This chapter gives an overview of the basic functionality provided by the AtlasRep package. The main concepts and interface functions are presented in the first three sections, and Section 2.4 shows a few small examples.
Let us first fix the setup for the examples shown in the package manual.
First of all, we load the AtlasRep package. Some of the examples require also the GAP packages CTblLib and TomLib, so we load also these packages.
gap> LoadPackage( "AtlasRep", false ); true gap> LoadPackage( "CTblLib", false ); true gap> LoadPackage( "TomLib", false ); true
Depending on the terminal capabilities, the output of DisplayAtlasInfo (3.5-1) may contain non-ASCII characters, which are not supported by the LaTeX and HTML versions of GAPDoc documents. The examples in this manual are used for tests of the package's functionality, thus we set the user preference DisplayFunction (see Section 4.2-11) to the value "Print" in order to produce output consisting only of ASCII characters, which is assumed to work in any terminal.
gap> origpref:= UserPreference( "AtlasRep", "DisplayFunction" );; gap> SetUserPreference( "AtlasRep", "DisplayFunction", "Print" );
The GAP output for the examples may look differently if data extensions have been loaded. In order to ignore these extensions in the examples, we unload them.
gap> priv:= Difference( > List( AtlasOfGroupRepresentationsInfo.notified, x -> x.ID ), > [ "core", "internal" ] );; gap> Perform( priv, AtlasOfGroupRepresentationsForgetData );
If the info level of InfoAtlasRep (7.1-1) is larger than zero then additional output appears on the screen. In order to avoid this output, we set the level to zero.
gap> globallevel:= InfoLevel( InfoAtlasRep );; gap> SetInfoLevel( InfoAtlasRep, 0 );
An important database to which the AtlasRep package gives access is the ATLAS of Group Representations [WWT+]. It contains generators and related data for several groups, mainly for extensions of simple groups (see Section 2.1-1) and for their maximal subgroups (see Section 2.1-2).
In general, these data are not part of the package. They are downloaded as soon as they are needed for the first time, see Section 4.2-1.
Each group that occurs in this database is specified by a name, which is a string similar to the name used in the ATLAS of Finite Groups [CCN+85]. For those groups whose character tables are contained in the GAP Character Table Library [Bre22], the names are equal to the Identifier (Reference: Identifier for character tables) values of these character tables. Examples of such names are "M24" for the Mathieu group \(M_{24}\), "2.A6" for the double cover of the alternating group \(A_6\), and "2.A6.2_1" for the double cover of the symmetric group \(S_6\). The names that actually occur are listed in the first column of the overview table that is printed by the function DisplayAtlasInfo (3.5-1), called without arguments, see below. The other columns of the table describe the data that are available in the database.
For example, DisplayAtlasInfo (3.5-1) may print the following lines. Omissions are indicated with
....
gap> DisplayAtlasInfo(); group | # | maxes | cl | cyc | out | fnd | chk | prs -------------------------+----+-------+----+-----+-----+-----+-----+---- ... 2.A5 | 26 | 3 | | | | | + | + 2.A5.2 | 11 | 4 | | | | | + | + 2.A6 | 18 | 5 | | | | | | 2.A6.2_1 | 3 | 6 | | | | | | 2.A7 | 24 | 2 | | | | | | 2.A7.2 | 7 | | | | | | | ... M22 | 58 | 8 | + | + | | + | + | + M22.2 | 46 | 7 | + | + | | + | + | + M23 | 66 | 7 | + | + | | + | + | + M24 | 62 | 9 | + | + | | + | + | + McL | 46 | 12 | + | + | | + | + | + McL.2 | 27 | 10 | | + | | + | + | + O7(3) | 28 | | | | | | | O7(3).2 | 3 | | | | | | | ... Suz | 30 | 17 | | + | 2 | + | + | ...
Called with a group name as the only argument, the function AtlasGroup (3.5-8) returns a group isomorphic to the group with the given name, or fail. If permutation generators are available in the database then a permutation group (of smallest available degree) is returned, otherwise a matrix group.
gap> g:= AtlasGroup( "M24" ); Group([ (1,4)(2,7)(3,17)(5,13)(6,9)(8,15)(10,19)(11,18)(12,21)(14,16) (20,24)(22,23), (1,4,6)(2,21,14)(3,9,15)(5,18,10)(13,17,16) (19,24,23) ]) gap> IsPermGroup( g ); NrMovedPoints( g ); Size( g ); true 24 244823040 gap> AtlasGroup( "J5" ); fail
Many maximal subgroups of extensions of simple groups can be constructed using the function AtlasSubgroup (3.5-9). Given the name of the extension of the simple group and the number of the conjugacy class of maximal subgroups, this function returns a representative from this class.
gap> g:= AtlasSubgroup( "M24", 1 ); Group([ (2,10)(3,12)(4,14)(6,9)(8,16)(15,18)(20,22)(21,24), (1,7,2,9) (3,22,10,23)(4,19,8,12)(5,14)(6,18)(13,16,17,24) ]) gap> IsPermGroup( g ); NrMovedPoints( g ); Size( g ); true 23 10200960 gap> AtlasSubgroup( "M24", 100 ); fail
The classes of maximal subgroups are ordered w. r. t. decreasing subgroup order. So the first class contains maximal subgroups of smallest index.
Note that groups obtained by AtlasSubgroup (3.5-9) may be not very suitable for computations in the sense that much nicer representations exist. For example, the sporadic simple O'Nan group \(O'N\) contains a maximal subgroup \(S\) isomorphic with the Janko group \(J_1\); the smallest permutation representation of \(O'N\) has degree \(122760\), and restricting this representation to \(S\) yields a representation of \(J_1\) of that degree. However, \(J_1\) has a faithful permutation representation of degree \(266\), which admits much more efficient computations. If you are just interested in \(J_1\) and not in its embedding into \(O'N\) then one possibility to get a nicer
faithful representation is to call SmallerDegreePermutationRepresentation (Reference: SmallerDegreePermutationRepresentation). In the abovementioned example, this works quite well; note that in general, we cannot expect that we get a representation of smallest degree in this way.
gap> s:= AtlasSubgroup( "ON", 3 ); <permutation group of size 175560 with 2 generators> gap> NrMovedPoints( s ); Size( s ); 122760 175560 gap> hom:= SmallerDegreePermutationRepresentation( s );; gap> NrMovedPoints( Image( hom ) ) < 2000; true
(Depending on random choices in the computations, one may or my not get the degree \(266\) representation.)
In this particular case, one could of course also ask directly for the group \(J_1\).
gap> j1:= AtlasGroup( "J1" ); <permutation group of size 175560 with 2 generators> gap> NrMovedPoints( j1 ); 266
If you have a group \(G\), say, and you are really interested in the embedding of a maximal subgroup of \(G\) into \(G\) then an easy way to get compatible generators is to create \(G\) with AtlasGroup (3.5-8) and then to call AtlasSubgroup (3.5-9) with first argument the group \(G\).
gap> g:= AtlasGroup( "ON" ); <permutation group of size 460815505920 with 2 generators> gap> s:= AtlasSubgroup( g, 3 ); <permutation group of size 175560 with 2 generators> gap> IsSubset( g, s ); true gap> IsSubset( g, j1 ); false
The function DisplayAtlasInfo (3.5-1), called with an admissible name of a group as the only argument, lists the ATLAS data available for this group.
gap> DisplayAtlasInfo( "A5" ); Representations for G = A5: (all refer to std. generators 1) --------------------------- 1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.) 2: G <= Sym(6) 2-trans., on cosets of D10 (2nd max.) 3: G <= Sym(10) rank 3, on cosets of S3 (3rd max.) 4: G <= GL(4a,2) character 4a 5: G <= GL(4b,2) character 2ab 6: G <= GL(4,3) character 4a 7: G <= GL(6,3) character 3ab 8: G <= GL(2a,4) character 2a 9: G <= GL(2b,4) character 2b 10: G <= GL(3,5) character 3a 11: G <= GL(5,5) character 5a 12: G <= GL(3a,9) character 3a 13: G <= GL(3b,9) character 3b 14: G <= GL(4,Z) character 4a 15: G <= GL(5,Z) character 5a 16: G <= GL(6,Z) character 3ab 17: G <= GL(3a,Field([Sqrt(5)])) character 3a 18: G <= GL(3b,Field([Sqrt(5)])) character 3b Programs for G = A5: (all refer to std. generators 1) -------------------- - class repres.* - presentation - maxes (all 3): 1: A4 2: D10 3: S3 - std. gen. checker: (check) (pres)
In order to fetch one of the listed permutation groups or matrix groups, you can call AtlasGroup (3.5-8) with second argument the function Position (Reference: Position) and third argument the position in the list.
gap> AtlasGroup( "A5", Position, 1 ); Group([ (1,2)(3,4), (1,3,5) ])
Note that this approach may yield a different group after a data extension has been loaded.
Alternatively, you can describe the desired group by conditions, such as the degree in the case of a permutation group, and the dimension and the base ring in the case of a matrix group.
gap> AtlasGroup( "A5", NrMovedPoints, 10 ); Group([ (2,4)(3,5)(6,8)(7,10), (1,2,3)(4,6,7)(5,8,9) ]) gap> AtlasGroup( "A5", Dimension, 4, Ring, GF(2) ); <matrix group of size 60 with 2 generators>
The same holds for the restriction to maximal subgroups: Use AtlasSubgroup (3.5-9) with the same arguments as AtlasGroup (3.5-8), except that additionally the number of the class of maximal subgroups is entered as the last argument. Note that the conditions refer to the group, not to the subgroup; it may happen that the subgroup moves fewer points than the big group.
gap> AtlasSubgroup( "A5", Dimension, 4, Ring, GF(2), 1 ); <matrix group of size 12 with 2 generators> gap> g:= AtlasSubgroup( "A5", NrMovedPoints, 10, 3 ); Group([ (2,4)(3,5)(6,8)(7,10), (1,4)(3,8)(5,7)(6,10) ]) gap> Size( g ); NrMovedPoints( g ); 6 9
Up to now, we have talked only about groups and subgroups. The AtlasRep package provides access to group generators, and in fact these generators have the property that mapping one set of generators to another set of generators for the same group defines an isomorphism. These generators are called standard generators, see Section 3.3.
So instead of thinking about several generating sets of a group \(G\), say, we can think about one abstract group \(G\), with one fixed set of generators, and mapping these generators to any set of generators provided by AtlasRep defines a representation of \(G\). This viewpoint had motivated the name ATLAS of Group Representations
for the core part of the database.
If you are interested in the generators provided by the database rather than in the groups they generate, you can use the function OneAtlasGeneratingSetInfo (3.5-6) instead of AtlasGroup (3.5-8), with the same arguments. This will yield a record that describes the representation in question. Calling the function AtlasGenerators (3.5-3) with this record will then yield a record with the additional component generators, which holds the list of generators.
gap> info:= OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, 10 ); rec( charactername := "1a+4a+5a", constituents := [ 1, 4, 5 ], contents := "core", groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p10B0.m1", "A5G1-p10B0.m2" ], 1, 10 ], isPrimitive := true, maxnr := 3, p := 10, rankAction := 3, repname := "A5G1-p10B0", repnr := 3, size := 60, stabilizer := "S3", standardization := 1, transitivity := 1, type := "perm" ) gap> info2:= AtlasGenerators( info ); rec( charactername := "1a+4a+5a", constituents := [ 1, 4, 5 ], contents := "core", generators := [ (2,4)(3,5)(6,8)(7,10), (1,2,3)(4,6,7)(5,8,9) ], groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p10B0.m1", "A5G1-p10B0.m2" ], 1, 10 ], isPrimitive := true, maxnr := 3, p := 10, rankAction := 3, repname := "A5G1-p10B0", repnr := 3, size := 60, stabilizer := "S3", standardization := 1, transitivity := 1, type := "perm" ) gap> info2.generators; [ (2,4)(3,5)(6,8)(7,10), (1,2,3)(4,6,7)(5,8,9) ]
The record info appears as the value of the attribute AtlasRepInfoRecord (3.5-10) in groups that are returned by AtlasGroup (3.5-8).
gap> g:= AtlasGroup( "A5", NrMovedPoints, 10 );; gap> AtlasRepInfoRecord( g ); rec( charactername := "1a+4a+5a", constituents := [ 1, 4, 5 ], contents := "core", groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p10B0.m1", "A5G1-p10B0.m2" ], 1, 10 ], isPrimitive := true, maxnr := 3, p := 10, rankAction := 3, repname := "A5G1-p10B0", repnr := 3, size := 60, stabilizer := "S3", standardization := 1, transitivity := 1, type := "perm" )
For computing certain group elements from standard generators, such as generators of a subgroup or class representatives, AtlasRep uses straight line programs, see Reference: Straight Line Programs. Essentially this means to evaluate words in the generators, which is similar to MappedWord (Reference: MappedWord) but can be more efficient.
It can be useful to deal with these straight line programs, see AtlasProgram (3.5-4). For example, an automorphism \(\alpha\), say, of the group \(G\), if available in AtlasRep, is given by a straight line program that defines the images of standard generators of \(G\). This way, one can for example compute the image of a subgroup \(U\) of \(G\) under \(\alpha\) by first applying the straight line program for \(\alpha\) to standard generators of \(G\), and then applying the straight line program for the restriction from \(G\) to \(U\).
gap> prginfo:= AtlasProgramInfo( "A5", "maxes", 1 ); rec( groupname := "A5", identifier := [ "A5", "A5G1-max1W1", 1 ], size := 12, standardization := 1, subgroupname := "A4", version := "1" ) gap> prg:= AtlasProgram( prginfo.identifier ); rec( groupname := "A5", identifier := [ "A5", "A5G1-max1W1", 1 ], program := <straight line program>, size := 12, standardization := 1, subgroupname := "A4", version := "1" ) gap> Display( prg.program ); # input: r:= [ g1, g2 ]; # program: r[3]:= r[1]*r[2]; r[4]:= r[2]*r[1]; r[5]:= r[3]*r[3]; r[1]:= r[5]*r[4]; # return values: [ r[1], r[2] ] gap> ResultOfStraightLineProgram( prg.program, info2.generators ); [ (1,10)(2,3)(4,9)(7,8), (1,2,3)(4,6,7)(5,8,9) ]
First we show the computation of class representatives of the Mathieu group \(M_{11}\), in a \(2\)-modular matrix representation. We start with the ordinary and Brauer character tables of this group.
gap> tbl:= CharacterTable( "M11" );; gap> modtbl:= tbl mod 2;; gap> CharacterDegrees( modtbl ); [ [ 1, 1 ], [ 10, 1 ], [ 16, 2 ], [ 44, 1 ] ]
The output of CharacterDegrees (Reference: CharacterDegrees) means that the \(2\)-modular irreducibles of \(M_{11}\) have degrees \(1\), \(10\), \(16\), \(16\), and \(44\).
Using DisplayAtlasInfo (3.5-1), we find out that matrix generators for the irreducible \(10\)-dimensional representation are available in the database.
gap> DisplayAtlasInfo( "M11", Characteristic, 2 ); Representations for G = M11: (all refer to std. generators 1) ---------------------------- 6: G <= GL(10,2) character 10a 7: G <= GL(32,2) character 16ab 8: G <= GL(44,2) character 44a 16: G <= GL(16a,4) character 16a 17: G <= GL(16b,4) character 16b
So we decide to work with this representation. We fetch the generators and compute the list of class representatives of \(M_{11}\) in the representation. The ordering of class representatives is the same as that in the character table of the ATLAS of Finite Groups ([CCN+85]), which coincides with the ordering of columns in the GAP table we have fetched above.
gap> info:= OneAtlasGeneratingSetInfo( "M11", Characteristic, 2, > Dimension, 10 );; gap> gens:= AtlasGenerators( info.identifier );; gap> ccls:= AtlasProgram( "M11", gens.standardization, "classes" ); rec( groupname := "M11", identifier := [ "M11", "M11G1-cclsW1", 1 ], outputs := [ "1A", "2A", "3A", "4A", "5A", "6A", "8A", "8B", "11A", "11B" ], program := <straight line program>, standardization := 1, version := "1" ) gap> reps:= ResultOfStraightLineProgram( ccls.program, gens.generators );;
If we would need only a few class representatives, we could use the GAP library function RestrictOutputsOfSLP (Reference: RestrictOutputsOfSLP) to create a straight line program that computes only specified outputs. Here is an example where only the class representatives of order eight are computed.
gap> ord8prg:= RestrictOutputsOfSLP( ccls.program, > Filtered( [ 1 .. 10 ], i -> ccls.outputs[i][1] = '8' ) ); <straight line program> gap> ord8reps:= ResultOfStraightLineProgram( ord8prg, gens.generators );; gap> List( ord8reps, m -> Position( reps, m ) ); [ 7, 8 ]
Let us check that the class representatives have the right orders.
gap> List( reps, Order ) = OrdersClassRepresentatives( tbl ); true
From the class representatives, we can compute the Brauer character we had started with. This Brauer character is defined on all classes of the \(2\)-modular table. So we first pick only those representatives, using the GAP function GetFusionMap (Reference: GetFusionMap); in this situation, it returns the class fusion from the Brauer table into the ordinary table.
gap> fus:= GetFusionMap( modtbl, tbl ); [ 1, 3, 5, 9, 10 ] gap> modreps:= reps{ fus };;
Then we call the GAP function BrauerCharacterValue (Reference: BrauerCharacterValue), which computes the Brauer character value from the matrix given.
gap> char:= List( modreps, BrauerCharacterValue ); [ 10, 1, 0, -1, -1 ] gap> Position( Irr( modtbl ), char ); 2
The second example shows the computation of a permutation representation from a matrix representation. We work with the \(10\)-dimensional representation used above, and consider the action on the \(2^{10}\) vectors of the underlying row space.
gap> grp:= Group( gens.generators );; gap> v:= GF(2)^10;; gap> orbs:= Orbits( grp, AsList( v ) );; gap> List( orbs, Length ); [ 1, 396, 55, 330, 66, 165, 11 ]
We see that there are six nontrivial orbits, and we can compute the permutation actions on these orbits directly using Action (Reference: Action homomorphisms). However, for larger examples, one cannot write down all orbits on the row space, so one has to use another strategy if one is interested in a particular orbit.
Let us assume that we are interested in the orbit of length \(11\). The point stabilizer is the first maximal subgroup of \(M_{11}\), thus the restriction of the representation to this subgroup has a nontrivial fixed point space. This restriction can be computed using the AtlasRep package.
gap> gens:= AtlasGenerators( "M11", 6, 1 );;
Now computing the fixed point space is standard linear algebra.
gap> id:= IdentityMat( 10, GF(2) );; gap> sub1:= Subspace( v, NullspaceMat( gens.generators[1] - id ) );; gap> sub2:= Subspace( v, NullspaceMat( gens.generators[2] - id ) );; gap> fix:= Intersection( sub1, sub2 ); <vector space of dimension 1 over GF(2)>
The final step is of course the computation of the permutation action on the orbit.
gap> orb:= Orbit( grp, Basis( fix )[1] );; gap> act:= Action( grp, orb );; Print( act, "\n" ); Group( [ ( 1, 2)( 4, 6)( 5, 8)( 7,10), ( 1, 3, 5, 9)( 2, 4, 7,11) ] )
Note that this group is not equal to the group obtained by fetching the permutation representation from the database. This is due to a different numbering of the points, thus the groups are permutation isomorphic, that is, they are conjugate in the symmetric group on eleven points.
gap> permgrp:= Group( AtlasGenerators( "M11", 1 ).generators );; gap> Print( permgrp, "\n" ); Group( [ ( 2,10)( 4,11)( 5, 7)( 8, 9), (1,4,3,8)(2,5,6,9) ] ) gap> permgrp = act; false gap> IsConjugate( SymmetricGroup(11), permgrp, act ); true
The straight line programs for applying outer automorphisms to standard generators can of course be used to define the automorphisms themselves as GAP mappings.
gap> DisplayAtlasInfo( "G2(3)", IsStraightLineProgram ); Programs for G = G2(3): (all refer to std. generators 1) ----------------------- - class repres. - presentation - repr. cyc. subg. - std. gen. checker - automorphisms: 2 - maxes (all 10): 1: U3(3).2 2: U3(3).2 3: (3^(1+2)+x3^2):2S4 4: (3^(1+2)+x3^2):2S4 5: L3(3).2 6: L3(3).2 7: L2(8).3 8: 2^3.L3(2) 9: L2(13) 10: 2^(1+4)+:3^2.2 gap> prog:= AtlasProgram( "G2(3)", "automorphism", "2" ).program;; gap> info:= OneAtlasGeneratingSetInfo( "G2(3)", Dimension, 7 );; gap> gens:= AtlasGenerators( info ).generators;; gap> imgs:= ResultOfStraightLineProgram( prog, gens );;
If we are not suspicious whether the script really describes an automorphism then we should tell this to GAP, in order to avoid the expensive checks of the properties of being a homomorphism and bijective (see Section Reference: Creating Group Homomorphisms). This looks as follows.
gap> g:= Group( gens );; gap> aut:= GroupHomomorphismByImagesNC( g, g, gens, imgs );; gap> SetIsBijective( aut, true );
If we are suspicious whether the script describes an automorphism then we might have the idea to check it with GAP, as follows.
gap> aut:= GroupHomomorphismByImages( g, g, gens, imgs );; gap> IsBijective( aut ); true
(Note that even for a comparatively small group such as \(G_2(3)\), this was a difficult task for GAP before version 4.3.)
Often one can form images under an automorphism \(\alpha\), say, without creating the homomorphism object. This is obvious for the standard generators of the group \(G\) themselves, but also for generators of a maximal subgroup \(M\) computed from standard generators of \(G\), provided that the straight line programs in question refer to the same standard generators. Note that the generators of \(M\) are given by evaluating words in terms of standard generators of \(G\), and their images under \(\alpha\) can be obtained by evaluating the same words at the images under \(\alpha\) of the standard generators of \(G\).
gap> max1:= AtlasProgram( "G2(3)", 1 ).program;; gap> mgens:= ResultOfStraightLineProgram( max1, gens );; gap> comp:= CompositionOfStraightLinePrograms( max1, prog );; gap> mimgs:= ResultOfStraightLineProgram( comp, gens );;
The list mgens is the list of generators of the first maximal subgroup of \(G_2(3)\), mimgs is the list of images under the automorphism given by the straight line program prog. Note that applying the program returned by CompositionOfStraightLinePrograms (Reference: CompositionOfStraightLinePrograms) means to apply first prog and then max1. Since we have already constructed the GAP object representing the automorphism, we can check whether the results are equal.
gap> mimgs = List( mgens, x -> x^aut ); true
However, it should be emphasized that using aut requires a huge machinery of computations behind the scenes, whereas applying the straight line programs prog and max1 involves only elementary operations with the generators. The latter is feasible also for larger groups, for which constructing the GAP automorphism might be too hard.
Let us suppose that we want to restrict a representation of the Mathieu group \(M_{12}\) to a non-maximal subgroup of the type \(L_2(11)\). The idea is that this subgroup can be found as a maximal subgroup of a maximal subgroup of the type \(M_{11}\), which is itself maximal in \(M_{12}\). For that, we fetch a representation of \(M_{12}\) and use a straight line program for restricting it to the first maximal subgroup, which has the type \(M_{11}\).
gap> info:= OneAtlasGeneratingSetInfo( "M12", NrMovedPoints, 12 ); rec( charactername := "1a+11a", constituents := [ 1, 2 ], contents := "core", groupname := "M12", id := "a", identifier := [ "M12", [ "M12G1-p12aB0.m1", "M12G1-p12aB0.m2" ], 1, 12 ], isPrimitive := true, maxnr := 1, p := 12, rankAction := 2, repname := "M12G1-p12aB0", repnr := 1, size := 95040, stabilizer := "M11", standardization := 1, transitivity := 5, type := "perm" ) gap> gensM12:= AtlasGenerators( info.identifier );; gap> restM11:= AtlasProgram( "M12", "maxes", 1 );; gap> gensM11:= ResultOfStraightLineProgram( restM11.program, > gensM12.generators ); [ (3,9)(4,12)(5,10)(6,8), (1,4,11,5)(2,10,8,3) ]
Now we cannot simply apply a straight line program for a group to some generators, since they are not necessarily standard generators of the group. We check this property using a semi-presentation for \(M_{11}\), see 6.1-7.
gap> checkM11:= AtlasProgram( "M11", "check" ); rec( groupname := "M11", identifier := [ "M11", "M11G1-check1", 1, 1 ] , program := <straight line decision>, standardization := 1, version := "1" ) gap> ResultOfStraightLineDecision( checkM11.program, gensM11 ); true
So we are lucky that applying the appropriate program for \(M_{11}\) will give us the required generators for \(L_2(11)\).
gap> restL211:= AtlasProgram( "M11", "maxes", 2 );; gap> gensL211:= ResultOfStraightLineProgram( restL211.program, gensM11 ); [ (3,9)(4,12)(5,10)(6,8), (1,11,9)(2,12,8)(3,6,10) ] gap> G:= Group( gensL211 );; Size( G ); IsSimple( G ); 660 true
In this case, we could also use the information that is stored about \(M_{11}\), as follows.
gap> DisplayAtlasInfo( "M11", IsStraightLineProgram ); Programs for G = M11: (all refer to std. generators 1) --------------------- - presentation - repr. cyc. subg. - std. gen. finder - class repres.: (direct) (composed) - maxes (all 5): 1: A6.2_3 1: A6.2_3 (std. 1) 2: L2(11) 2: L2(11) (std. 1) 3: 3^2:Q8.2 4: S5 4: S5 (std. 1) 5: 2.S4 - standardizations of maxes: from 1st max., version 1 to A6.2_3, std. 1 from 2nd max., version 1 to L2(11), std. 1 from 4th max., version 1 to A5.2, std. 1 - std. gen. checker: (check) (pres)
The entry std.1
in the line about the maximal subgroup of type \(L_2(11)\) means that a straight line program for computing standard generators (in standardization 1) of the subgroup. This program can be fetched as follows.
gap> restL211std:= AtlasProgram( "M11", "maxes", 2, 1 );; gap> ResultOfStraightLineProgram( restL211std.program, gensM11 ); [ (3,9)(4,12)(5,10)(6,8), (1,11,9)(2,12,8)(3,6,10) ]
We see that we get the same generators for the subgroup as above. (In fact the second approach first applies the same program as is given by restL211.program, and then applies a program to the results that does nothing.)
Usually representations are not given in terms of standard generators. For example, let us take the \(M_{11}\) type group returned by the GAP function MathieuGroup (Reference: MathieuGroup).
gap> G:= MathieuGroup( 11 );; gap> gens:= GeneratorsOfGroup( G ); [ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ] gap> ResultOfStraightLineDecision( checkM11.program, gens ); false
If we want to compute an \(L_2(11)\) type subgroup of this group, we can use a black box program for computing standard generators, and then apply the straight line program for computing the restriction.
gap> find:= AtlasProgram( "M11", "find" ); rec( groupname := "M11", identifier := [ "M11", "M11G1-find1", 1, 1 ], program := <black box program>, standardization := 1, version := "1" ) gap> stdgens:= ResultOfBBoxProgram( find.program, Group( gens ) );; gap> List( stdgens, Order ); [ 2, 4 ] gap> ResultOfStraightLineDecision( checkM11.program, stdgens ); true gap> gensL211:= ResultOfStraightLineProgram( restL211.program, stdgens );; gap> List( gensL211, Order ); [ 2, 3 ] gap> G:= Group( gensL211 );; Size( G ); IsSimple( G ); 660 true
Note that applying the black box program several times may yield different group elements, because computations of random elements are involved, see ResultOfBBoxProgram (6.2-4). All what the black box program promises is to construct standard generators, and these are defined only up to conjugacy in the automorphism group of the group in question.
The GAP Library of Tables of Marks (the GAP package TomLib, [NMP18]) provides, for many almost simple groups, information for constructing representatives of all conjugacy classes of subgroups. If this information is compatible with the standard generators of the ATLAS of Group Representations then we can use it to restrict any representation from the ATLAS to prescribed subgroups. This is useful in particular for those subgroups for which the ATLAS of Group Representations itself does not contain a straight line program.
gap> tom:= TableOfMarks( "A5" ); TableOfMarks( "A5" ) gap> info:= StandardGeneratorsInfo( tom ); [ rec( ATLAS := true, description := "|a|=2, |b|=3, |ab|=5", generators := "a, b", script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ] ], standardization := 1 ) ]
The true value of the component ATLAS indicates that the information stored on tom refers to the standard generators of type \(1\) in the ATLAS of Group Representations.
We want to restrict a \(4\)-dimensional integral representation of \(A_5\) to a Sylow \(2\) subgroup of \(A_5\), and use RepresentativeTomByGeneratorsNC (Reference: RepresentativeTomByGeneratorsNC) for that.
gap> info:= OneAtlasGeneratingSetInfo( "A5", Ring, Integers, Dimension, 4 );; gap> stdgens:= AtlasGenerators( info.identifier ); rec( charactername := "4a", constituents := [ 4 ], contents := "core", dim := 4, generators := [ [ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ -1, -1, -1, -1 ] ], [ [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 1, 0, 0, 0 ] ] ], groupname := "A5", id := "", identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ], repname := "A5G1-Zr4B0", repnr := 14, ring := Integers, size := 60, standardization := 1, type := "matint" ) gap> orders:= OrdersTom( tom ); [ 1, 2, 3, 4, 5, 6, 10, 12, 60 ] gap> pos:= Position( orders, 4 ); 4 gap> sub:= RepresentativeTomByGeneratorsNC( tom, pos, stdgens.generators ); <matrix group of size 4 with 2 generators> gap> GeneratorsOfGroup( sub ); [ [ [ 1, 0, 0, 0 ], [ -1, -1, -1, -1 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ], [ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ -1, -1, -1, -1 ] ] ]
The sporadic simple Mathieu group \(M_{22}\) contains a unique class of subgroups of index \(770\) (and order \(576\)). This can be seen for example using GAP's Library of Tables of Marks.
gap> tom:= TableOfMarks( "M22" ); TableOfMarks( "M22" ) gap> subord:= Size( UnderlyingGroup( tom ) ) / 770; 576 gap> ord:= OrdersTom( tom );; gap> tomstabs:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = subord ); [ 144 ]
The permutation representation of \(M_{22}\) on the right cosets of such a subgroup \(S\) is contained in the ATLAS of Group Representations.
gap> DisplayAtlasInfo( "M22", NrMovedPoints, 770 ); Representations for G = M22: (all refer to std. generators 1) ---------------------------- 12: G <= Sym(770) rank 9, on cosets of (A4xA4):4 < 2^4:A6
Now we verify the information shown about the point stabilizer and about the maximal overgroups of \(S\) in \(M_{22}\).
gap> maxtom:= MaximalSubgroupsTom( tom ); [ [ 155, 154, 153, 152, 151, 150, 146, 145 ], [ 22, 77, 176, 176, 231, 330, 616, 672 ] ] gap> List( tomstabs, i -> List( maxtom[1], j -> ContainedTom( tom, i, j ) ) ); [ [ 0, 10, 0, 0, 0, 0, 0, 0 ] ]
We see that the only maximal subgroups of \(M_{22}\) that contain \(S\) have index \(77\) in \(M_{22}\). According to the ATLAS of Finite Groups, these maximal subgroups have the structure \(2^4:A_6\). From that and from the structure of \(A_6\), we conclude that \(S\) has the structure \(2^4:(3^2:4)\).
Alternatively, we look at the permutation representation of degree \(770\). We fetch it from the ATLAS of Group Representations. There is exactly one nontrivial block system for this representation, with \(77\) blocks of length \(10\).
gap> g:= AtlasGroup( "M22", NrMovedPoints, 770 ); <permutation group of size 443520 with 2 generators> gap> allbl:= AllBlocks( g );; gap> List( allbl, Length ); [ 10 ]
Furthermore, GAP computes that the point stabilizer \(S\) has the structure \((A_4 \times A_4):4\).
gap> stab:= Stabilizer( g, 1 );; gap> StructureDescription( stab : nice ); "(A4 x A4) : C4" gap> blocks:= Orbit( g, allbl[1], OnSets );; gap> act:= Action( g, blocks, OnSets );; gap> StructureDescription( Stabilizer( act, 1 ) ); "(C2 x C2 x C2 x C2) : A6"
The ATLAS of Group Representations contains three degree \(462\) permutation representations of the group \(M_{22}\).
gap> DisplayAtlasInfo( "M22", NrMovedPoints, 462 ); Representations for G = M22: (all refer to std. generators 1) ---------------------------- 7: G <= Sym(462a) rank 5, on cosets of 2^4:A5 < 2^4:A6 8: G <= Sym(462b) rank 8, on cosets of 2^4:A5 < L3(4), 2^4:S5 9: G <= Sym(462c) rank 8, on cosets of 2^4:A5 < L3(4), 2^4:A6
The point stabilizers in these three representations have the structure \(2^4:A_5\). Using GAP's Library of Tables of Marks, we can show that these stabilizers are exactly the three classes of subgroups of order \(960\) in \(M_{22}\). For that, we first verify that the group generators stored in GAP's table of marks coincide with the standard generators used by the ATLAS of Group Representations.
gap> tom:= TableOfMarks( "M22" ); TableOfMarks( "M22" ) gap> genstom:= GeneratorsOfGroup( UnderlyingGroup( tom ) );; gap> checkM22:= AtlasProgram( "M22", "check" ); rec( groupname := "M22", identifier := [ "M22", "M22G1-check1", 1, 1 ] , program := <straight line decision>, standardization := 1, version := "1" ) gap> ResultOfStraightLineDecision( checkM22.program, genstom ); true
There are indeed three classes of subgroups of order \(960\) in \(M_{22}\).
gap> ord:= OrdersTom( tom );; gap> tomstabs:= Filtered( [ 1 .. Length( ord ) ], i -> ord[i] = 960 ); [ 147, 148, 149 ]
Now we compute representatives of these three classes in the three representations 462a, 462b, and 462c. We see that each of the three classes occurs as a point stabilizer in exactly one of the three representations.
gap> atlasreps:= AllAtlasGeneratingSetInfos( "M22", NrMovedPoints, 462 ); [ rec( charactername := "1a+21a+55a+154a+231a", constituents := [ 1, 2, 5, 7, 9 ], contents := "core", groupname := "M22", id := "a", identifier := [ "M22", [ "M22G1-p462aB0.m1", "M22G1-p462aB0.m2" ], 1, 462 ], isPrimitive := false, p := 462, rankAction := 5, repname := "M22G1-p462aB0", repnr := 7, size := 443520, stabilizer := "2^4:A5 < 2^4:A6", standardization := 1, transitivity := 1, type := "perm" ), rec( charactername := "1a+21a^2+55a+154a+210a", constituents := [ 1, [ 2, 2 ], 5, 7, 8 ], contents := "core", groupname := "M22", id := "b", identifier := [ "M22", [ "M22G1-p462bB0.m1", "M22G1-p462bB0.m2" ], 1, 462 ], isPrimitive := false, p := 462, rankAction := 8, repname := "M22G1-p462bB0", repnr := 8, size := 443520, stabilizer := "2^4:A5 < L3(4), 2^4:S5", standardization := 1, transitivity := 1, type := "perm" ), rec( charactername := "1a+21a^2+55a+154a+210a", constituents := [ 1, [ 2, 2 ], 5, 7, 8 ], contents := "core", groupname := "M22", id := "c", identifier := [ "M22", [ "M22G1-p462cB0.m1", "M22G1-p462cB0.m2" ], 1, 462 ], isPrimitive := false, p := 462, rankAction := 8, repname := "M22G1-p462cB0", repnr := 9, size := 443520, stabilizer := "2^4:A5 < L3(4), 2^4:A6", standardization := 1, transitivity := 1, type := "perm" ) ] gap> atlasreps:= List( atlasreps, AtlasGroup );; gap> tomstabreps:= List( atlasreps, G -> List( tomstabs, > i -> RepresentativeTomByGenerators( tom, i, GeneratorsOfGroup( G ) ) ) );; gap> List( tomstabreps, x -> List( x, NrMovedPoints ) ); [ [ 462, 462, 461 ], [ 460, 462, 462 ], [ 462, 461, 462 ] ]
More precisely, we see that the point stabilizers in the three representations 462a, 462b, 462c lie in the subgroup classes \(149\), \(147\), \(148\), respectively, of the table of marks.
The point stabilizers in the representations 462b and 462c are isomorphic, but not isomorphic with the point stabilizer in 462a.
gap> stabs:= List( atlasreps, G -> Stabilizer( G, 1 ) );; gap> List( stabs, IdGroup ); [ [ 960, 11358 ], [ 960, 11357 ], [ 960, 11357 ] ] gap> List( stabs, PerfectIdentification ); [ [ 960, 2 ], [ 960, 1 ], [ 960, 1 ] ]
The three representations are imprimitive. The containment of the point stabilizers in maximal subgroups of \(M_{22}\) can be computed using the table of marks of \(M_{22}\).
gap> maxtom:= MaximalSubgroupsTom( tom ); [ [ 155, 154, 153, 152, 151, 150, 146, 145 ], [ 22, 77, 176, 176, 231, 330, 616, 672 ] ] gap> List( tomstabs, i -> List( maxtom[1], j -> ContainedTom( tom, i, j ) ) ); [ [ 21, 0, 0, 0, 1, 0, 0, 0 ], [ 21, 6, 0, 0, 0, 0, 0, 0 ], [ 0, 6, 0, 0, 0, 0, 0, 0 ] ]
We see:
The point stabilizers in 462a (subgroups in the class \(149\) of the table of marks) are contained only in maximal subgroups in class \(154\); these groups have the structure \(2^4:A_6\).
The point stabilizers in 462b (subgroups in the class \(147\)) are contained in maximal subgroups in the classes \(155\) and \(151\); these groups have the structures \(L_3(4)\) and \(2^4:S_5\), respectively.
The point stabilizers in 462c (subgroups in the class \(148\)) are contained in maximal subgroups in the classes \(155\) and \(154\).
We identify the supergroups of the point stabilizers by computing the block systems.
gap> bl:= List( atlasreps, AllBlocks );; gap> List( bl, Length ); [ 1, 3, 2 ] gap> List( bl, l -> List( l, Length ) ); [ [ 6 ], [ 21, 21, 2 ], [ 21, 6 ] ]
Note that the two block systems with blocks of length \(21\) for 462b belong to the same supergroups (of the type \(L_3(4)\)); each of these subgroups fixes two different subsets of \(21\) points.
The representation 462a is multiplicity-free, that is, it splits into a sum of pairwise nonisomorphic irreducible representations. This can be seen from the fact that the rank of this permutation representation (that is, the number of orbits of the point stabilizer) is five; each permutation representation with this property is multiplicity-free.
The other two representations have rank eight. We have seen the ranks in the overview that was shown by DisplayAtlasInfo (3.5-1) in the beginning. Now we compute the ranks from the permutation groups.
gap> List( atlasreps, RankAction ); [ 5, 8, 8 ]
In fact the two representations 462b and 462c have the same permutation character. We check this by computing the possible permutation characters of degree \(462\) for \(M_{22}\), and decomposing them into irreducible characters, using the character table from GAP's Character Table Library.
gap> t:= CharacterTable( "M22" );; gap> perms:= PermChars( t, 462 ); [ Character( CharacterTable( "M22" ), [ 462, 30, 3, 2, 2, 2, 3, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "M22" ), [ 462, 30, 12, 2, 2, 2, 0, 0, 0, 0, 0, 0 ] ) ] gap> MatScalarProducts( t, Irr( t ), perms ); [ [ 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0 ], [ 1, 2, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0 ] ]
In particular, we see that the rank eight characters are not multiplicity-free.
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