[1X5 Technicalities of the [5XAtlasRep[1X Package[0X This chapter describes those parts of the [5XGAP[0m interface to the [5XATLAS[0m of Group Representations that do not belong to the user interface (cf. Chapter [14X2[0m). Besides global variables used for administrational purposes (see Section [14X5.1[0m) and several sanity checks (see Section [14X5.8[0m), they can be regarded as the interface between the data actually contained in the files and the corresponding [5XGAP[0m objects (see Section [14X5.2[0m, [14X5.3[0m, [14X5.4[0m, and [14X5.5[0m), and the interface between the remote and the local version of the database (see Section [14X5.6[0m and [14X5.7[0m). The former interface contains functions to read and write files in [5XMeatAxe[0m format, which may be interesting for users familiar with [5XMeatAxe[0m 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 [11Xgap[0m directory of the package, but it may happen that this will be changed in future versions of the package. [1X5.1 Global Variables Used by the [5XAtlasRep[1X Package[0X For debugging purposes, the functions from the [5XGAP[0m interface to the [5XATLAS[0m of Group Representations print information depending on the info level of the info classes [2XInfoAtlasRep[0m ([14X5.1-1[0m), [2XInfoCMeatAxe[0m ([14X5.1-2[0m), and [2XInfoBBox[0m ([14X5.1-3[0m) (cf. [14X'Reference: Info Functions'[0m). The info level of an info class can be changed using [2XSetInfoLevel[0m ([14XReference: SetInfoLevel[0m). For example, the info level of [2XInfoAtlasRep[0m ([14X5.1-1[0m) can be set to the nonnegative integer [3Xn[0m using [10XSetInfoLevel( InfoAtlasRep, [3Xn[0m[10X )[0m. Information about files being read can be obtained by setting the value of the global variable [10XInfoRead1[0m to [2XPrint[0m ([14XReference: Print[0m). [1X5.1-1 InfoAtlasRep[0m [2X> InfoAtlasRep____________________________________________________[0Xinfo class If the info level of [2XInfoAtlasRep[0m is at least 1 then information about [9Xfail[0m results of functions in the [5XAtlasRep[0m 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. [1X5.1-2 InfoCMeatAxe[0m [2X> InfoCMeatAxe____________________________________________________[0Xinfo class If the info level of [2XInfoCMeatAxe[0m is at least 1 then information about [9Xfail[0m results of [10XC[0m-[5XMeatAxe[0m functions is printed. The default level is zero, no information is printed on this level. [1X5.1-3 InfoBBox[0m [2X> InfoBBox________________________________________________________[0Xinfo class If the info level of [2XInfoBBox[0m is at least 1 then information about [9Xfail[0m results of functions dealing with black box programs (see Section [14X4.2[0m) is printed. The default level is 0, no information is printed on this level. [1X5.1-4 CMeatAxe.FastRead[0m [2X> CMeatAxe.FastRead__________________________________________[0Xglobal variable If this component is bound and has the value [9Xtrue[0m then [2XScanMeatAxeFile[0m ([14X5.3-1[0m) reads text files via [2XStringFile[0m ([14XGAPDoc: StringFile[0m). Otherwise each file containing a matrix over a finite field is read line by line via [2XReadLine[0m ([14XReference: ReadLine[0m), and the [5XGAP[0m matrix is constructed line by line, in a compressed representation (see [14X'Reference: Row Vectors over Finite Fields'[0m and [14X'Reference: Matrices over Finite Fields'[0m), which makes it possible to read large matrices in a reasonable amount of space. The [2XStringFile[0m ([14XGAPDoc: StringFile[0m) approach is faster but needs more intermediate space when text files containing matrices over finite fields are read. [1X5.1-5 AtlasOfGroupRepresentationsInfo[0m [2X> AtlasOfGroupRepresentationsInfo____________________________[0Xglobal variable This is a record that is defined in the file [11Xgap/types.g[0m of the package, with the following components. Components corresponding to [13Xuser parameters[0m (see Section [14X1.7[0m) are [8X[10Xremote[0m[8X[0m a boolean that controls what files are available; if the value is [9Xtrue[0m then [5XGAP[0m is allowed to try remotely accessing any [5XATLAS[0m file from the servers (see below) and thus all files listed in the global table of contents are available, if the value is [9Xfalse[0m then [5XGAP[0m may access only those files that are stored in the database directories of the local [5XGAP[0m installation (see Section [14X1.7-1[0m), [8X[10Xservers[0m[8X[0m a list of pairs [10X[ [0m[3Xserver[0m[10X, [0m[3Xpath[0m[10X ][0m, where [3Xserver[0m is a string denoting the [11Xhttp[0m address of a server where files can be fetched that are not stored in the local database, and [3Xpath[0m is a string describing the path where the data directories on the server reside, [8X[10Xwget[0m[8X[0m a boolean that controls whether the [5XGAP[0m package [5XIO[0m[Neu07] or the external program [11Xwget[0m is used to fetch data files, see [14X1.7-3[0m, [8X[10Xcompress[0m[8X[0m a boolean that controls whether [5XMeatAxe[0m format text files are stored in compressed form; if the value is [9Xtrue[0m then these files are compressed with [11Xgzip[0m after they have been fetched from a server, see Section [14X1.7-4[0m, [8X[10XdisplayFunction[0m[8X[0m the function that is used by [2XDisplayAtlasInfo[0m ([14X2.5-1[0m) for printing the formatted data, see Section [14X1.7-5[0m, [8X[10XaccessFunctions[0m[8X[0m a list of records, each describing how to access the data files, see Sections [14X1.7-6[0m and [14X5.2[0m. [8X[10Xmarkprivate[0m[8X[0m a string used in [2XDisplayAtlasInfo[0m ([14X2.5-1[0m) to mark private data, see Section [14X3.2[0m, and [13XSystem components[0m (which are computed automatically) are [8X[10XGAPnames[0m[8X[0m a list of pairs, each containing the [5XGAP[0m name and the [5XATLAS[0m-file name of a group, see Section [14X2.2[0m, [8X[10Xgroupnames[0m[8X[0m 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 [5XATLAS[0m-file name used for G, see Section [14X5.6[0m, [8X[10Xringinfo[0m[8X[0m 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; [2XDisplayAtlasInfo[0m ([14X2.5-1[0m) displays this information for example for representations over proper extensions of the rational number field only if the representation is mentioned in the [10Xringinfo[0m list, [8X[10Xprivate[0m[8X[0m a list of pairs of strings used for administrating private data (see Chapter [14X3[0m); the value is changed by [2XAtlasOfGroupRepresentationsNotifyPrivateDirectory[0m ([14X3.1-1[0m) and [2XAtlasOfGroupRepresentationsForgetPrivateDirectory[0m ([14X3.1-2[0m), [8X[10XTableOfContents[0m[8X[0m a record with at most the components [10Xlocal[0m, [10Xremote[0m, [10Xtypes[0m, and the names of private data directories. The values of the components [10Xlocal[0m and [10Xremote[0m can be computed automatically by [2XReloadAtlasTableOfContents[0m ([14X1.6-1[0m), the value of the component [10Xtypes[0m is set in [2XAGRDeclareDataType[0m ([14X5.5-1[0m), and the values of the components for local data directories are created by [2XAtlasOfGroupRepresentationsNotifyPrivateDirectory[0m ([14X3.1-1[0m). [1X5.2 How to Customize the Access to Data files[0X We discuss the three steps listed in Section [14X1.7-6[0m. 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 [10XaccessFunctions[0m component of [2XAtlasOfGroupRepresentationsInfo[0m ([14X5.1-5[0m), the components of which are as follows. [8X [10Xlocation( [3Xfilename[0m[8X[10X, [3Xgroupname[0m[8X[10X, [3Xdirname[0m[8X[10X, [3Xtype[0m[8X[10X )[0m[8X [0m Let [3Xfilename[0m be the default filename (without path) of the required file, or a list of such filenames. Let [3Xgroupname[0m be the [5XATLAS[0m name of the group to which the data in these files belong, [3Xdirname[0m be the default directory name (one of [10X"datagens"[0m, [10X"dataword"[0m, or the [3Xdirid[0m value of a private directory, see [2XAtlasOfGroupRepresentationsNotifyPrivateDirectory[0m ([14X3.1-1[0m)), and [3Xtype[0m be the data type (see [2XAGRDeclareDataType[0m ([14X5.5-1[0m)). 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 [9Xfail[0m if this function does not feel responsible for these file(s). In the latter case, the [10Xlocation[0m function in another record will know a path. The file(s) is/are regarded as not locally available if all installed [10Xlocation[0m functions return either [9Xfail[0m or paths of nonexisting files, in the sense of [2XIsExistingFile[0m ([14XReference: IsExistingFile[0m). [8X [10Xfetch( [3Xfilepath[0m[8X[10X, [3Xfilename[0m[8X[10X, [3Xgroupname[0m[8X[10X, [3Xdirname[0m[8X[10X, [3Xtype[0m[8X[10X )[0m[8X [0m This function is called when a file is not locally available and if the [10Xlocation[0m function in the current record has returned a path or a list of paths. The arguments [3Xdirname[0m and [3Xtype[0m must be the same as for the [10Xlocation[0m function, and [3Xfilepath[0m and [3Xfilename[0m must be strings ([13Xnot[0m lists of strings). The return value must be [9Xtrue[0m if the function succeeded with making the file locally available (including postprocessing if applicable), and [9Xfalse[0m otherwise. [8X[10Xcontents( [3Xfilepath[0m[8X[10X, [3Xtype[0m[8X[10X )[0m[8X[0m This function is called when the [10Xlocation[0m function in the current record has returned the path(s) [3Xfilepath[0m, and if either these are paths of existing files or the [10Xfetch[0m function in the current record has been called for these paths, and the return value was [9Xtrue[0m. The argument [3Xtype[0m must be the same as for the [10Xlocation[0m and the [10Xfetch[0m functions. The return value must be the contents of the file(s), in the sense that the [5XGAP[0m 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. [8X[10Xdescription[0m[8X[0m 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 [2XAtlasOfGroupRepresentationsShowUserParameters[0m ([14X1.7-8[0m) is called. [8X[10Xactive[0m[8X[0m The current [10XaccessFunctions[0m record is ignored by [2XAGRFileContents[0m ([14X5.6-2[0m) if the value is not [9Xtrue[0m. In [2XAGRFileContents[0m ([14X5.6-2[0m), the records in the [10XaccessFunctions[0m component of [2XAtlasOfGroupRepresentationsInfo[0m ([14X5.1-5[0m) are considered in reversed order. By default, the [10XaccessFunctions[0m list contains three records. Only for one of them, the [10Xactive[0m component has the value [9Xtrue[0m. 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 [5XMeatAxe[0m binary files are transferred and read instead of [5XMeatAxe[0m 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. [1X5.3 Reading and Writing MeatAxe Format Files[0X [1X5.3-1 ScanMeatAxeFile[0m [2X> ScanMeatAxeFile( [0X[3Xfilename[, q][, "string"][0X[2X ) _____________________[0Xfunction [6XReturns:[0X the matrix or list of permutations stored in the file or encoded by the string. Let [3Xfilename[0m be the name of a [5XGAP[0m readable file (see [14X'Reference: Filename'[0m) that contains a matrix or a permutation or a list of permutations in [5XMeatAxe[0m text format (see the section about the program [11Xzcv[0m in the [5XMeatAxe[0m manual [Rin98]), and let [3Xq[0m be a prime power. [2XScanMeatAxeFile[0m returns the corresponding [5XGAP[0m matrix or list of permutations, respectively. If the file contains a matrix then the way how it is read by [2XScanMeatAxeFile[0m depends on the value of the global variable [2XCMeatAxe.FastRead[0m ([14X5.1-4[0m). If the parameter [3Xq[0m is given then the result matrix is represented over the field with [3Xq[0m elements, the default for [3Xq[0m is the field size stored in the file. If the file contains a list of permutations then it is read with [2XStringFile[0m ([14XGAPDoc: StringFile[0m); the parameter [3Xq[0m, if given, is ignored in this case. If the string [10X"string"[0m is entered as the third argument then the first argument must be a string as obtained by reading a file in [5XMeatAxe[0m text format as a text stream (see [2XInputTextFile[0m ([14XReference: InputTextFile[0m)). Also in this case, [2XScanMeatAxeFile[0m returns the corresponding [5XGAP[0m matrix or list of permutations, respectively. [1X5.3-2 MeatAxeString[0m [2X> MeatAxeString( [0X[3Xmat, q[0X[2X ) _________________________________________[0Xoperation [2X> MeatAxeString( [0X[3Xperms, degree[0X[2X ) __________________________________[0Xoperation [2X> MeatAxeString( [0X[3Xperm, q, dims[0X[2X ) __________________________________[0Xoperation [6XReturns:[0X a string encoding the [5XGAP[0m objects given as input in [5XMeatAxe[0m format. In the first form, for a matrix [3Xmat[0m whose entries lie in the finite field with [3Xq[0m elements, [2XMeatAxeString[0m returns a string that encodes [3Xmat[0m as a matrix over [10XGF([3Xq[0m[10X)[0m, in [5XMeatAxe[0m text format. In the second form, for a nonempty list [3Xperms[0m of permutations that move only points up to the positive integer [3Xdegree[0m, [2XMeatAxeString[0m returns a string that encodes [3Xperms[0m as permutations of degree [3Xdegree[0m, in [5XMeatAxe[0m text format (see [Rin98]). In the third form, for a permutation [3Xperm[0m with largest moved point n, say, a prime power [3Xq[0m, and a list [3Xdims[0m of length 2 containing two positive integers larger than or equal to n, [2XMeatAxeString[0m returns a string that encodes [3Xperm[0m as a matrix over [10XGF([3Xq[0m[10X)[0m, of dimensions [3Xdims[0m, whose first n rows and columns describe the permutation matrix corresponding to [3Xperm[0m, and the remaining rows and columns are zero. When strings are printed to files using [2XPrintTo[0m ([14XReference: PrintTo[0m) or [2XAppendTo[0m ([14XReference: AppendTo[0m) then line breaks are inserted whenever lines exceed the number of characters given by the second entry of the list returned by [2XSizeScreen[0m ([14XReference: SizeScreen[0m), see [14X'Reference: Operations for Output Streams'[0m. This behaviour is not desirable for creating data files. So the recommended functions for printing the result of [2XMeatAxeString[0m to a file are [2XFileString[0m ([14XGAPDoc: FileString[0m) and [2XWriteAll[0m ([14XReference: WriteAll[0m). [4X--------------------------- Example ----------------------------[0X [4Xgap> mat:= [ [ 1, -1 ], [ 0, 1 ] ] * Z(3)^0;;[0X [4Xgap> str:= MeatAxeString( mat, 3 );[0X [4X"1 3 2 2\n12\n01\n"[0X [4Xgap> mat = ScanMeatAxeFile( str, "string" );[0X [4Xtrue[0X [4Xgap> str:= MeatAxeString( mat, 9 );[0X [4X"1 9 2 2\n12\n01\n"[0X [4Xgap> mat = ScanMeatAxeFile( str, "string" );[0X [4Xtrue[0X [4Xgap> perms:= [ (1,2,3)(5,6) ];;[0X [4Xgap> str:= MeatAxeString( perms, 6 );[0X [4X"12 1 6 1\n2\n3\n1\n4\n6\n5\n"[0X [4Xgap> perms = ScanMeatAxeFile( str, "string" );[0X [4Xtrue[0X [4Xgap> str:= MeatAxeString( perms, 8 );[0X [4X"12 1 8 1\n2\n3\n1\n4\n6\n5\n7\n8\n"[0X [4Xgap> perms = ScanMeatAxeFile( str, "string" );[0X [4Xtrue[0X [4Xgap> perm:= (1,2,4);;[0X [4Xgap> str:= MeatAxeString( perm, 3, [ 5, 6 ] );[0X [4X"2 3 5 6\n2\n4\n3\n1\n5\n"[0X [4Xgap> mat:= ScanMeatAxeFile( str, "string" );; Print( mat, "\n" );[0X [4X[ [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [0X [4X [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ], [0X [4X [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [0X [4X [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], [0X [4X [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ] ][0X [4Xgap> MeatAxeString( mat, 3 ) = str;[0X [4Xtrue[0X [4X------------------------------------------------------------------[0X [1X5.3-3 FFList[0m [2X> FFList( [0X[3XF[0X[2X ) ______________________________________________________[0Xfunction [6XReturns:[0X a list of elements in the given finite field. [2X> FFLists____________________________________________________[0Xglobal variable [2XFFList[0m is a utility program for the conversion of vectors and matrices from [5XMeatAxe[0m format to [5XGAP[0m format and vice versa. It is used by [2XScanMeatAxeFile[0m ([14X5.3-1[0m) and [2XMeatAxeString[0m ([14X5.3-2[0m). For a finite field [3XF[0m, [2XFFList[0m returns a list [3Xl[0m giving the correspondence between the [5XMeatAxe[0m numbering and the [5XGAP[0m numbering of the elements in [3XF[0m. The element of [3XF[0m corresponding to [5XMeatAxe[0m number [3Xn[0m is [3Xl[0m[ [3Xn[0m+1 ], and the [5XMeatAxe[0m number of the field element [3Xz[0m is [10XPosition( [0m[3Xl[0m[10X, [0m[3Xz[0m[10X ) - 1[0m. The global variable [2XFFLists[0m is used to store the information about [3XF[0m once it has been computed. [4X--------------------------- Example ----------------------------[0X [4Xgap> FFList( GF(4) );[0X [4X[ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2 ][0X [4Xgap> IsBound( FFLists[4] );[0X [4Xtrue[0X [4X------------------------------------------------------------------[0X [1X5.3-4 CMtxBinaryFFMatOrPerm[0m [2X> CMtxBinaryFFMatOrPerm( [0X[3Xelm, def, outfile[0X[2X ) _______________________[0Xfunction Let the pair ([3Xelm[0m, [3Xdef[0m) 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 [3Xoutfile[0m be a string. [2XCMtxBinaryFFMatOrPerm[0m writes the [10XC[0m-[5XMeatAxe[0m 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 [3Xoutfile[0m. (The binary format is described in the [10XC[0m-[5XMeatAxe[0m manual [Rin98].) [4X--------------------------- Example ----------------------------[0X [4Xgap> tmpdir:= DirectoryTemporary();;[0X [4Xgap> mat:= Filename( tmpdir, "mat" );;[0X [4Xgap> q:= 4;;[0X [4Xgap> mats:= GeneratorsOfGroup( GL(10,q) );;[0X [4Xgap> CMtxBinaryFFMatOrPerm( mats[1], q, Concatenation( mat, "1" ) );[0X [4Xgap> CMtxBinaryFFMatOrPerm( mats[2], q, Concatenation( mat, "2" ) );[0X [4Xgap> prm:= Filename( tmpdir, "prm" );;[0X [4Xgap> n:= 200;;[0X [4Xgap> perms:= GeneratorsOfGroup( SymmetricGroup( n ) );;[0X [4Xgap> CMtxBinaryFFMatOrPerm( perms[1], n, Concatenation( prm, "1" ) );[0X [4Xgap> CMtxBinaryFFMatOrPerm( perms[2], n, Concatenation( prm, "2" ) );[0X [4X------------------------------------------------------------------[0X [1X5.3-5 FFMatOrPermCMtxBinary[0m [2X> FFMatOrPermCMtxBinary( [0X[3Xfname[0X[2X ) ___________________________________[0Xfunction [6XReturns:[0X the matrix or permutation stored in the file. Let [3Xfname[0m be the name of a file that contains the [10XC[0m-[5XMeatAxe[0m binary format of a matrix over a finite field or of a permutation, as is described in [Rin98]. [2XFFMatOrPermCMtxBinary[0m returns the corresponding [5XGAP[0m matrix or permutation. [4X--------------------------- Example ----------------------------[0X [4Xgap> FFMatOrPermCMtxBinary( Concatenation( mat, "1" ) ) = mats[1];[0X [4Xtrue[0X [4Xgap> FFMatOrPermCMtxBinary( Concatenation( mat, "2" ) ) = mats[2];[0X [4Xtrue[0X [4Xgap> FFMatOrPermCMtxBinary( Concatenation( prm, "1" ) ) = perms[1];[0X [4Xtrue[0X [4Xgap> FFMatOrPermCMtxBinary( Concatenation( prm, "2" ) ) = perms[2];[0X [4Xtrue[0X [4X------------------------------------------------------------------[0X [1X5.4 Reading and Writing [5XATLAS[1X Straight Line Programs[0X [1X5.4-1 ScanStraightLineProgram[0m [2X> ScanStraightLineProgram( [0X[3Xfilename[, "string"][0X[2X ) __________________[0Xfunction [6XReturns:[0X a record containing the straight line program. Let [3Xfilename[0m 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. [8X[10X#[3Xanything[0m[8X[10X[0m[8X[0m lines starting with a hash sign [10X#[0m are ignored, [8X[10Xecho [3Xanything[0m[8X[10X[0m[8X[0m lines starting with [10Xecho[0m are ignored for the [10Xprogram[0m 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, [8X[10Xinp [3Xn[0m[8X[10X[0m[8X[0m means that there are [3Xn[0m inputs, referred to via the labels [10X1[0m, [10X2[0m, ..., [3Xn[0m, [8X[10Xinp [3Xk[0m[8X[10X [3Xa1[0m[8X[10X [3Xa2[0m[8X[10X ... [3Xak[0m[8X[10X[0m[8X[0m means that the next [3Xk[0m inputs are referred to via the labels [3Xa1[0m, [3Xa2[0m, ..., [3Xak[0m, [8X[10Xcjr [3Xa[0m[8X[10X [3Xb[0m[8X[10X[0m[8X[0m means that [3Xa[0m is replaced by [10X[3Xb[0m[10X^(-1) * [3Xa[0m[10X * [3Xb[0m[10X[0m, [8X[10Xcj [3Xa[0m[8X[10X [3Xb[0m[8X[10X [3Xc[0m[8X[10X[0m[8X[0m means that [3Xc[0m is defined as [10X[3Xb[0m[10X^(-1) * [3Xa[0m[10X * [3Xb[0m[10X[0m, [8X[10Xcom [3Xa[0m[8X[10X [3Xb[0m[8X[10X [3Xc[0m[8X[10X[0m[8X[0m means that [3Xc[0m is defined as [10X[3Xa[0m[10X^(-1) * [3Xb[0m[10X^(-1) * [3Xa[0m[10X * [3Xb[0m[10X[0m, [8X[10Xiv [3Xa[0m[8X[10X [3Xb[0m[8X[10X[0m[8X[0m means that [3Xb[0m is defined as [10X[3Xa[0m[10X^(-1)[0m, [8X[10Xmu [3Xa[0m[8X[10X [3Xb[0m[8X[10X [3Xc[0m[8X[10X[0m[8X[0m means that [3Xc[0m is defined as [10X[3Xa[0m[10X * [3Xb[0m[10X[0m, [8X[10Xpwr [3Xa[0m[8X[10X [3Xb[0m[8X[10X [3Xc[0m[8X[10X[0m[8X[0m means that [3Xc[0m is defined as [10X[3Xb[0m[10X^[3Xa[0m[10X[0m, [8X[10Xcp [3Xa[0m[8X[10X [3Xb[0m[8X[10X[0m[8X[0m means that [3Xb[0m is defined as a copy of [3Xa[0m, [8X[10Xoup [3Xl[0m[8X[10X[0m[8X[0m means that there are [3Xl[0m outputs, stored in the labels [10X1[0m, [10X2[0m, ..., [3Xl[0m, and [8X[10Xoup [3Xl[0m[8X[10X [3Xb1[0m[8X[10X [3Xb2[0m[8X[10X ... [3Xbl[0m[8X[10X[0m[8X[0m means that the next [3Xl[0m outputs are stored in the labels [3Xb1[0m, [3Xb2[0m, ... [3Xbl[0m. Each of the labels [3Xa[0m, [3Xb[0m, [3Xc[0m can be any nonempty sequence of digits and alphabet characters, except that the first argument of [10Xpwr[0m must denote an integer. If the [10Xinp[0m or [10Xoup[0m statements are missing then the input or output, respectively, is assumed to be given by the labels [10X1[0m and [10X2[0m. There can be multiple [10Xinp[0m lines at the beginning of the program and multiple [10Xoup[0m lines at the end of the program. Only the first [10Xinp[0m or [10Xoup[0m line may omit the names of the elements. For example, an empty file [3Xfilename[0m or an empty string [3Xstring[0m represent a straight line program with two inputs that are returned as outputs. No command except [10Xcjr[0m may overwrite its own input. For example, the line [10Xmu a b a[0m is not legal. (This is not checked.) [2XScanStraightLineProgram[0m returns a record containing as the value of its component [10Xprogram[0m the corresponding [5XGAP[0m straight line program (see [2XIsStraightLineProgram[0m ([14XReference: IsStraightLineProgram[0m)) if the input string satisfies the syntax rules stated above, and returns [9Xfail[0m otherwise. In the latter case, information about the first corrupted line of the program is printed if the info level of [2XInfoCMeatAxe[0m ([14X5.1-2[0m) is at least 1. If the string [10X"string"[0m is entered as the second argument then the first argument must be a string as obtained by reading a file in [5XMeatAxe[0m text format as a text stream (see [2XInputTextFile[0m ([14XReference: InputTextFile[0m)). Also in this case, [2XScanStraightLineProgram[0m returns either a record with the corresponding [5XGAP[0m straight line program or [9Xfail[0m. 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 [10Xoutputs[0m. 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 [14X2.4[0m 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. [4X--------------------------- Example ----------------------------[0X [4Xecho "Classes 1A 2A 3A 5A 5B"[0X [4Xoup 5 3 1 2 4 5[0X [4X------------------------------------------------------------------[0X This example means that the list of outputs of the program contains elements of the classes [10X1A[0m, [10X2A[0m, [10X3A[0m, [10X5A[0m, and [10X5B[0m (in this order), and that inside the program, these elements are referred to by the names [10X3[0m, [10X1[0m, [10X2[0m, [10X4[0m, and [10X5[0m. [1X5.4-2 AtlasStringOfProgram[0m [2X> AtlasStringOfProgram( [0X[3Xprog[, outputnames][0X[2X ) ______________________[0Xfunction [2X> AtlasStringOfProgram( [0X[3Xprog[, "mtx"][0X[2X ) ____________________________[0Xfunction [6XReturns:[0X a string encoding the straight line program/decision in the format used in [5XATLAS[0m files. For a straight line program or straight line decision [3Xprog[0m (see [2XIsStraightLineProgram[0m ([14XReference: IsStraightLineProgram[0m) and [2XIsStraightLineDecision[0m ([14X4.1-1[0m)), this function returns a string describing the input format of an equivalent straight line program or straight line decision as used in the [5XATLAS[0m of Group Representations, that is, the lines are of the form described in [2XScanStraightLineProgram[0m ([14X5.4-1[0m). A list of strings that is given as the optional second argument [3Xoutputnames[0m is interpreted as the class names corresponding to the outputs; this argument has the effect that appropriate [10Xecho[0m statements appear in the result string. If the string [10X"mtx"[0m is given as the second argument then the result has the format used in the [10XC[0m-[5XMeatAxe[0m (see [Rin98]) rather than the format described in Section [14X5.4[0m. (Note that the [10XC[0m-[5XMeatAxe[0m format does not make sense if the argument [3Xoutputnames[0m is given, and that this format does not support [10Xinp[0m and [10Xoup[0m statements.) The argument [3Xprog[0m must not be a black box program (see [2XIsBBoxProgram[0m ([14X4.2-1[0m)). [4X--------------------------- Example ----------------------------[0X [4Xgap> str:= "inp 2\nmu 1 2 3\nmu 3 1 2\niv 2 1\noup 2 1 2";;[0X [4Xgap> prg:= ScanStraightLineProgram( str, "string" );[0X [4Xrec( program := <straight line program> )[0X [4Xgap> prg:= prg.program;;[0X [4Xgap> Display( prg );[0X [4X# input:[0X [4Xr:= [ g1, g2 ];[0X [4X# program:[0X [4Xr[3]:= r[1]*r[2];[0X [4Xr[2]:= r[3]*r[1];[0X [4Xr[1]:= r[2]^-1;[0X [4X# return values:[0X [4X[ r[1], r[2] ][0X [4Xgap> StringOfResultOfStraightLineProgram( prg, [ "a", "b" ] );[0X [4X"[ (aba)^-1, aba ]"[0X [4Xgap> AtlasStringOfProgram( prg );[0X [4X"inp 2\nmu 1 2 3\nmu 3 1 2\niv 2 1\noup 2\n"[0X [4Xgap> prg:= StraightLineProgram( "(a^2b^3)^-1", [ "a", "b" ] );[0X [4X<straight line program>[0X [4Xgap> Print( AtlasStringOfProgram( prg ) );[0X [4Xinp 2[0X [4Xpwr 2 1 4[0X [4Xpwr 3 2 5[0X [4Xmu 4 5 3[0X [4Xiv 3 4[0X [4Xoup 1 4[0X [4Xgap> prg:= StraightLineProgram( [ [2,3], [ [3,1,1,4], [1,2,3,1] ] ], 2 );[0X [4X<straight line program>[0X [4Xgap> Print( AtlasStringOfProgram( prg ) );[0X [4Xinp 2[0X [4Xpwr 3 2 3[0X [4Xpwr 4 1 5[0X [4Xmu 3 5 4[0X [4Xpwr 2 1 6[0X [4Xmu 6 3 5[0X [4Xoup 2 4 5[0X [4Xgap> Print( AtlasStringOfProgram( prg, "mtx" ) );[0X [4X# inputs are expected in 1 2[0X [4Xzsm pwr3 2 3[0X [4Xzsm pwr4 1 5[0X [4Xzmu 3 5 4[0X [4Xzsm pwr2 1 6[0X [4Xzmu 6 3 5[0X [4Xecho "outputs are in 4 5"[0X [4Xgap> str:= "inp 2\nchor 1 2\nchor 2 3\nmu 1 2 3\nchor 3 5";;[0X [4Xgap> prg:= ScanStraightLineDecision( str );;[0X [4Xgap> AtlasStringOfProgram( prg.program );[0X [4X"inp 2\nchor 1 2\nchor 2 3\nmu 1 2 3\nchor 3 5\n"[0X [4X------------------------------------------------------------------[0X [1X5.5 Data Types Used in the [5XATLAS[1X of Group Representations[0X Each representation or program that is administrated by the [5XAtlasRep[0m package belongs to a unique [13Xdata type[0m. 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 [2XDisplayAtlasInfo[0m ([14X2.5-1[0m) when called without arguments or with only argument a list of group names, -- a line format of its own for the output produced by [2XDisplayAtlasInfo[0m ([14X2.5-1[0m) when called with first argument a group name, -- an input format of its own for [2XAtlasProgram[0m ([14X2.5-3[0m), -- an input format of its own for [2XOneAtlasGeneratingSetInfo[0m ([14X2.5-4[0m), and -- specific tests for the data of this data type; these functions are used by the global tests described in Section [14X5.8[0m. Formally, a data type is defined by a record whose components are used by the interface functions. The details are described in the following. [1X5.5-1 AGRDeclareDataType[0m [2X> AGRDeclareDataType( [0X[3Xkind, name, record[0X[2X ) _________________________[0Xfunction Let [3Xkind[0m be one of the strings [10X"rep"[0m or [10X"prg"[0m, and [3Xrecord[0m be a record. [2XAGRDeclareDataType[0m declares a new data type of representations (if [3Xkind[0m is [10X"rep"[0m) or of programs (if [3Xkind[0m is [10X"prg"[0m). For each group used in the [5XAtlasRep[0m package, the record that contains the information about the data will have a component [3Xname[0m whose value is a list containing the data about the new type. Examples of [3Xname[0m are [10X"perm"[0m, [10X"matff"[0m, and [10X"classes"[0m. [13XMandatory components[0m of [3Xrecord[0m are [8X[10XFilenameFormat[0m[8X[0m 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 [2XAGRParseFilenameFormat[0m ([14X5.6-1[0m), 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.) [8X[10XAddFileInfo[0m[8X[0m 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 [2XAGRParseFilenameFormat[0m ([14X5.6-1[0m) for the given type, and a filename). This function adds the necessary parts of the data entry to the list, and returns [9Xtrue[0m if the data belongs to the type, otherwise [9Xfalse[0m 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). [8X[10XReadAndInterpretDefault[0m[8X[0m This is the function that does the work for the default [10Xcontents[0m value of the [10XaccessFunctions[0m component of [2XAtlasOfGroupRepresentationsInfo[0m ([14X5.1-5[0m), see Section [14X5.2[0m. This function must take a path and return the [5XGAP[0m object given by this file. [8X[10XAddDescribingComponents[0m[8X (for [10Xrep[0m[8X only)[0m This function takes two arguments, a record (that will be returned by [2XAtlasGenerators[0m ([14X2.5-2[0m), [2XOneAtlasGeneratingSetInfo[0m ([14X2.5-4[0m), or [2XAllAtlasGeneratingSetInfos[0m ([14X2.5-5[0m)) and the type record [3Xrecord[0m. It sets the components [10Xp[0m, [10Xdim[0m, [10Xid[0m, and [10Xring[0m that are promised for return values of the abovementioned three functions. [8X[10XDisplayGroup[0m[8X (for [10Xrep[0m[8X only)[0m This defines the format of the lines printed by [2XDisplayAtlasInfo[0m ([14X2.5-1[0m) for a given group. The value must be a function that takes a list as returned by the function given in the component [10XAddFileInfo[0m, and returns the string to be printed for the representation in question. [13XOptional components[0m of [3Xrecord[0m are [8X[10XDisplayOverviewInfo[0m[8X[0m This is used to introduce a new column in the output of [2XDisplayAtlasInfo[0m ([14X2.5-1[0m) 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 [10X"r"[0m or [10X"l"[0m, 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 [5XAtlasRep[0m package and a group name), and returns a list of length two, containing the string to be printed as the column value and [9Xtrue[0m or [9Xfalse[0m, depending on whether private data is involved or not. (The default is to print no column for the data type.) [8X[10XDisplayPRG[0m[8X (for [10Xprg[0m[8X only)[0m This is used in [2XDisplayAtlasInfo[0m ([14X2.5-1[0m) for [5XATLAS[0m 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 [9Xtrue[0m 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.) [8X[10XAccessGroupCondition[0m[8X (for [10Xrep[0m[8X only)[0m This is used in [2XDisplayAtlasInfo[0m ([14X2.5-1[0m) and [2XOneAtlasGeneratingSetInfo[0m ([14X2.5-4[0m). The value must be a function that takes two arguments (a list as returned by [2XAGRParseFilenameFormat[0m ([14X5.6-1[0m), and a list of conditions), and returns [9Xtrue[0m or [9Xfalse[0m, depending on whether the first argument satisfies the conditions. (The default value is [2XReturnFalse[0m ([14XReference: ReturnFalse[0m).) The function must support conditions such as [10X[ IsPermGroup, true ][0m and [10X[ NrMovedPoints, [ 5, 6 ] ][0m, in general a list of functions followed by a prescribed value, a list of prescribed values, another (unary) function, or the string [10X"minimal"[0m. For an overview of the interesting functions, see [2XDisplayAtlasInfo[0m ([14X2.5-1[0m). [8X[10XAccessPRG[0m[8X (for [10Xprg[0m[8X only)[0m This is used in [2XAtlasProgram[0m ([14X2.5-3[0m). 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 [9Xtrue[0m for the required standardization, and a list of conditions given by the optional arguments of [2XAtlasProgram[0m ([14X2.5-3[0m)), and returns either [9Xfail[0m or a list that together with the group name forms the identifier of a program that matches the conditions. (The default value is [2XReturnFail[0m ([14XReference: ReturnFail[0m).) [8X[10XAtlasProgram[0m[8X (for [10Xprg[0m[8X only)[0m This is used in [2XAtlasProgram[0m ([14X2.5-3[0m) to create the result value from the identifier. (The default value is [10XAtlasProgramDefault[0m, 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.) [8X[10XTOCEntryString[0m[8X[0m This is used in [2XStoreAtlasTableOfContents[0m ([14X1.6-2[0m). The value must be a function that takes two arguments (the name [3Xname[0m of the type and a list as returned by [2XAGRParseFilenameFormat[0m ([14X5.6-1[0m) and returns a string that describes the appropriate function call. (The default value is [10XTOCEntryStringDefault[0m.) [8X[10XPostprocessFileInfo[0m[8X[0m This is used in the construction of a table of contents via [2XReloadAtlasTableOfContents[0m ([14X1.6-1[0m), 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.) [8X[10XSortTOCEntries[0m[8X[0m This is used in the construction of a table of contents (see [2XReloadAtlasTableOfContents[0m ([14X1.6-1[0m)), for sorting the entries after they have been added and after the value of the component [10XPostprocessFileInfo[0m has been called. The value must be a function that takes a list as returned by [2XAGRParseFilenameFormat[0m ([14X5.6-1[0m), and returns the sorting key. (There is no default value, which means that no sorting is needed.) [8X[10XTestFileHeaders[0m[8X (for [10Xrep[0m[8X only)[0m This is used in the function [2XAtlasOfGroupRepresentationsTestFileHeaders[0m ([14X5.8-5[0m). The value must be a function that takes the same four arguments as [2XAGRFileContents[0m ([14X5.6-2[0m), except that the first argument [10X"datagens"[0m can be replaced by [10X"local"[0m and that the third argument is a list as returned by [2XAGRParseFilenameFormat[0m ([14X5.6-1[0m). (The default value is [2XReturnTrue[0m ([14XReference: ReturnTrue[0m).) [8X[10XTestFiles[0m[8X (for [10Xrep[0m[8X only)[0m This is used in the function [2XAtlasOfGroupRepresentationsTestFiles[0m ([14X5.8-7[0m). The format of the value and the default are the same as for the value of the component [10XTestFileHeaders[0m. [8X[10XTestWords[0m[8X (for [10Xprg[0m[8X only)[0m This is used in the function [2XAtlasOfGroupRepresentationsTestWords[0m ([14X5.8-6[0m). The value must be a function that takes five arguments where the first four are the same arguments as for [2XAGRFileContents[0m ([14X5.6-2[0m), except that the first argument [10X"dataword"[0m can be replaced by [10X"local"[0m, and the fifth argument is [9Xtrue[0m or [9Xfalse[0m, indicating verbose mode or not. [1X5.6 Filenames Used in the [5XATLAS[1X of Group Representations[0X The data of each local [5XGAP[0m version of the [5XATLAS[0m of Group Representations is either private (see Chapter [14X3[0m) or is stored in the two directories [11Xdatagens[0m and [11Xdataword[0m. In the following, we describe the format of filenames in the latter two directories, as a reference of the "official" part of the [5XATLAS[0m. In the directory [11Xdatagens[0m, the generators for the [13Xrepresentations[0m available are stored, the directory [11Xdataword[0m contains the [13Xprograms[0m 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 [14X2.3[0m). The name of each data file in the [5XATLAS[0m 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 [10X-[0m. The first part is always of the form [3Xgroupname[0m[10XG[0m[3Xi[0m, where the integer [3Xi[0m denotes the [3Xi[0m-th set of standard generators for the group G, say, with [5XATLAS[0m-file name [3Xgroupname[0m (see [14X2.2[0m). The translations of the name [3Xgroupname[0m to the name(s) used within [5XGAP[0m is given by the component [10XGAPnames[0m of [2XAtlasOfGroupRepresentationsInfo[0m ([14X5.1-5[0m). The filenames in the directory [11Xdataword[0m have one of the following forms. In each of these cases, the suffix [10XW[0m[3Xn[0m means that [3Xn[0m is the version number of the program. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-cycW[3Xn[0m[8X[10X[0m[8X[0m 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 [10XCo1G1-cycW1[0m. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-cclsW[3Xn[0m[8X[10X[0m[8X[0m In this case, the file contains a straight line program that returns a list of conjugacy class representatives of G. An example is [10XRuG1-cclsW1[0m. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10XcycW[3Xn[0m[8X[10X-cclsW[3Xm[0m[8X[10X[0m[8X[0m In this case, the file contains a straight line program that takes the return value of the program in the file [3Xgroupname[0m[10XG[0m[3Xi[0m[10X-cycW[0m[3Xn[0m (see above), and returns a list of conjugacy class representatives of G. An example is [10XM11G1cycW1-cclsW1[0m. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-max[3Xk[0m[8X[10XW[3Xn[0m[8X[10X[0m[8X[0m In this case, the file contains a straight line program that takes generators of G w.r.t. the [3Xi[0m-th set of standard generators, and returns a list of generators (in general [13Xnot[0m standard generators) for a subgroup U in the [3Xk[0m-th class of maximal subgroups of G. An example is [10XJ1G1-max7W1[0m. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10Xmax[3Xk[0m[8X[10XW[3Xn[0m[8X[10X-[3Xsubgroupname[0m[8X[10XG[3Xj[0m[8X[10XW[3Xm[0m[8X[10X[0m[8X[0m In this case, the file contains a straight line program that takes the return value of the program in the file [3Xgroupname[0m[10XG[0m[3Xi[0m[10X-max[0m[3Xk[0m[10XW[0m[3Xn[0m (see above), which are generators for a group U, say; [3Xsubgroupname[0m is a name for U, and the return value is a list of standard generators for U, w.r.t. the [3Xj[0m-th set of standard generators. (Of course this implies that the groups in the [3Xk[0m-th class of maximal subgroups of G are isomorphic to the group with name [3Xsubgroupname[0m.) An example is [10XJ1G1max1W1-L211G1W1[0m; 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. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-a[3Xoutname[0m[8X[10XW[3Xn[0m[8X[10X[0m[8X[0m In this case, the file contains a straight line program that takes generators of G w.r.t. the [3Xi[0m-th set of standard generators, and returns the list of their images under the outer automorphism alpha of G given by the name [3Xoutname[0m; 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 [10X2_1[0m or [10X2a[0m, [10X4_2[0m or [10X4b[0m, [10X3_3[0m or [10X3c[0m ... 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 [3Xk[0m[10Xp[0m[3Xd[0m, with [3Xk[0m one of the above forms and [3Xd[0m an integer then [3Xd[0m denotes the number of dashes appended to the automorphism described by [3Xk[0m; if [3Xd[0m = 1 then [3Xd[0m can be omitted. Examples are [10XA5G1-aW1[0m, [10XL34G1-a2_1W1[0m, [10XU43G1-a2_3pW1[0m, and [10XO8p3G1-a2_2p5W1[0m; 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. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-G[3Xj[0m[8X[10XW[3Xn[0m[8X[10X[0m[8X[0m In this case, the file contains a straight line program that takes generators of G w.r.t. the [3Xi[0m-th set of standard generators, and returns standard generators of G w.r.t. the [3Xj[0m-th set of standard generators. An example is [10XL35G1-G2W1[0m. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-check[3Xn[0m[8X[10X[0m[8X[0m In this case, the file contains a straight line decision that takes generators of G, and returns [9Xtrue[0m if these generators are standard generators w.r.t. the [3Xi[0m-th standardization, and [9Xfalse[0m otherwise. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-P[3Xn[0m[8X[10X[0m[8X[0m In this case, the file contains a straight line decision that takes some group elements, and returns [9Xtrue[0m if these elements are standard generators for [3XG[0m, w.r.t. the [3Xi[0m-th standardization, and [9Xfalse[0m otherwise. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-find[3Xn[0m[8X[10X[0m[8X[0m 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 [3XG[0m, w.r.t. the [3Xi[0m-th standardization. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-X[3Xdescr[0m[8X[10XW[3Xn[0m[8X[10X[0m[8X[0m In this case, the file contains a straight line program that takes generators of G w.r.t. the [3Xi[0m-th set of standard generators, and whose return value corresponds to [3Xdescr[0m. This format is used only in private extensions (see Chapter [14X3[0m), such a script can be accessed with [3Xdescr[0m as the third argument of [2XAtlasProgram[0m ([14X2.5-3[0m). The filenames in the directory [11Xdatagens[0m have one of the following forms. In each of these cases, [3Xid[0m is a (possibly empty) string that starts with a lowercase alphabet letter (see [2XIsLowerAlphaChar[0m ([14XReference: IsLowerAlphaChar[0m)), and [3Xm[0m is a nonnegative integer, meaning that the generators are written w.r.t. the [3Xm[0m-th basis (the meaning is defined by the [5XATLAS[0m developers). [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-f[3Xq[0m[8X[10Xr[3Xdim[0m[8X[10X[3Xid[0m[8X[10XB[3Xm[0m[8X[10X.m[3Xnr[0m[8X[10X[0m[8X[0m a file in [5XMeatAxe[0m text file format containing the [3Xnr[0m-th generator of a matrix representation over the field with [3Xq[0m elements, of dimension [3Xdim[0m. An example is [10XS5G1-f2r4aB0.m1[0m. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-p[3Xn[0m[8X[10X[3Xid[0m[8X[10XB[3Xm[0m[8X[10X.m[3Xnr[0m[8X[10X[0m[8X[0m a file in [5XMeatAxe[0m text file format containing the [3Xnr[0m-th generator of a permutation representation on [3Xn[0m points. An example is [10XM11G1-p11B0.m1[0m. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-Ar[3Xdim[0m[8X[10X[3Xid[0m[8X[10XB[3Xm[0m[8X[10X.g[0m[8X[0m a [5XGAP[0m readable file containing all generators of a matrix representation of dimension [3Xdim[0m over an algebraic number field not specified further. An example is [10XA5G1-Ar3aB0.g[0m. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-Zr[3Xdim[0m[8X[10X[3Xid[0m[8X[10XB[3Xm[0m[8X[10X.g[0m[8X[0m a [5XGAP[0m readable file containing all generators of a matrix representation over the integers, of dimension [3Xdim[0m. An example is [10XA5G1-Zr4B0.g[0m. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-Hr[3Xdim[0m[8X[10X[3Xid[0m[8X[10XB[3Xm[0m[8X[10X.g[0m[8X[0m a [5XGAP[0m readable file containing all generators of a matrix representation over a quaternion algebra over an algebraic number field, of dimension [3Xdim[0m. An example is [10X2A6G1-Hr2aB0.g[0m. [8X[10X[3Xgroupname[0m[8X[10XG[3Xi[0m[8X[10X-Z[3Xn[0m[8X[10Xr[3Xdim[0m[8X[10X[3Xid[0m[8X[10XB[3Xm[0m[8X[10X.g[0m[8X[0m a [5XGAP[0m readable file containing all generators of a matrix representation of dimension [3Xdim[0m over the ring of integers mod [3Xn[0m. An example is [10X2A8G1-Z4r4aB0.g[0m. [1X5.6-1 AGRParseFilenameFormat[0m [2X> AGRParseFilenameFormat( [0X[3Xstring, format[0X[2X ) _________________________[0Xfunction [6XReturns:[0X a list of strings and integers if [3Xstring[0m matches [3Xformat[0m, and [9Xfail[0m otherwise. Let [3Xstring[0m be a filename, and [3Xformat[0m 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 [11Xtrue[0m or [11Xfalse[0m, and such that each entry f_i is a function for parsing a filename, such as the currently undocumented functions [10XParseForwards[0m and [10XParseBackwards[0m. [2XAGRParseFilenameFormat[0m returns a list of strings and integers such that the concatenation of their [2XString[0m ([14XReference: String[0m) values yields [3Xstring[0m if [3Xstring[0m matches [3Xformat[0m, and [9Xfail[0m otherwise. Matching is defined as follows. Splitting [3Xstring[0m at each minus character ([10X-[0m) yields m parts s_1, s_2, ..., s_m. The string [3Xstring[0m matches [3Xformat[0m 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-[9Xfail[0m result. [4X--------------------------- Example ----------------------------[0X [4Xgap> format:= [ [ [ IsChar, "G", IsDigitChar ],[0X [4X> [ "p", IsDigitChar, IsLowerAlphaOrDigitChar,[0X [4X> "B", IsDigitChar, ".m", IsDigitChar ] ],[0X [4X> [ ParseBackwards, ParseForwards ] ];;[0X [4Xgap> AGRParseFilenameFormat( "A6G1-p10B0.m1", format );[0X [4X[ "A6", "G", 1, "p", 10, "", "B", 0, ".m", 1 ][0X [4Xgap> AGRParseFilenameFormat( "A6G1-p15aB0.m1", format );[0X [4X[ "A6", "G", 1, "p", 15, "a", "B", 0, ".m", 1 ][0X [4Xgap> AGRParseFilenameFormat( "A6G1-f2r16B0.m1", format );[0X [4Xfail[0X [4X------------------------------------------------------------------[0X [1X5.6-2 AGRFileContents[0m [2X> AGRFileContents( [0X[3Xdirname, groupname, filename, type[0X[2X ) ____________[0Xfunction [6XReturns:[0X the [5XGAP[0m object obtained from reading and interpreting the file(s) with name(s) [3Xfilename[0m. Let [3Xdirname[0m and [3Xgroupname[0m be strings, [3Xfilename[0m be a string or a list of strings, and [3Xtype[0m be a data type (see [2XAGRDeclareDataType[0m ([14X5.5-1[0m)). [3Xdirname[0m must be one of [10X"datagens"[0m, [10X"dataword"[0m, or the [3Xdirid[0m value of a private directory, see [2XAtlasOfGroupRepresentationsNotifyPrivateDirectory[0m ([14X3.1-1[0m). If [3Xgroupname[0m is the [5XATLAS[0m-file name of a group G (see Section [14X2.2[0m), and if [3Xfilename[0m is either the name of an accessible file in the [3Xdirname[0m directory of the [5XATLAS[0m, or a list of such filenames, with data concerning G and for the data type [10Xtype[0m, then [2XAGRFileContents[0m returns the contents of the corresponding file(s), in the sense that the file(s) (or equivalent ones, see Section [14X1.7-6[0m) is/are read, and the result is interpreted if necessary; otherwise [9Xfail[0m is returned. Note that if [3Xfilename[0m refers to file(s) already stored in the [3Xdirname[0m directory then [2XAGRFileContents[0m does [13Xnot[0m check whether the table of contents of the [5XATLAS[0m of Group Representations actually contains [3Xfilename[0m. [1X5.7 The Tables of Contents of the [5XATLAS[1X of Group Representations[0X The list of data currently available is stored in several [13Xtables of contents[0m, one for the local [5XGAP[0m data, one for the data on remote servers, and one for each private data directory. These tables of contents are created by [2XReloadAtlasTableOfContents[0m ([14X1.6-1[0m). It is assumed that the local data directories contain only files that are also available on servers. Private extensions to the database (cf. Section [14X1.8[0m and Chapter [14X3[0m) 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 [5XATLAS[0m-file names of the groups (see Section [14X2.2[0m) and [10Xlastupdated[0m, 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 [14X5.5[0m) for which data are available. Note that the name mapping between the [5XATLAS[0m-file and [5XGAP[0m names of the groups is provided by the [10Xgroupnames[0m component of [2XAtlasOfGroupRepresentationsInfo[0m ([14X5.1-5[0m), and information about the base rings of matrix representations is provided by the [10Xringinfo[0m component. Group names are notified with [2XAGRGNAN[0m ([14X5.7-1[0m), and base ring information can be notified with [2XAGRRNG[0m ([14X5.7-2[0m); these two administrational functions may be useful for private extensions of the package (see Chapter [14X3[0m). [1X5.7-1 AGRGNAN[0m [2X> AGRGNAN( [0X[3Xgapname, atlasname[, size[, maxessize[, "all"[, compatinfo]]]][0X[2X ) [0Xfunction Let [3Xgapname[0m be a string denoting a [5XGAP[0m group name, and [3Xatlasname[0m be a string denoting the corresponding [5XATLAS[0m-file name used in filenames of the [5XATLAS[0m of Group Representations. The following optional arguments are supported. [8X[3Xsize[0m[8X[0m the order of the corresponding group, [8X[3Xmaxessizes[0m[8X[0m a (not necessarily dense) list of orders of the maximal subgroups of this group [8X[3Xcomplete[0m[8X[0m the string [10X"all"[0m if the [3Xmaxessizes[0m list is complete, [8X[3Xcompatinfo[0m[8X[0m a list of entries of the form [10X[ std, factname, factstd, flag ][0m meaning that mapping standard generators of standardization [10Xstd[0m to the factor group with [5XGAP[0m group name [10Xfactname[0m, via the natural epimorphism, yields standard generators of standardization [10Xfactstd[0m if [10Xflag[0m is [9Xtrue[0m. [2XAGRGNAN[0m adds the list of its arguments to the list stored in the [10XGAPnames[0m component of [2XAtlasOfGroupRepresentationsInfo[0m ([14X5.1-5[0m), making the [5XATLAS[0m data involving [3Xatlasname[0m accessible for the group with name [3Xgapname[0m. An example of a valid call is [10XAGRGNAN("A6.2_2","PGL29",360)[0m, see also Section [14X3.3[0m. [1X5.7-2 AGRRNG[0m [2X> AGRRNG( [0X[3Xfilename, descr[0X[2X ) ________________________________________[0Xfunction Let [3Xfilename[0m 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 [3Xdescr[0m be a string describing this ring [3XR[0m, say. [2XAGRRNG[0m adds the triple [10X[ [3Xfilename[0m[10X, [3Xdescr[0m[10X, [3XR[0m[10X ][0m to the list stored in the [10Xringinfo[0m component of [2XAtlasOfGroupRepresentationsInfo[0m ([14X5.1-5[0m). An example of a valid call is [10XAGRRNG("A5G1-Ar3aB0","Field([Sqrt(5)])")[0m. [1X5.8 Sanity Checks for the [5XATLAS[1X of Group Representations[0X The fact that the [5XATLAS[0m of Group Representations is designed as an open database (see Section [14X1.7-1[0m) makes it especially desirable to have consistency checks available which can be run automatically whenever new data are added by the developers of the [5XATLAS[0m. The tests described in the following can also be used for private extensions of the package (see Chapter [14X3[0m). The file [11Xtst/testall.g[0m of the package contains [2XReadTest[0m ([14XReference: ReadTest[0m) statements for executing a collection of such sanity checks; one can run them by starting [5XGAP[0m in the [11Xtst[0m directory, and then calling [10XRead( "testall.g" )[0m. If no problem occurs then [5XGAP[0m prints only lines starting with one of the following. [4X--------------------------- Example ----------------------------[0X [4X+ $Id:[0X [4X+ GAP4stones:[0X [4X------------------------------------------------------------------[0X 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 [11Xtst/docxpl.tst[0m of the package. The file [11Xtst/atlasrep.tst[0m contains calls to the functions [2XAtlasOfGroupRepresentationsTestGroupOrders[0m ([14X5.8-1[0m), which checks the consistency of the stored group orders and the actual data, [2XAtlasOfGroupRepresentationsTestFileHeaders[0m ([14X5.8-5[0m), which checks the consistency of the names of [5XMeatAxe[0m text files and the first lines of the files, and [2XAtlasOfGroupRepresentationsTestWords[0m ([14X5.8-6[0m), which checks whether the available programs do what they promise. The calls to [2XAtlasOfGroupRepresentationsTestFiles[0m ([14X5.8-7[0m), [2XAtlasOfGroupRepresentationsTestClassScripts[0m ([14X5.8-8[0m), and [2XAGR_TestMinimalDegrees[0m ([14X4.3-5[0m) are not part of the tests that are run by reading [11Xtst/testall.g[0m. All these tests apply only to the [13Xlocal[0m table of contents (see Section [14X5.7[0m), that is, only those data files are checked that are actually available in the local [5XGAP[0m 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. [1X5.8-1 AtlasOfGroupRepresentationsTestGroupOrders[0m [2X> AtlasOfGroupRepresentationsTestGroupOrders( [0X[3X[0X[2X ) ___________________[0Xfunction [6XReturns:[0X [9Xfalse[0m if a contradiction was found, [9Xtrue[0m otherwise. This function checks whether the group orders stored in the [10XGAPnames[0m component of [2XAtlasOfGroupRepresentationsInfo[0m ([14X5.1-5[0m) coincide with the orders computed from an [5XATLAS[0m 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 ([10X.[0m), 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. [1X5.8-2 AtlasOfGroupRepresentationsTestSubgroupOrders[0m [2X> AtlasOfGroupRepresentationsTestSubgroupOrders( [0X[3X[0X[2X ) ________________[0Xfunction [6XReturns:[0X [9Xfalse[0m if a contradiction was found, [9Xtrue[0m otherwise. This function checks whether the orders of maximal subgroups stored in the [10XGAPnames[0m component of [2XAtlasOfGroupRepresentationsInfo[0m ([14X5.1-5[0m) coincide with the orders computed from an [5XATLAS[0m 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. [1X5.8-3 AtlasOfGroupRepresentationsTestStdCompatibility[0m [2X> AtlasOfGroupRepresentationsTestStdCompatibility( [0X[3X[0X[2X ) ______________[0Xfunction [6XReturns:[0X [9Xfalse[0m if a contradiction was found, [9Xtrue[0m otherwise. This function checks whether the compatibility info stored in the [10XGAPnames[0m component of [2XAtlasOfGroupRepresentationsInfo[0m ([14X5.1-5[0m) coincide with computed values. The following criterion is used for computing the value for a group G. Use the [5XGAP[0m 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. [1X5.8-4 AtlasOfGroupRepresentationsTestCompatibleMaxes[0m [2X> AtlasOfGroupRepresentationsTestCompatibleMaxes( [0X[3X[0X[2X ) _______________[0Xfunction [6XReturns:[0X [9Xfalse[0m if a contradiction was found, [9Xtrue[0m otherwise. This function checks whether the information about maximal subgroups stored in the [10Xmaxext[0m components of the records stored in the [10XTableOfContents.remote[0m component of [2XAtlasOfGroupRepresentationsInfo[0m ([14X5.1-5[0m) 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 [2XAtlasOfGroupRepresentationsTestStdCompatibility[0m ([14X5.8-3[0m)) and if there are a presentation for F and a permutation representation of G then it is checked whether the [10X"maxes"[0m 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. [1X5.8-5 AtlasOfGroupRepresentationsTestFileHeaders[0m [2X> AtlasOfGroupRepresentationsTestFileHeaders( [0X[3X[tocid[, groupname]][0X[2X ) [0Xfunction [6XReturns:[0X [9Xfalse[0m if an error occurs, otherwise [9Xtrue[0m. First suppose that this function is called with two arguments [3Xtocid[0m, the identifier of a directory (see Section [2Xsect:Adding a Private Data Directory[0m ([14X3.1[0m)), and [3Xgroupname[0m, an [5XATLAS[0m-file name that occurs as a component name in the table of contents of the directory. The function checks for those data files for [3Xgroupname[0m in the [3Xtocid[0m directory that are in [5XMeatAxe[0m text format whether the filename and the header line are consistent; it checks the data file in [5XGAP[0m format whether the file name is consistent with the contents of the file. If only one argument [3Xtocid[0m is given then all representations available for [3Xgroupname[0m are checked with the three argument version. If only one argument [3Xtocid[0m is given then all available groups in the directory with identifier [3Xtocid[0m are checked; the contents of the local [10Xdataword[0m directory can be checked by entering [10X"local"[0m, which is also the default for [3Xtocid[0m. [1X5.8-6 AtlasOfGroupRepresentationsTestWords[0m [2X> AtlasOfGroupRepresentationsTestWords( [0X[3X[tocid[, groupname]][0X[2X ) _____[0Xfunction [6XReturns:[0X [9Xfalse[0m if an error occurs, otherwise [9Xtrue[0m. Called with one argument [3Xtocid[0m, a string, [2XAtlasOfGroupRepresentationsTestWords[0m processes all programs that are stored in the directory with identifier [3Xtocid[0m (see Section [2Xsect:Adding a Private Data Directory[0m ([14X3.1[0m)), using the function stored in the [10XTestWords[0m component of the data type in question. The contents of the local [11Xdataword[0m directory can be checked by entering [10X"local"[0m, which is also the default. If the string [3Xgroupname[0m, an [5XATLAS[0m-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. [1X5.8-7 AtlasOfGroupRepresentationsTestFiles[0m [2X> AtlasOfGroupRepresentationsTestFiles( [0X[3X[tocid[, groupname]][0X[2X ) _____[0Xfunction [6XReturns:[0X [9Xfalse[0m if an error occurs, otherwise [9Xtrue[0m. This function is an analogue of [2XAtlasOfGroupRepresentationsTestFileHeaders[0m ([14X5.8-5[0m). It checks whether reading [5XMeatAxe[0m text files with [2XScanMeatAxeFile[0m ([14X5.3-1[0m) returns non-[9Xfail[0m results. It does not check whether the first line of a [5XMeatAxe[0m text file is consistent with the filename, since this is tested by [2XAtlasOfGroupRepresentationsTestFileHeaders[0m ([14X5.8-5[0m). [1X5.8-8 AtlasOfGroupRepresentationsTestClassScripts[0m [2X> AtlasOfGroupRepresentationsTestClassScripts( [0X[3X[groupname][0X[2X ) _______[0Xfunction [6XReturns:[0X [9Xfalse[0m if an error occurs, otherwise [9Xtrue[0m. First suppose that [2XAtlasOfGroupRepresentationsTestClassScripts[0m is called with one argument [3Xgroupname[0m, the name of a component in [10XAtlasOfGroupRepresentationsInfo.TableOfContents.( "local" )[0m. If the [5XGAP[0m table library contains an ordinary character table with [2XIdentifier[0m ([14XReference: Identifier!for character tables[0m) value the [5XGAP[0m name corresponding to [3Xgroupname[0m 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.