Private Extensions of the `AtlasRep` Package

It may be interesting to use the functions of the &GAP; interface also for representations or programs that are not part of the &ATLAS; of Group Representations. This chapter describes how to achieve this.

The main idea is that users can notify directories containing the private data files, which may consist of

new faithful representations and programs for groups that are declared already in the official &ATLAS; of Group Representations,
the declaration of groups that are not declared in the official &ATLAS; of Group Representations, and representations and programs for them, and
the definition of new kinds of representations and programs.

The first two issues are dealt with in Section and Section . The last is described in Section .

Finally, an example of using private extensions is given in Section .

Several of the sanity checks for the official part of the `AtlasRep` package make sense also for private extensions, see Section for more information.

Adding a Private Data Directory
<#Include Label="[3]{access}"> <#Include Label="AtlasOfGroupRepresentationsNotifyPrivateDirectory"> <#Include Label="AtlasOfGroupRepresentationsForgetPrivateDirectory">

The Effect of Private Extensions on the User Interface

First suppose that only new groups or new data for known groups are added.

In this case, lists the private representations and programs in the same way as the official data, except that private parts are marked with the string stored in the component markprivate of ; by default, this is a star *. The ordering of representations listed by (and referred to by ) will in general change when private directories are notified. If several private directories are used then the ordering of data may depend on the ordering of notifications. For the other interface functions described in Chapter , the only difference is that also the private data can be accessed. In particular the free format groupnameGi-XdescrWn for straight line programs (see Section ) may be used in private directories; the data can be accessed with , where the last two arguments are the strings "other" and descr.

If also private data types are introduced (see Section ) then additional columns or rows can appear in the output of , and new inputs can become meaningful for all interface functions. Examples for these changes can be found in Section .

An Example of Extending the `AtlasRep` Package

In the beginning we set the info level of to 1.

gap> level:= InfoLevel( InfoAtlasRep );; gap> SetInfoLevel( InfoAtlasRep, 1 );

Let us assume that the directory privdir contains data for the cyclic group C_4 of order 4 and for the alternating group A_5 on 5 points, respectively. Note that it is obvious what the term standard generators means for the group C_4.

Further let us assume that privdir contains the following files.

C4G1-p4B0.m1
a faithful permutation representation of C_4 on 4 points,
C4G1-max1W1
the straight line program that returns the square of its unique input,
C4G1-a2W1
the straight line program that raises its unique input to the third power,
C4G1-XtestW1
the straight line program that returns the square of its unique input,
A5G1-p60B0.m1
and A5G1-p60B0.m2
the regular permutation representation of A_5.

The directory and the files can be created as follows.

gap> prv:= DirectoryTemporary( "privdir" );; gap> FileString( Filename( prv, "C4G1-p4B0.m1" ), > MeatAxeString( [ (1,2,3,4) ], 4 ) );; gap> FileString( Filename( prv, "C4G1-max1W1" ), > "inp 1\npwr 2 1 2\noup 1 2\n" );; gap> FileString( Filename( prv, "C4G1-XtestW1" ), > "inp 1\npwr 2 1 2\noup 1 2\n" );; gap> FileString( Filename( prv, "C4G1-a2W1" ), > "inp 1\npwr 3 1 2\noup 1 2\n" );; gap> FileString( Filename( prv, "C4G1-Ar1aB0.g" ), > "return rec( generators:= [ [[E(4)]] ] );\n" );; gap> points:= Elements( AlternatingGroup( 5 ) );; gap> FileString( Filename( prv, "A5G1-p60B0.m1" ), > MeatAxeString( [ Permutation( (1,2)(3,4), points, OnRight ) ], 60 ) );; gap> FileString( Filename( prv, "A5G1-p60B0.m2" ), > MeatAxeString( [ Permutation( (1,3,5), points, OnRight ) ], 60 ) );;

(We could also introduce intermediate directories C4 and A5, say, each with the data for one group only. Here we do not show this because creating directories programmatically seems to be possible only with the &GAP; package `IO`.)

The official part of the `AtlasRep` package does not contain information about C_4, so we first notify this group, in the file privdir/toc.g. Besides the name of the group, we store the following information: the group order, the number of (classes of) maximal subgroups, their orders, their structures, and describing data about the two permutation representations. (The group A_5 is known with name A5 in the official part of the `AtlasRep` package, so it cannot be notified again.)

gap> FileString( Filename( prv, "toc.g" ), Concatenation( [ > "AGR.GNAN(\"C4\",\"C4\");\n", > "AGR.GRS(\"C4\",4);\n", > "AGR.MXN(\"C4\",1);\n", > "AGR.MXO(\"C4\",[2]);\n", > "AGR.MXS(\"C4\",[\"C2\"]);\n", > "AGR.API(\"C4G1-p4B0\",[1,4,\"imprim\",\"1 < C2\"]);\n", > "AGR.API(\"A5G1-p60B0\",[1,60,\"imprim\",\"1 < A4\"]);\n", > ] ) );;

Then we notify the private directory.

gap> AtlasOfGroupRepresentationsNotifyPrivateDirectory( prv, "priv", true ); true

Now we can use the interface functions for accessing the data in the private directory.

gap> DisplayAtlasInfo( [ "C4" ] ); group | # | maxes | cl | cyc | out | fnd | chk | prs ------+---+-------+----+-----+-----+-----+-----+---- C4* | 2 | 1 | | | 2 | | | gap> DisplayAtlasInfo( "C4" ); Representations for G = C4: (all refer to std. generators 1) --------------------------- 1: G <= Sym(4)* rank 4, on cosets of 1 < C2 2: G <= GL(1a,C)*

Programs for G = C4: (all refer to std. generators 1) -------------------- automorphisms: 2* maxes (all 1): 1*: C2 other scripts: "test"* gap> DisplayAtlasInfo( "C4", IsPermGroup, true ); Representations for G = C4: (all refer to std. generators 1) --------------------------- 1: G <= Sym(4)* rank 4, on cosets of 1 < C2 gap> DisplayAtlasInfo( "C4", IsMatrixGroup ); Representations for G = C4: (all refer to std. generators 1) --------------------------- 2: G <= GL(1a,C)* gap> DisplayAtlasInfo( "C4", Dimension, 2 ); gap> DisplayAtlasInfo( "A5", NrMovedPoints, 60 ); Representations for G = A5: (all refer to std. generators 1) --------------------------- 4: G <= Sym(60)* rank 60, on cosets of 1 < A4 gap> info:= OneAtlasGeneratingSetInfo( "C4" ); rec( groupname := "C4", id := "", identifier := [ [ "priv", "C4" ], [ "C4G1-p4B0.m1" ], 1, 4 ], isPrimitive := false, p := 4, rankAction := 4, repname := "C4G1-p4B0", repnr := 1, size := 4, stabilizer := "1 < C2", standardization := 1, transitivity := 1, type := "perm" ) gap> AtlasGenerators( info.identifier ); rec( generators := [ (1,2,3,4) ], groupname := "C4", id := "", identifier := [ [ "priv", "C4" ], [ "C4G1-p4B0.m1" ], 1, 4 ], isPrimitive := false, p := 4, rankAction := 4, repname := "C4G1-p4B0", repnr := 1, size := 4, stabilizer := "1 < C2", standardization := 1, transitivity := 1, type := "perm" ) gap> AtlasProgram( "C4", 1 ); rec( groupname := "C4", identifier := [ [ "priv", "C4" ], "C4G1-max1W1", 1 ], program := <straight line program>, size := 2, standardization := 1, subgroupname := "C2" ) gap> AtlasProgram( "C4", "maxes", 1 ); rec( groupname := "C4", identifier := [ [ "priv", "C4" ], "C4G1-max1W1", 1 ], program := <straight line program>, size := 2, standardization := 1, subgroupname := "C2" ) gap> AtlasProgram( "C4", "maxes", 2 ); fail gap> AtlasGenerators( "C4", 1 ); rec( generators := [ (1,2,3,4) ], groupname := "C4", id := "", identifier := [ [ "priv", "C4" ], [ "C4G1-p4B0.m1" ], 1, 4 ], isPrimitive := false, p := 4, rankAction := 4, repname := "C4G1-p4B0", repnr := 1, size := 4, stabilizer := "1 < C2", standardization := 1, transitivity := 1, type := "perm" ) gap> AtlasGenerators( "C4", 2 ); rec( dim := 1, generators := [ [ [ E(4) ] ] ], groupname := "C4", id := "a", identifier := [ [ "priv", "C4" ], "C4G1-Ar1aB0.g", 1, 1 ], repname := "C4G1-Ar1aB0", repnr := 2, size := 4, standardization := 1, type := "matalg" ) gap> AtlasGenerators( "C4", 3 ); fail gap> AtlasProgram( "C4", "other", "test" ); rec( groupname := "C4", identifier := [ [ "priv", "C4" ], "C4G1-XtestW1", 1 ], program := <straight line program>, standardization := 1 )

We can restrict the data shown by to the private directory, as follows.

gap> DisplayAtlasInfo( "contents", "priv" ); group | # | maxes | cl | cyc | out | fnd | chk | prs -------------------------+---+-------+----+-----+-----+-----+-----+---- A5* | 1 | | | | | | | C4* | 2 | 1 | | | 2 | | |

For checking the data in the private directory, we apply the relevant sanity checks (see Section ).

gap> if not IsBound( AGR.Test ) then > ReadPackage( "atlasrep", "gap/test.g" ); > fi; gap> AGR.Test.Words( "priv" ); true gap> AGR.Test.FileHeaders( "priv" ); true gap> AGR.Test.Files( "priv" ); true gap> AGR.Test.BinaryFormat( "priv" ); true gap> AGR.Test.Primitivity( "priv" ); true gap> AGR.Test.Characters( "priv" ); true

Finally, we uninstall the private directory, and reset the info level that had been set to 1 in the beginning. (Also the group name C4 is removed this way, which is an advantage of using a toc.g file over calling AGRGNAN directly.), Note that we need not remove the data in the temporary directory, &GAP; will do this automatically.

gap> AtlasOfGroupRepresentationsForgetPrivateDirectory( "priv" ); gap> SetInfoLevel( InfoAtlasRep, level );