 lists tests which apply only to the core part of the database.

All these tests apply only to *locally* available files (see Section 7.8), no files are downloaded during the tests. Thus the required space and time for running these tests depend on the amount of locally available data.

Some of the tests 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. These tests are experimental in the sense that they involve several heuristics. Depending on the data to which they are applied, it may happen that the tests run out of space or do not finish in acceptable time. Please inform the package maintainer if you run into such problems.

The following tests can be used to check the data that belong to a given part of the database (core data or extension). 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 `AtlasOfGroupRepresentationsNotifyData`

(5.1-1). The contents of the core part can be checked by entering `"core"`

, 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. (For some of them, the global option `TryToExtendData`

can be entered in order to try the computation of not yet stored data.)

`AGR.Test.GroupOrders()`

checks whether the group orders stored in the

`GAPnames`

component of`AtlasOfGroupRepresentationsInfo`

(7.1-5) coincide with the group orders computed from an**ATLAS**permutation representation of degree up to`AGR.Test.MaxTestDegree`

, from the available character table or table of marks with the given name, or from the structure of the name, in the sense that splitting the name at the first dot (`.`

) or colon (`:`

) and applying the same criteria to derive the group order from the two parts may yield enough information.`AGR.Test.Words( [`

\(tocid\)`] )`

processes the straight line programs that belong to \(tocid\), using the function stored in the

`TestWords`

component of the data type in question.The straight line programs for the cases listed in

`AGR.Test.HardCases.TestWords`

are omitted.`AGR.Test.ClassScripts( [`

\(tocid\)`] )`

checks whether the straight line programs that belong to \(tocid\) and 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( [`

\(tocid\)`][:TryToExtendData] )`

checks whether all straight line programs that belong to \(tocid\) and that compute class representatives from representatives of cyclic subgroups possess a corresponding straight line program (

*anywhere*in the database) for computing representatives of cyclic subgroups.`AGR.Test.FileHeaders( [`

\(tocid\)`] )`

checks whether the

**MeatAxe**text files that belong to \(tocid\) have a header line that is consistent with the filename, and whether the contents of all**GAP**format data files that belong to \(tocid\) is consistent with the filename.`AGR.Test.Files( [`

\(tocid\)`] )`

checks whether the

**MeatAxe**text files that belong to \(tocid\) 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 files that belong to \(tocid\) satisfy that applying first`CMtxBinaryFFMatOrPerm`

(7.3-4) and then`FFMatOrPermCMtxBinary`

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

\(tocid\)`][:TryToExtendData] )`

checks the stored primitivity information for the permutation representations that belong to \(tocid\). That is, the number of orbits, in case of a transitive action the transitivity, the rank, the information about the point stabilizers are computed if possible, and compared with the stored information.

`AGR.Test.Characters( [`

\(tocid\)`][:TryToExtendData] )`

checks the character information (that belongs to \(tocid\)) for the matrix and permutation representations.

`AGR.Test.StdCompatibility( [`

\(tocid\)`][:TryToExtendData] )`

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`

(7.1-5) and belongs to \(tocid\) 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.KernelGenerators( [`

\(tocid\)`][:TryToExtendData] )`

checks whether the straight line programs (that belong to \(tocid\)) for computing generators of kernels of natural epimorphisms between

**ATLAS**groups compute generators of normal subgroups of the right group orders. If it is known that the given standard generators of the given group are compatible with some standard generators of the factor group in question (see the section about`AGR.Test.StdCompatibility`

) then it is also checked whether evaluating the straight line program at these standard generators of the factor group yields only the identity.Note that the verification of normal subgroups of matrix groups may be

*very*time and space consuming if the package**recog**[NSA+18] is not available.The function also tries to

*find*words for computing kernel generators of those epimorphisms for which no straight line programs are stored; the candidates are given by stored factor fusions between the character tables from the**GAP**Character Table Library.`AGR.Test.MaxesOrders( [`

\(tocid\)`] )`

checks whether the orders of maximal subgroups stored in the component

`GAPnames`

of`AtlasOfGroupRepresentationsInfo`

(7.1-5) coincide with the orders computed from the restriction of an**ATLAS**permutation representation of degree up to`AGR.Test.MaxTestDegree`

(using a straight line program that belongs to \(tocid\)), from the character table, or the table of marks with the given name, or from the information about maximal subgroups of the 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`

(7.1-5) coincide with the names computed from the**GAP**character table with the given name.`AGR.Test.MaxesStandardization( [`

\(tocid\)`] )`

checks whether the straight line programs (that belong to \(tocid\)) for standardizing the generators of maximal subgroups are correct: If a semi-presentation is available for the maximal subgroup and the standardization in question then it is used, otherwise an explicit isomorphism is tried.

`AGR.Test.CompatibleMaxes( [`

\(tocid\)`][:TryToExtendData] )`

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.

The tests described in this section are intended for checking data that do not belong to a particular part of the **AtlasRep** database. Therefore *all* locally available data are used in these tests. 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.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.MinimalDegrees()`

checks that the (permutation and matrix) representations available in the database do not have smaller degree than the minimum claimed in Section 6.3.

Finally, we reset the user preference and the info level which had been set at the beginning of Chapter 2.

gap> SetUserPreference( "AtlasRep", "DisplayFunction", origpref ); gap> SetInfoLevel( InfoAtlasRep, globallevel );

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