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.
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).
> InfoAtlasRep | ( info class ) |
If the info level of InfoAtlasRep
is at least 1 then information about fail
results of functions in the AtlasRep package is printed. If the info level is at least 2 then information about calls to external programs is printed. The default level is 0, no information is printed on this level.
> InfoCMeatAxe | ( info class ) |
If the info level of InfoCMeatAxe
is at least 1 then information about fail
results of C
-MeatAxe functions is printed. The default level is zero, no information is printed on this level.
> InfoBBox | ( info class ) |
If the info level of InfoBBox
is at least 1 then information about fail
results of functions dealing with black box programs (see Section 4.2) is printed. The default level is 0, no information is printed on this level.
> CMeatAxe.FastRead | ( global variable ) |
If this component is bound and has the value true
then ScanMeatAxeFile
(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.
> 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).
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.
> 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.
> 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 |
> 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 |
> 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" ) ); |
> 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 |
> 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
.
> 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" |
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.
> 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 containing the data about the new type. Examples of name are "perm"
, "matff"
, and "classes"
.
Mandatory components of record are
FilenameFormat
This defines the format of the filenames containing data of the type in question. The value must be a list that can be used as the second argument of 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
(5.8-5). 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
(5.8-7). 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
(5.8-6). 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.
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 groupnameG
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 groupnameG
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 groupnameG
i-max
kW
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 kp
d, with k one of the above forms and d an integer then d denotes the number of dashes appended to the automorphism described by k; if d = 1 then d can be omitted. Examples are A5G1-aW1
, L34G1-a2_1W1
, U43G1-a2_3pW1
, and O8p3G1-a2_2p5W1
; these file names describe the outer order 2 automorphism of A_5 (induced by the action of S_5) and the order 2 automorphisms of L_3(4), U_4(3), and O_8^+(3) induced by the actions of L_3(4).2_1, U_4(3).2_2^', and O_8^+(3).2_2^{'''''}, respectively.
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
.
> 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 |
> 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.
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).
> AGRGNAN ( gapname, atlasname[, size[, maxessize[, "all"[, compatinfo]]]] ) | ( 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. The following optional arguments are supported.
the order of the corresponding group,
a (not necessarily dense) list of orders of the maximal subgroups of this group
the string "all"
if the maxessizes list is complete,
a list of entries of the form [ std, factname, factstd, flag ]
meaning that mapping standard generators of standardization std
to the factor group with GAP group name factname
, via the natural epimorphism, yields standard generators of standardization factstd
if flag
is true
.
AGRGNAN
adds the list of its arguments 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.
> 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)])")
.
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
(5.8-5), which checks the consistency of the names of MeatAxe text files and the first lines of the files, and AtlasOfGroupRepresentationsTestWords
(5.8-6), which checks whether the available programs do what they promise.
The calls to AtlasOfGroupRepresentationsTestFiles
(5.8-7), AtlasOfGroupRepresentationsTestClassScripts
(5.8-8), 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.
> 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, or 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.
> AtlasOfGroupRepresentationsTestSubgroupOrders ( ) | ( function ) |
Returns: false
if a contradiction was found, true
otherwise.
This function checks whether the orders of maximal subgroups 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, or from the information about maximal subgroups of a factor group modulo a central subgroup that is contained in the derived subgroup.
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.
> AtlasOfGroupRepresentationsTestStdCompatibility ( ) | ( function ) |
Returns: false
if a contradiction was found, true
otherwise.
This function checks whether the compatibility info stored in the GAPnames
component of AtlasOfGroupRepresentationsInfo
(5.1-5) coincide 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 Ncosets 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.
A message is printed for each group name for which no compatibility info was stored and now can be stored, for which the stored info cannot be verified, for which a contradiction was found.
> AtlasOfGroupRepresentationsTestCompatibleMaxes ( ) | ( function ) |
Returns: false
if a contradiction was found, true
otherwise.
This function checks whether the information about maximal subgroups stored in the maxext
components of the records stored in the TableOfContents.remote
component of AtlasOfGroupRepresentationsInfo
(5.1-5) 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 AtlasOfGroupRepresentationsTestStdCompatibility
(5.8-3)) and if there are a presentation for F and a permutation representation of G then it is checked whether the "maxes"
type scripts for F can be used to compute also generators for the maximal subgroups of G; if not then words in terms of the standard generators are computed such that the results of the script for F together with the images of these words describe the corresponding maximal subgroup of G.
A message is printed for each group name for which no compatibility info was stored and now can be stored, for which the stored info cannot be verified, for which a contradiction was found.
> 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.
> 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.
> AtlasOfGroupRepresentationsTestFiles ( [tocid[, groupname]] ) | ( function ) |
Returns: false
if an error occurs, otherwise true
.
This function is an analogue of AtlasOfGroupRepresentationsTestFileHeaders
(5.8-5). 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
(5.8-5).
> 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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