Spracherkennung für: .tst vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen]
# This file was created automatically, do not edit!
#############################################################################
##
#W docxpl2.tst GAP 4 package CTblLib Thomas Breuer
##
## This file contains the GAP code of those examples in the package
## documentation files that do not involve the visual mode used by the
## Browse package and that run not too long.
##
## In order to run the tests, one starts GAP from the 'tst' subdirectory
## of the 'pkg/ctbllib' directory, and calls 'Test( "docxpl2.tst" );'.
##
gap> LoadPackage( "CTblLib", false );
true
gap> save:= SizeScreen();;
gap> SizeScreen( [ 72 ] );;
gap> START_TEST( "docxpl2.tst" );
## doc/introduc.xml (26-29)
gap> InstalledPackageVersion( "ctbllib" ) <> fail;
true
## doc/introduc.xml (453-459)
gap> m:= CharacterTable( "M" );;
gap> prim:= List( Maxes( m ),
> nam -> TrivialCharacter( CharacterTable( nam ) )^m );;
gap> Length( prim );
46
## doc/tutorial.xml (27-30)
gap> origpref:= UserPreference( "AtlasRep", "DisplayFunction" );;
gap> SetUserPreference( "AtlasRep", "DisplayFunction", "Print" );
## doc/tutorial.xml (172-179)
gap> CharacterTable( "J1" );
CharacterTable( "J1" )
gap> CharacterTable( "L2(11)" );
CharacterTable( "L2(11)" )
gap> CharacterTable( "S5" );
CharacterTable( "A5.2" )
## doc/tutorial.xml (196-201)
gap> AllCharacterTableNames( Size, 168 );
[ "(2^2xD14):3", "2^3.7.3", "L3(2)", "L3(4)M7", "L3(4)M8" ]
gap> OneCharacterTableName( NrConjugacyClasses, n -> n <= 4 );
"S3"
## doc/tutorial.xml (233-238)
gap> tom:= TableOfMarks( "M11" );
TableOfMarks( "M11" )
gap> t:= CharacterTable( tom );
CharacterTable( "M11" )
## doc/tutorial.xml (258-265)
gap> t:= CharacterTable( "M11" );
CharacterTable( "M11" )
gap> HasMaxes( t );
true
gap> Maxes( t );
[ "A6.2_3", "L2(11)", "3^2:Q8.2", "A5.2", "2.S4" ]
## doc/tutorial.xml (272-275)
gap> CharacterTable( "M11M2" );
CharacterTable( "L2(11)" )
## doc/tutorial.xml (283-288)
gap> NamesOfFusionSources( t );
[ "A5.2", "A6.2_3", "P48/G1/L1/V1/ext2", "P48/G1/L1/V2/ext2",
"L2(11)", "2.S4", "3^5:M11", "3^6.M11", "3^(2+5+10).(M11x2S4)",
"s4", "3^2:Q8.2", "M11N2", "5:4", "11:5" ]
## doc/tutorial.xml (295-298)
gap> List( ComputedClassFusions( t ), r -> r.name );
[ "A11", "M12", "M23", "HS", "McL", "ON", "3^5:M11", "B" ]
## doc/tutorial.xml (321-330)
gap> t1:= CharacterTable( "A5" );;
gap> t2:= CharacterTable( "PSL", 2, 4 );;
gap> t3:= CharacterTable( "PSL", 2, 5 );;
gap> TransformingPermutationsCharacterTables( t1, t2 );
rec( columns := (), group := Group([ (4,5) ]), rows := () )
gap> TransformingPermutationsCharacterTables( t1, t3 );
rec( columns := (2,4)(3,5), group := Group([ (2,3) ]),
rows := (2,5,3,4) )
## doc/tutorial.xml (342-352)
gap> t:= CharacterTable( "M12" );
CharacterTable( "M12" )
gap> mx:= Maxes( t );
[ "M11", "M12M2", "A6.2^2", "M12M4", "L2(11)", "3^2.2.S4", "M12M7",
"2xS5", "M8.S4", "4^2:D12", "A4xS3" ]
gap> s1:= CharacterTable( mx[1] );
CharacterTable( "M11" )
gap> s2:= CharacterTable( mx[2] );
CharacterTable( "M12M2" )
## doc/tutorial.xml (366-403)
gap> GetFusionMap( s1, t );
[ 1, 3, 4, 7, 8, 10, 12, 12, 15, 14 ]
gap> GetFusionMap( s2, t );
[ 1, 3, 4, 6, 8, 10, 11, 11, 14, 15 ]
gap> Display( t );
M12
2 6 4 6 1 2 5 5 1 2 1 3 3 1 . .
3 3 1 1 3 2 . . . 1 1 . . . . .
5 1 1 . . . . . 1 . . . . 1 . .
11 1 . . . . . . . . . . . . 1 1
1a 2a 2b 3a 3b 4a 4b 5a 6a 6b 8a 8b 10a 11a 11b
2P 1a 1a 1a 3a 3b 2b 2b 5a 3b 3a 4a 4b 5a 11b 11a
3P 1a 2a 2b 1a 1a 4a 4b 5a 2a 2b 8a 8b 10a 11a 11b
5P 1a 2a 2b 3a 3b 4a 4b 1a 6a 6b 8a 8b 2a 11a 11b
11P 1a 2a 2b 3a 3b 4a 4b 5a 6a 6b 8a 8b 10a 1a 1a
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 11 -1 3 2 -1 -1 3 1 -1 . -1 1 -1 . .
X.3 11 -1 3 2 -1 3 -1 1 -1 . 1 -1 -1 . .
X.4 16 4 . -2 1 . . 1 1 . . . -1 A /A
X.5 16 4 . -2 1 . . 1 1 . . . -1 /A A
X.6 45 5 -3 . 3 1 1 . -1 . -1 -1 . 1 1
X.7 54 6 6 . . 2 2 -1 . . . . 1 -1 -1
X.8 55 -5 7 1 1 -1 -1 . 1 1 -1 -1 . . .
X.9 55 -5 -1 1 1 3 -1 . 1 -1 -1 1 . . .
X.10 55 -5 -1 1 1 -1 3 . 1 -1 1 -1 . . .
X.11 66 6 2 3 . -2 -2 1 . -1 . . 1 . .
X.12 99 -1 3 . 3 -1 -1 -1 -1 . 1 1 -1 . .
X.13 120 . -8 3 . . . . . 1 . . . -1 -1
X.14 144 4 . . -3 . . -1 1 . . . -1 1 1
X.15 176 -4 . -4 -1 . . 1 -1 . . . 1 . .
A = E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9
= (-1+Sqrt(-11))/2 = b11
## doc/tutorial.xml (409-416)
gap> IsDuplicateTable( s2 );
true
gap> IdentifierOfMainTable( s2 );
"M11"
gap> IdentifiersOfDuplicateTables( s1 );
[ "HSM9", "M12M2", "ONM11" ]
## doc/tutorial.xml (463-476)
gap> isambivalent:= tbl -> PowerMap( tbl, -1 )
> = [ 1 .. NrConjugacyClasses( tbl ) ];;
gap> AllCharacterTableNames( IsSimple, true, IsDuplicateTable, false,
> IsAbelian, false, isambivalent, true );
[ "3D4(2)", "3D4(3)", "3D4(4)", "A10", "A14", "A5", "A6", "J1", "J2",
"L2(101)", "L2(109)", "L2(113)", "L2(121)", "L2(125)", "L2(13)",
"L2(16)", "L2(17)", "L2(25)", "L2(29)", "L2(32)", "L2(37)",
"L2(41)", "L2(49)", "L2(53)", "L2(61)", "L2(64)", "L2(73)",
"L2(8)", "L2(81)", "L2(89)", "L2(97)", "O12+(2)", "O12-(2)",
"O12-(3)", "O7(5)", "O8+(2)", "O8+(3)", "O8+(7)", "O8-(2)",
"O8-(3)", "O9(3)", "S10(2)", "S12(2)", "S4(4)", "S4(5)", "S4(8)",
"S4(9)", "S6(2)", "S6(4)", "S6(5)", "S8(2)" ]
## doc/tutorial.xml (502-536)
gap> isppure:= function( p )
> return tbl -> Size( tbl ) mod p = 0 and
> ForAll( OrdersClassRepresentatives( tbl ),
> n -> n mod p <> 0 or IsPrimePowerInt( n ) );
> end;;
gap> for i in [ 2, 3, 5, 7, 11, 13 ] do
> Print( i, "\n",
> AllCharacterTableNames( IsSimple, true, IsAbelian, false,
> IsDuplicateTable, false, isppure( i ), true ),
> "\n" );
> od;
2
[ "A5", "A6", "L2(16)", "L2(17)", "L2(31)", "L2(32)", "L2(64)",
"L2(8)", "L3(2)", "L3(4)", "Sz(32)", "Sz(8)" ]
3
[ "A5", "A6", "L2(17)", "L2(19)", "L2(27)", "L2(53)", "L2(8)",
"L2(81)", "L3(2)", "L3(4)" ]
5
[ "A5", "A6", "A7", "L2(11)", "L2(125)", "L2(25)", "L2(49)", "L3(4)",
"M11", "M22", "S4(7)", "Sz(32)", "Sz(8)", "U4(2)", "U4(3)" ]
7
[ "A7", "A8", "A9", "G2(3)", "HS", "J1", "J2", "L2(13)", "L2(49)",
"L2(8)", "L2(97)", "L3(2)", "L3(4)", "M22", "O8+(2)", "S6(2)",
"Sz(8)", "U3(3)", "U3(5)", "U4(3)", "U6(2)" ]
11
[ "A11", "A12", "A13", "Co2", "HS", "J1", "L2(11)", "L2(121)",
"L2(23)", "L5(3)", "M11", "M12", "M22", "M23", "M24", "McL",
"O10+(3)", "O12+(3)", "ON", "Suz", "U5(2)", "U6(2)" ]
13
[ "2E6(2)", "2F4(2)'", "3D4(2)", "A13", "A14", "A15", "F4(2)",
"Fi22", "G2(3)", "G2(4)", "L2(13)", "L2(25)", "L2(27)", "L3(3)",
"L4(3)", "O7(3)", "O8+(3)", "S4(5)", "S6(3)", "Suz", "Sz(8)",
"U3(4)" ]
## doc/tutorial.xml (589-609)
gap> fun:= function( tbl )
> local result, p, bl;
>
> result:= false;
> for p in PrimeDivisors( Size( tbl ) ) do
> bl:= PrimeBlocks( tbl, p );
> if Length( bl.defect ) = 1 then
> result:= true;
> Print( "only one block: ", Identifier( tbl ), ", p = ", p, "\n" );
> fi;
> od;
>
> return result;
> end;;
gap> AllCharacterTableNames( IsSimple, true, IsAbelian, false,
> IsDuplicateTable, false, fun, true );
only one block: M22, p = 2
only one block: M24, p = 2
[ "M22", "M24" ]
## doc/tutorial.xml (634-637)
gap> CharacterTable( "ONN3" );
CharacterTable( "3^4:2^(1+4)D10" )
## doc/tutorial.xml (645-658)
gap> 3t:= CharacterTable( "3.ON" );;
gap> orders:= OrdersClassRepresentatives( 3t );;
gap> ord3:= PositionsProperty( orders, x -> x = 3 );
[ 2, 3, 7 ]
gap> sizes:= SizesCentralizers( 3t ){ ord3 };
[ 1382446517760, 1382446517760, 3240 ]
gap> Size( 3t );
1382446517760
gap> Collected( Factors( sizes[3] ) );
[ [ 2, 3 ], [ 3, 4 ], [ 5, 1 ] ]
gap> 9 in orders;
false
## doc/tutorial.xml (680-699)
gap> tbl:= CharacterTable( "2.A6" );;
gap> HasMaxes( tbl );
true
gap> maxes:= Maxes( tbl );
[ "2.A5", "2.A6M2", "3^2:8", "2.Symm(4)", "2.A6M5" ]
gap> mx:= List( maxes, CharacterTable );;
gap> prim1:= List( mx, s -> TrivialCharacter( s )^tbl );;
gap> Display( tbl,
> rec( chars:= prim1, centralizers:= false, powermap:= false ) );
2.A6
1a 2a 4a 3a 6a 3b 6b 8a 8b 5a 10a 5b 10b
Y.1 6 6 2 3 3 . . . . 1 1 1 1
Y.2 6 6 2 . . 3 3 . . 1 1 1 1
Y.3 10 10 2 1 1 1 1 2 2 . . . .
Y.4 15 15 3 3 3 . . 1 1 . . . .
Y.5 15 15 3 . . 3 3 1 1 . . . .
## doc/tutorial.xml (704-707)
gap> PermCharInfo( tbl, prim1 ).ATLAS;
[ "1a+5a", "1a+5b", "1a+9a", "1a+5a+9a", "1a+5b+9a" ]
## doc/tutorial.xml (712-728)
gap> tom:= TableOfMarks( tbl );
TableOfMarks( "2.A6" )
gap> allperm:= PermCharsTom( tbl, tom );;
gap> prim2:= allperm{ MaximalSubgroupsTom( tom )[1] };;
gap> Display( tbl,
> rec( chars:= prim2, centralizers:= false, powermap:= false ) );
2.A6
1a 2a 4a 3a 6a 3b 6b 8a 8b 5a 10a 5b 10b
Y.1 6 6 2 3 3 . . . . 1 1 1 1
Y.2 6 6 2 . . 3 3 . . 1 1 1 1
Y.3 10 10 2 1 1 1 1 2 2 . . . .
Y.4 15 15 3 . . 3 3 1 1 . . . .
Y.5 15 15 3 3 3 . . 1 1 . . . .
## doc/tutorial.xml (741-746)
gap> FusionToTom( tbl );
rec( map := [ 1, 2, 5, 4, 8, 3, 7, 11, 11, 6, 13, 6, 13 ],
name := "2.A6", perm := (4,5),
text := "fusion map is unique up to table autom." )
## doc/tutorial.xml (776-818)
gap> t:= CharacterTable( "Fi23" );
CharacterTable( "Fi23" )
gap> mx:= Maxes( t );
[ "2.Fi22", "O8+(3).3.2", "2^2.U6(2).2", "S8(2)", "S3xO7(3)",
"2..11.m23", "3^(1+8).2^(1+6).3^(1+2).2S4", "Fi23M8", "A12.2",
"(2^2x2^(1+8)).(3xU4(2)).2", "2^(6+8):(A7xS3)", "S4xS6(2)",
"S4(4).4", "L2(23)" ]
gap> m:= CharacterTable( mx[7] );
CharacterTable( "3^(1+8).2^(1+6).3^(1+2).2S4" )
gap> n:= ClassPositionsOfPCore( m, 3 );
[ 1 .. 6 ]
gap> f:= m / n;
CharacterTable( "3^(1+8).2^(1+6).3^(1+2).2S4/[ 1, 2, 3, 4, 5, 6 ]" )
gap> reg:= 0 * [ 1 .. NrConjugacyClasses( f ) ];;
gap> reg[1]:= Size( f );;
gap> infl:= reg{ GetFusionMap( m, f ) };
[ 165888, 165888, 165888, 165888, 165888, 165888, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
gap> ind:= Induced( m, t, [ infl ] );
[ ClassFunction( CharacterTable( "Fi23" ),
[ 207766624665600, 0, 0, 0, 603832320, 127567872, 6635520, 663552,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0 ] ) ]
gap> PermCharInfo( t, ind ).contained;
[ [ 1, 0, 0, 0, 864, 1538, 3456, 13824, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ]
gap> PositionsProperty( OrdersClassRepresentatives( t ), x -> x = 3 );
[ 5, 6, 7, 8 ]
## doc/tutorial.xml (835-841)
gap> nam:= OneCharacterTableName( CommutatorLength, x -> x > 1
> : OrderedBy:= Size );
"3.(A4x3):2"
gap> Size( CharacterTable( nam ) );
216
## doc/tutorial.xml (845-849)
gap> OneSmallGroup( Size, [ 2 .. 100 ],
> G -> CommutatorLength( G ) > 1, true );
<pc group of size 96 with 6 generators>
## doc/tutorial.xml (860-866)
gap> OneCharacterTableName( IsSimple, true, IsAbelian, false,
> IsDuplicateTable, false,
> CommutatorLength, x -> x > 1
> : OrderedBy:= Size );
fail
## doc/tutorial.xml (870-878)
gap> nam:= OneCharacterTableName( IsPerfect, true,
> IsDuplicateTable, false,
> CommutatorLength, x -> x > 1
> : OrderedBy:= Size );
"P1/G1/L1/V1/ext2"
gap> Size( CharacterTable( nam ) );
960
## doc/tutorial.xml (883-893)
gap> for n in [ 2 .. 960 ] do
> for i in [ 1 .. NrPerfectGroups( n ) ] do
> g:= PerfectGroup( n, i);
> if CommutatorLength( g ) <> 1 then
> Print( [ n, i ], "\n" );
> fi;
> od;
> od;
[ 960, 2 ]
## doc/tutorial.xml (938-948)
gap> t:= CharacterTable( "J4" );;
gap> deg1333:= Filtered( Irr( t ), x -> x[1] = 1333 );;
gap> antisym:= AntiSymmetricParts( t, deg1333, 2 );;
gap> List( antisym, x -> Position( Irr( t ), x ) );
[ 7, 6 ]
gap> ComplexConjugate( antisym[1] ) = antisym[2];
true
gap> chi:= antisym[1];; chi[1];
887778
## doc/tutorial.xml (957-964)
gap> s:= CharacterTable( Maxes( t )[1] );;
gap> Size( s ) = 2^11 * Size( CharacterTable( "M24" ) );
true
gap> rest:= RestrictedClassFunction( chi, s );;
gap> smod11:= s mod 11;;
gap> rest:= RestrictedClassFunction( rest, smod11 );;
## doc/tutorial.xml (969-975)
gap> dec:= Decomposition( Irr( smod11 ), [ rest ], "nonnegative" )[1];;
gap> Sum( dec );
9
gap> constpos:= PositionsProperty( dec, x -> x <> 0 );
[ 15, 36, 46, 53, 55, 58, 63, 67, 69 ]
## doc/tutorial.xml (979-989)
gap> smod11fuss:= GetFusionMap( smod11, s );;
gap> sfust:= GetFusionMap( s, t );;
gap> fus:= CompositionMaps( sfust, smod11fuss );;
gap> inv:= Filtered( InverseMap( fus ), IsList );
[ [ 3, 4, 5 ], [ 2, 6, 7 ], [ 8, 9 ], [ 10, 11, 16 ],
[ 12, 14, 15, 17, 18, 21 ], [ 13, 19, 20, 22 ], [ 26, 27, 28, 30 ],
[ 25, 29, 31 ], [ 34, 39 ], [ 35, 37, 38 ], [ 40, 42 ], [ 41, 43 ],
[ 44, 47, 48 ], [ 45, 49, 50 ], [ 46, 51 ], [ 56, 57 ], [ 63, 64 ],
[ 69, 70 ] ]
## doc/tutorial.xml (993-1001)
gap> const:= Irr( smod11 ){ constpos };;
gap> zero:= 0 * TrivialCharacter( smod11 );;
gap> comb:= List( Combinations( const ), x -> Sum( x, zero ) );;
gap> cand:= Filtered( comb,
> x -> ForAll( inv, l -> Length( Set( x{ l } ) ) = 1 ) );;
gap> List( cand, x -> x[1] );
[ 0, 887778 ]
## doc/tutorial.xml (1054-1093)
gap> GenProjNotProj:= function( modtbl )
> local p, tbl, X, PIMs, n, psingular, list, labels, i, j, psi,
> pos, dec, poss;
>
> p:= UnderlyingCharacteristic( modtbl );
> tbl:= OrdinaryCharacterTable( modtbl );
> X:= Irr( tbl );
> PIMs:= TransposedMat( DecompositionMatrix( modtbl ) ) * X;
> n:= Length( X );
> psingular:= Difference( [ 1 .. n ], GetFusionMap( modtbl, tbl ) );
> list:= [];
> labels:= [];
> for i in [ 1 .. n ] do
> for j in [ 1 .. i ] do
> psi:= List( [ 1 .. n ], x -> X[i][x] * X[j][x] );
> if IsZero( psi{ psingular } ) then
> # This is a generalized projective character.
> pos:= Position( list, psi );
> if pos = fail then
> Add( list, psi );
> Add( labels, [ [ j, i ] ] );
> else
> Add( labels[ pos ], [ j, i ] );
> fi;
> fi;
> od;
> od;
>
> if Length( list ) > 0 then
> # Decompose the generalized projective tensor products
> # into the projective indecomposables.
> dec:= Decomposition( PIMs, list, "nonnegative" );
> poss:= Positions( dec, fail );
> return Set( Concatenation( labels{ poss } ) );
> else
> return [];
> fi;
> end;;
## doc/tutorial.xml (1102-1111)
gap> tbl:= CharacterTable( "J2" );;
gap> modtbl:= tbl mod 2;;
gap> pairs:= GenProjNotProj( modtbl );
[ [ 6, 12 ] ]
gap> irr:= Irr( tbl );;
gap> PIMs:= TransposedMat( DecompositionMatrix( modtbl ) ) * irr;;
gap> SolutionMat( PIMs, irr[6] * irr[12] );
[ 0, 0, 0, 1, 1, 1, 0, 0, -2, 3 ]
## doc/tutorial.xml (1234-1296)
gap> ApplyCriteria:= "dummy";; # Avoid a syntax error ...
gap> ApplyCriteria:= function( tbl )
> local id, ord, invpos, cen, facttbl, factfus, invmap, factord,
> factinvpos, imgs;
> id:= ReplacedString( Identifier( tbl ), " ", "" );
> ord:= OrdersClassRepresentatives( tbl );
> invpos:= PositionsProperty( ord, x -> x <= 2 );
> if Length( invpos ) <= 3 then
> # There are at most 2 involution classes.
> Print( "#I ", id, ": ",
> "done (", Length( invpos ) - 1, " inv. class(es))\n" );
> return true;
> elif Length( invpos ) = 4 and
> ClassMultiplicationCoefficient( tbl, invpos[2], invpos[3],
> invpos[4] ) <> 0 then
> Print( "#I ", id, ": ",
> "done (3 inv. classes, nonzero str. const.)\n" );
> return true;
> fi;
> cen:= Intersection( invpos, ClassPositionsOfCentre( tbl ) );
> if Length( cen ) > 1 then
> # Consider the factor modulo the largest central el. ab. 2-group.
> facttbl:= tbl / cen;
> factfus:= GetFusionMap( tbl, facttbl );
> invmap:= InverseMap( factfus );
> factord:= OrdersClassRepresentatives( facttbl );
> factinvpos:= PositionsProperty( factord, x -> x <= 2 );
> if ForAll( factinvpos,
> i -> invmap[i] in invpos or
> ( IsList( invmap[i] ) and
> IsSubset( invpos, invmap[i] ) ) ) then
> # All involutions of the factor group lift to involutions.
> if ApplyCriteria( facttbl ) = true then
> Print( "#I ", id, ": ",
> "done (all inv. in ",
> ReplacedString( Identifier( facttbl ), " ", "" ),
> " lift to inv.)\n" );
> return true;
> fi;
> fi;
> imgs:= Set( factfus{ invpos } );
> if Length( imgs ) = 2 and
> ForAll( imgs,
> i -> invmap[i] in invpos or
> ( IsList( invmap[i] ) and
> IsSubset( invpos, invmap[i] ) ) ) then
> # There is a C2 subgroup of the factor
> # such that its involution lifts to involutions,
> # and the lifts of the C2 cover all involution classes of 'tbl'.
> Print( "#I ", id, ": ",
> "done (all inv. in ", id,
> " are lifts of a C2\n",
> "#I in the factor modulo ",
> ReplacedString( String( cen ), " ", "" ), ")\n" );
> return true;
> fi;
> fi;
> Print( "#I ", id, ": ",
> "OPEN (", Length( invpos ) - 1, " inv. class(es))\n" );
> return false;
> end;;
## doc/tutorial.xml (1304-1340)
gap> SizeScreen( [ 72 ] );;
gap> spor:= AllCharacterTableNames( IsSporadicSimple, true,
> IsDuplicateTable, false );
[ "B", "Co1", "Co2", "Co3", "F3+", "Fi22", "Fi23", "HN", "HS", "He",
"J1", "J2", "J3", "J4", "Ly", "M", "M11", "M12", "M22", "M23",
"M24", "McL", "ON", "Ru", "Suz", "Th" ]
gap> Filtered( spor,
> x -> not ApplyCriteria( CharacterTable( x ) ) );
#I B: OPEN (4 inv. class(es))
#I Co1: OPEN (3 inv. class(es))
#I Co2: done (3 inv. classes, nonzero str. const.)
#I Co3: done (2 inv. class(es))
#I F3+: done (2 inv. class(es))
#I Fi22: done (3 inv. classes, nonzero str. const.)
#I Fi23: done (3 inv. classes, nonzero str. const.)
#I HN: done (2 inv. class(es))
#I HS: done (2 inv. class(es))
#I He: done (2 inv. class(es))
#I J1: done (1 inv. class(es))
#I J2: done (2 inv. class(es))
#I J3: done (1 inv. class(es))
#I J4: done (2 inv. class(es))
#I Ly: done (1 inv. class(es))
#I M: done (2 inv. class(es))
#I M11: done (1 inv. class(es))
#I M12: done (2 inv. class(es))
#I M22: done (1 inv. class(es))
#I M23: done (1 inv. class(es))
#I M24: done (2 inv. class(es))
#I McL: done (1 inv. class(es))
#I ON: done (1 inv. class(es))
#I Ru: done (2 inv. class(es))
#I Suz: done (2 inv. class(es))
#I Th: done (1 inv. class(es))
[ "B", "Co1" ]
## doc/tutorial.xml (1357-1378)
gap> t:= CharacterTable( "B" );;
gap> invpos:= Positions( OrdersClassRepresentatives( t ), 2 );
[ 2, 3, 4, 5 ]
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> s:= First( mx,
> x -> Size( x ) = 20 * Size( CharacterTable( "HS.2" ) ) );
CharacterTable( "5:4xHS.2" )
gap> fus:= GetFusionMap( s, t );;
gap> prod:= ClassPositionsOfDirectProductDecompositions( s );
[ [ [ 1, 40 .. 157 ], [ 1 .. 39 ] ] ]
gap> fusinB:= List( prod[1], l -> fus{ l } );
[ [ 1, 18, 8, 3, 8 ],
[ 1, 3, 4, 6, 8, 9, 14, 19, 18, 18, 25, 22, 31, 36, 43, 51, 50, 54,
57, 81, 100, 2, 5, 8, 11, 16, 21, 20, 24, 34, 33, 48, 52, 59,
76, 106, 100, 100, 137 ] ]
gap> IsSubset( fusinB[2], invpos );
true
gap> h:= CharacterTable( "HS.2" );;
gap> fusinB[2]{ Positions( OrdersClassRepresentatives( h ), 2 ) };
[ 3, 4, 2, 5 ]
## doc/tutorial.xml (1389-1404)
gap> tom:= TableOfMarks( h );
TableOfMarks( "HS.2" )
gap> ord:= OrdersTom( tom );;
gap> invpos:= Positions( ord, 2 );
[ 2, 3, 534, 535 ]
gap> 8pos:= Positions( ord, 8 );;
gap> filt:= Filtered( 8pos,
> x -> ForAll( invpos,
> y -> Length( IntersectionsTom( tom, x, y ) ) >= y
> and IntersectionsTom( tom, x, y )[y] <> 0 ) );
[ 587, 589, 590, 593, 595 ]
gap> reps:= List( filt, i -> RepresentativeTom( tom, i ) );;
gap> ForAll( reps, IsElementaryAbelian );
true
## doc/tutorial.xml (1418-1432)
gap> t:= CharacterTable( "Co1" );;
gap> invpos:= Positions( OrdersClassRepresentatives( t ), 2 );
[ 2, 3, 4 ]
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> s:= First( mx, x -> Size( x ) = 3 * Factorial( 9 ) );
CharacterTable( "A9xS3" )
gap> fus:= GetFusionMap( s, t );;
gap> prod:= ClassPositionsOfDirectProductDecompositions( s );
[ [ [ 1 .. 3 ], [ 1, 4 .. 52 ] ] ]
gap> List( prod[1], l -> fus{ l } );
[ [ 1, 8, 2 ],
[ 1, 3, 4, 5, 7, 6, 13, 14, 15, 19, 24, 28, 36, 37, 39, 50, 61, 61
] ]
## doc/tutorial.xml (1515-1540)
gap> sporcov:= AllCharacterTableNames( IsSporadicSimple, true,
> IsDuplicateTable, false, OfThose, SchurCover );
[ "12.M22", "2.B", "2.Co1", "2.HS", "2.J2", "2.M12", "2.Ru", "3.F3+",
"3.J3", "3.McL", "3.ON", "6.Fi22", "6.Suz", "Co2", "Co3", "Fi23",
"HN", "He", "J1", "J4", "Ly", "M", "M11", "M23", "M24", "Th" ]
gap> Filtered( sporcov, x -> '.' in x );
[ "12.M22", "2.B", "2.Co1", "2.HS", "2.J2", "2.M12", "2.Ru", "3.F3+",
"3.J3", "3.McL", "3.ON", "6.Fi22", "6.Suz" ]
gap> relevant:= [ "2.M22", "4.M22", "2.B", "2.Co1", "2.HS", "2.J2",
> "2.M12", "2.Ru", "2.Fi22", "2.Suz" ];;
gap> Filtered( relevant,
> x -> not ApplyCriteria( CharacterTable( x ) ) );
#I 2.M22: done (3 inv. classes, nonzero str. const.)
#I 4.M22: done (2 inv. class(es))
#I 2.B: OPEN (5 inv. class(es))
#I 2.Co1: OPEN (4 inv. class(es))
#I 2.HS: done (3 inv. classes, nonzero str. const.)
#I 2.J2: done (3 inv. classes, nonzero str. const.)
#I 2.M12: done (3 inv. classes, nonzero str. const.)
#I 2.Ru: done (3 inv. classes, nonzero str. const.)
#I 2.Fi22/[1,2]: done (3 inv. classes, nonzero str. const.)
#I 2.Fi22: done (all inv. in 2.Fi22/[1,2] lift to inv.)
#I 2.Suz: done (3 inv. classes, nonzero str. const.)
[ "2.B", "2.Co1" ]
## doc/tutorial.xml (1550-1559)
gap> t:= CharacterTable( "B" );;
gap> 2t:= CharacterTable( "2.B" );;
gap> invpost:= Positions( OrdersClassRepresentatives( t ), 2 );
[ 2, 3, 4, 5 ]
gap> invpos2t:= Positions( OrdersClassRepresentatives( 2t ), 2 );
[ 2, 3, 4, 5, 7 ]
gap> GetFusionMap( 2t, t ){ invpos2t };
[ 1, 2, 3, 3, 5 ]
## doc/tutorial.xml (1572-1575)
gap> ClassMultiplicationCoefficient( t, 2, 3, 5 );
120
## doc/tutorial.xml (1585-1594)
gap> t:= CharacterTable( "Co1" );;
gap> 2t:= CharacterTable( "2.Co1" );;
gap> invpost:= Positions( OrdersClassRepresentatives( t ), 2 );
[ 2, 3, 4 ]
gap> invpos2t:= Positions( OrdersClassRepresentatives( 2t ), 2 );
[ 2, 3, 4, 6 ]
gap> GetFusionMap( 2t, t ){ invpos2t };
[ 1, 2, 2, 4 ]
## doc/tutorial.xml (1605-1608)
gap> ClassMultiplicationCoefficient( t, 2, 2, 4 );
264
## doc/tutorial.xml (1707-1771)
gap> list:= [
> [ "A5", "2.A5" ],
> [ "L3(2)", "2.L3(2)" ],
> [ "L3(4)", "2.L3(4)", "2^2.L3(4)", "4_1.L3(4)", "4_2.L3(4)",
> "(2x4).L3(4)", "4^2.L3(4)" ],
> [ "A8", "2.A8" ],
> [ "U4(2)", "2.U4(2)"],
> [ "U6(2)", "2.U6(2)", "2^2.U6(2)" ],
> [ "A6", "2.A6" ],
> [ "Sz(8)", "2.Sz(8)", "2^2.Sz(8)" ],
> [ "S6(2)", "2.S6(2)" ],
> [ "O8+(2)", "2.O8+(2)", "2^2.O8+(2)" ],
> [ "G2(4)", "2.G2(4)" ],
> [ "F4(2)", "2.F4(2)" ],
> [ "2E6(2)", "2.2E6(2)", "2^2.2E6(2)" ] ];;
gap> Filtered( Concatenation( list ),
> x -> not ApplyCriteria( CharacterTable( x ) ) );
#I A5: done (1 inv. class(es))
#I 2.A5: done (1 inv. class(es))
#I L3(2): done (1 inv. class(es))
#I 2.L3(2): done (1 inv. class(es))
#I L3(4): done (1 inv. class(es))
#I 2.L3(4): done (3 inv. classes, nonzero str. const.)
#I 2^2.L3(4)/[1,2,3,4]: done (1 inv. class(es))
#I 2^2.L3(4): done (all inv. in 2^2.L3(4)/[1,2,3,4] lift to inv.)
#I 4_1.L3(4): done (2 inv. class(es))
#I 4_2.L3(4): done (2 inv. class(es))
#I (2x4).L3(4): done (all inv. in (2x4).L3(4) are lifts of a C2
#I in the factor modulo [1,2,3,4])
#I 4^2.L3(4): done (all inv. in 4^2.L3(4) are lifts of a C2
#I in the factor modulo [1,2,3,4])
#I A8: done (2 inv. class(es))
#I 2.A8: done (2 inv. class(es))
#I U4(2): done (2 inv. class(es))
#I 2.U4(2): done (2 inv. class(es))
#I U6(2): done (3 inv. classes, nonzero str. const.)
#I 2.U6(2)/[1,2]: done (3 inv. classes, nonzero str. const.)
#I 2.U6(2): done (all inv. in 2.U6(2)/[1,2] lift to inv.)
#I 2^2.U6(2)/[1,2,3,4]: done (3 inv. classes, nonzero str. const.)
#I 2^2.U6(2): done (all inv. in 2^2.U6(2)/[1,2,3,4] lift to inv.)
#I A6: done (1 inv. class(es))
#I 2.A6: done (1 inv. class(es))
#I Sz(8): done (1 inv. class(es))
#I 2.Sz(8): done (2 inv. class(es))
#I 2^2.Sz(8)/[1,2,3,4]: done (1 inv. class(es))
#I 2^2.Sz(8): done (all inv. in 2^2.Sz(8)/[1,2,3,4] lift to inv.)
#I S6(2): OPEN (4 inv. class(es))
#I 2.S6(2): OPEN (3 inv. class(es))
#I O8+(2): OPEN (5 inv. class(es))
#I 2.O8+(2): OPEN (5 inv. class(es))
#I 2^2.O8+(2): OPEN (5 inv. class(es))
#I G2(4): done (2 inv. class(es))
#I 2.G2(4): done (3 inv. classes, nonzero str. const.)
#I F4(2): OPEN (4 inv. class(es))
#I 2.F4(2)/[1,2]: OPEN (4 inv. class(es))
#I 2.F4(2): OPEN (9 inv. class(es))
#I 2E6(2): done (3 inv. classes, nonzero str. const.)
#I 2.2E6(2)/[1,2]: done (3 inv. classes, nonzero str. const.)
#I 2.2E6(2): done (all inv. in 2.2E6(2)/[1,2] lift to inv.)
#I 2^2.2E6(2)/[1,2,3,4]: done (3 inv. classes, nonzero str. const.)
#I 2^2.2E6(2): done (all inv. in 2^2.2E6(2)/[1,2,3,4] lift to inv.)
[ "S6(2)", "2.S6(2)", "O8+(2)", "2.O8+(2)", "2^2.O8+(2)", "F4(2)",
"2.F4(2)" ]
## doc/tutorial.xml (1788-1802)
gap> t:= CharacterTable( "S6(2)" );;
gap> invpos:= Positions( OrdersClassRepresentatives( t ), 2 );
[ 2, 3, 4, 5 ]
gap> mx:= List( Maxes( t ), CharacterTable );;
gap> s:= First( mx,
> x -> Size( x ) = 2^6 * Size( CharacterTable( "L3(2)" ) ) );
CharacterTable( "2^6:L3(2)" )
gap> corepos:= ClassPositionsOfPCore( s, 2 );
[ 1 .. 5 ]
gap> OrdersClassRepresentatives( t ){ corepos };
[ 1, 2, 2, 2, 2 ]
gap> GetFusionMap( s, t ){ corepos };
[ 1, 3, 4, 2, 5 ]
## doc/tutorial.xml (1812-1820)
gap> 2t:= CharacterTable( "2.S6(2)" );;
gap> invpost:= Positions( OrdersClassRepresentatives( t ), 2 );
[ 2, 3, 4, 5 ]
gap> invpos2t:= Positions( OrdersClassRepresentatives( 2t ), 2 );
[ 2, 4, 6 ]
gap> GetFusionMap( 2t, t ){ invpos2t };
[ 1, 3, 5 ]
## doc/tutorial.xml (1831-1834)
gap> ClassMultiplicationCoefficient( t, 3, 5, 5 );
15
## doc/tutorial.xml (1842-1858)
gap> t:= CharacterTable( "O8+(2)" );;
gap> tom:= TableOfMarks( t );
TableOfMarks( "O8+(2)" )
gap> ord:= OrdersTom( tom );;
gap> invpos:= Positions( ord, 2 );
[ 2, 3, 4, 5, 6 ]
gap> 8pos:= Positions( ord, 8 );;
gap> filt:= Filtered( 8pos,
> x -> ForAll( invpos,
> y -> Length( IntersectionsTom( tom, x, y ) ) >= y
> and IntersectionsTom( tom, x, y )[y] <> 0 ) );
[ 151, 153 ]
gap> reps:= List( filt, i -> RepresentativeTom( tom, i ) );;
gap> ForAll( reps, IsElementaryAbelian );
true
## doc/tutorial.xml (1869-1877)
gap> 2t:= CharacterTable( "2.O8+(2)" );;
gap> invpost:= Positions( OrdersClassRepresentatives( t ), 2 );
[ 2, 3, 4, 5, 6 ]
gap> invpos2t:= Positions( OrdersClassRepresentatives( 2t ), 2 );
[ 2, 3, 4, 5, 8 ]
gap> GetFusionMap( 2t, t ){ invpos2t };
[ 1, 2, 3, 3, 6 ]
## doc/tutorial.xml (1887-1890)
gap> ClassMultiplicationCoefficient( t, 2, 3, 6 );
4
## doc/tutorial.xml (1901-1907)
gap> v4t:= CharacterTable( "2^2.O8+(2)" );;
gap> invposv4t:= Positions( OrdersClassRepresentatives( v4t ), 2 );
[ 2, 3, 4, 5, 12 ]
gap> GetFusionMap( v4t, t ){ invposv4t };
[ 1, 1, 1, 2, 6 ]
## doc/tutorial.xml (1916-1919)
gap> ClassMultiplicationCoefficient( t, 2, 6, 6 );
27
## doc/tutorial.xml (1931-1947)
gap> t:= CharacterTable( "F4(2)" );;
gap> invpost:= Positions( OrdersClassRepresentatives( t ), 2 );
[ 2, 3, 4, 5 ]
gap> "S8(2)" in Maxes( t );
true
gap> s:= CharacterTable( "S8(2)M4" );
CharacterTable( "2^10.A8" )
gap> corepos:= ClassPositionsOfPCore( s, 2 );
[ 1 .. 7 ]
gap> OrdersClassRepresentatives( s ){ corepos };
[ 1, 2, 2, 2, 2, 2, 2 ]
gap> poss:= PossibleClassFusions( s, t );;
gap> List( poss, map -> map{ corepos } );
[ [ 1, 4, 2, 3, 4, 5, 5 ], [ 1, 4, 2, 3, 4, 5, 5 ],
[ 1, 4, 3, 2, 4, 5, 5 ], [ 1, 4, 3, 2, 4, 5, 5 ] ]
## doc/tutorial.xml (1956-1962)
gap> 2t:= CharacterTable( "2.F4(2)" );;
gap> invpos2t:= Positions( OrdersClassRepresentatives( 2t ), 2 );
[ 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
gap> GetFusionMap( 2t, t ){ invpos2t };
[ 1, 2, 2, 3, 3, 4, 4, 5, 5 ]
## doc/../gap4/ctadmin.gd (648-656)
gap> s5:= CharacterTable( "A5.2" );
CharacterTable( "A5.2" )
gap> sym5:= CharacterTable( "Symmetric", 5 );
CharacterTable( "Sym(5)" )
gap> TransformingPermutationsCharacterTables( s5, sym5 );
rec( columns := (2,3,4,7,5), group := Group(()),
rows := (1,7,3,4,6,5,2) )
## doc/../gap4/ctadmin.gd (664-669)
gap> CharacterTable( "J5" );
fail
gap> CharacterTable( "A5" ) mod 2;
BrauerTable( "A5", 2 )
## doc/../gap4/ctadmin.gd (700-707)
gap> BrauerTable( "A5", 2 );
BrauerTable( "A5", 2 )
gap> BrauerTable( "J5", 2 ); # no ordinary table with name J5
fail
gap> BrauerTable( "M", 2 ); # Brauer table not known
fail
## doc/../gap4/ctadmin.gd (1024-1026)
gap> names:= AllCharacterTableNames();;
## doc/../gap4/ctadmin.gd (1031-1034)
gap> simpnames:= AllCharacterTableNames( IsSimple, true,
> IsAbelian, false );;
## doc/../gap4/ctadmin.gd (1039-1043)
gap> AllCharacterTableNames( IsSimple, true, IsAbelian, false,
> Size, [ 1 .. 100 ] );
[ "A5", "A6M2", "Alt(5)" ]
## doc/../gap4/ctadmin.gd (1056-1059)
gap> AllCharacterTableNames( Size, IsPrimeInt );
[ "2.Alt(2)", "Alt(3)", "C2", "C3", "Sym(2)" ]
## doc/../gap4/ctadmin.gd (1064-1068)
gap> AllCharacterTableNames( Identifier,
> x -> PositionSublist( x, "L8" ) <> fail );
[ "L8(2)", "P1L82", "P2L82" ]
## doc/../gap4/ctadmin.gd (1078-1087)
gap> CTblLib.SupportedAttributes;
[ "AbelianInvariants", "HasFusionToTom", "Identifier",
"IdentifiersOfDuplicateTables", "InfoText", "IsAbelian",
"IsAlmostSimple", "IsAtlasCharacterTable", "IsDuplicateTable",
"IsNontrivialDirectProduct", "IsPerfect", "IsQuasisimple",
"IsSimple", "IsSporadicSimple", "KnowsDeligneLusztigNames",
"KnowsSomeGroupInfo", "Maxes", "NamesOfFusionSources",
"NrConjugacyClasses", "Size" ]
## doc/../gap4/ctadmin.gd (1109-1122)
gap> AllCharacterTableNames( IsSporadicSimple, true,
> Size, [ 1 .. 10^6 ],
> IsDuplicateTable, false );
[ "J1", "J2", "M11", "M12", "M22" ]
gap> AllCharacterTableNames( IsSporadicSimple, true,
> Size, [ 1 .. 10^6 ],
> IsDuplicateTable, false : OrderedBy:= Size );
[ "M11", "M12", "J1", "M22", "J2" ]
gap> AllCharacterTableNames( IsSporadicSimple, true,
> Size, [ 1 .. 10^6 ],
> IsDuplicateTable, false : OrderedBy:= NrConjugacyClasses );
[ "M11", "M22", "J1", "M12", "J2" ]
## doc/../gap4/ctadmin.gd (1136-1140)
gap> maxesnames:= AllCharacterTableNames( IsSporadicSimple, true,
> HasMaxes, true,
> OfThose, Maxes );;
## doc/../gap4/ctadmin.gd (1180-1185)
gap> OneCharacterTableName( IsSimple, true, Size, 60 );
"A5"
gap> OneCharacterTableName( IsSimple, true, Size, 20 );
fail
## doc/../gap4/ctadmin.gd (1225-1236)
gap> tbl:= CharacterTable( "Alternating", 5 );;
gap> NameOfEquivalentLibraryCharacterTable( tbl );
"A5"
gap> NamesOfEquivalentLibraryCharacterTables( tbl );
[ "A5", "A6M2", "Alt(5)" ]
gap> tbl:= CharacterTable( "Cyclic", 17 );;
gap> NameOfEquivalentLibraryCharacterTable( tbl );
fail
gap> NamesOfEquivalentLibraryCharacterTables( tbl );
[ ]
## doc/../gap4/ctadmin.gi (5428-5433)
gap> TableOfMarks( CharacterTable( "A5" ) );
TableOfMarks( "A5" )
gap> TableOfMarks( CharacterTable( "M" ) );
fail
## doc/../gap4/tomlib_only.g (113-116)
gap> CharacterTable( TableOfMarks( "A5" ) );
CharacterTable( "A5" )
## doc/../gap4/tomlib_only.g (50-57)
gap> tbl:= CharacterTable( "A5" );
CharacterTable( "A5" )
gap> tom:= TableOfMarks( "A5" );
TableOfMarks( "A5" )
gap> FusionCharTableTom( tbl, tom );
[ 1, 2, 3, 5, 5 ]
## doc/../gap4/ctadmin.gd (1835-1840)
gap> FusionToTom( CharacterTable( "2.A6" ) );
rec( map := [ 1, 2, 5, 4, 8, 3, 7, 11, 11, 6, 13, 6, 13 ],
name := "2.A6", perm := (4,5),
text := "fusion map is unique up to table autom." )
## doc/../gap4/ctadmin.gd (2621-2626)
gap> NameOfLibraryCharacterTable( "A5" );
"A5"
gap> NameOfLibraryCharacterTable( "S5" );
"A5.2"
## doc/../gap4/ctadmin.gd (1972-2008)
gap> GroupInfoForCharacterTable( CharacterTable( "A5" ) );
[ [ "AlternatingGroup", [ 5 ] ], [ "AtlasGroup", [ "A5" ] ],
[ "AtlasStabilizer", [ "A6", "A6G1-p6aB0" ] ],
[ "AtlasStabilizer", [ "A6", "A6G1-p6bB0" ] ],
[ "AtlasStabilizer", [ "L2(11)", "L211G1-p11aB0" ] ],
[ "AtlasStabilizer", [ "L2(11)", "L211G1-p11bB0" ] ],
[ "AtlasStabilizer", [ "L2(19)", "L219G1-p57aB0" ] ],
[ "AtlasStabilizer", [ "L2(19)", "L219G1-p57bB0" ] ],
[ "AtlasSubgroup", [ "A5.2", 1 ] ], [ "AtlasSubgroup", [ "A6", 1 ] ]
, [ "AtlasSubgroup", [ "A6", 2 ] ],
[ "AtlasSubgroup", [ "J2", 9 ] ],
[ "AtlasSubgroup", [ "L2(109)", 4 ] ],
[ "AtlasSubgroup", [ "L2(109)", 5 ] ],
[ "AtlasSubgroup", [ "L2(11)", 1 ] ],
[ "AtlasSubgroup", [ "L2(11)", 2 ] ],
[ "AtlasSubgroup", [ "S6(3)", 11 ] ],
[ "GroupForTom", [ "2^4:A5", 68 ] ],
[ "GroupForTom", [ "2^4:A5`", 56 ] ], [ "GroupForTom", [ "A5" ] ],
[ "GroupForTom", [ "A5xA5", 85 ] ], [ "GroupForTom", [ "A6", 21 ] ],
[ "GroupForTom", [ "J2", 99 ] ],
[ "GroupForTom", [ "L2(109)", 25 ] ],
[ "GroupForTom", [ "L2(11)", 15 ] ],
[ "GroupForTom", [ "L2(125)", 18 ] ],
[ "GroupForTom", [ "L2(16)", 18 ] ],
[ "GroupForTom", [ "L2(19)", 17 ] ],
[ "GroupForTom", [ "L2(29)", 19 ] ],
[ "GroupForTom", [ "L2(31)", 25 ] ],
[ "GroupForTom", [ "S5", 18 ] ], [ "PSL", [ 2, 4 ] ],
[ "PSL", [ 2, 5 ] ], [ "PerfectGroup", [ 60, 1 ] ],
[ "PrimitiveGroup", [ 5, 4 ] ], [ "PrimitiveGroup", [ 6, 1 ] ],
[ "PrimitiveGroup", [ 10, 1 ] ], [ "SmallGroup", [ 60, 5 ] ],
[ "TransitiveGroup", [ 5, 4 ] ], [ "TransitiveGroup", [ 6, 12 ] ],
[ "TransitiveGroup", [ 10, 7 ] ], [ "TransitiveGroup", [ 12, 33 ] ],
[ "TransitiveGroup", [ 15, 5 ] ], [ "TransitiveGroup", [ 20, 15 ] ],
[ "TransitiveGroup", [ 30, 9 ] ] ]
## doc/../gap4/ctadmin.gd (2031-2036)
gap> KnowsSomeGroupInfo( CharacterTable( "A5" ) );
true
gap> KnowsSomeGroupInfo( CharacterTable( "M" ) );
false
## doc/../gap4/ctadmin.gd (2065-2068)
gap> CharacterTableForGroupInfo( [ "AlternatingGroup", [ 5 ] ] );
CharacterTable( "A5" )
## doc/../gap4/ctadmin.gd (2101-2106)
gap> GroupForGroupInfo( [ "AlternatingGroup", [ 5 ] ] );
Alt( [ 1 .. 5 ] )
gap> GroupForGroupInfo( [ "PrimitiveGroup", [ 5, 4 ] ] );
A(5)
## doc/../gap4/ctadmin.gd (2141-2149)
gap> g:= GroupForTom( "A5" ); u:= GroupForTom( "A5", 2 );
Group([ (2,4)(3,5), (1,2,5) ])
Group([ (2,3)(4,5) ])
gap> IsSubset( g, u );
true
gap> GroupForTom( "J4" );
fail
## doc/../gap4/ctadmin.gd (2187-2190)
gap> AtlasStabilizer( "A5","A5G1-p5B0");
Group([ (1,2)(3,4), (2,3,4) ])
## doc/../gap4/ctadmin.gd (2211-2217)
gap> mx:= Maxes( CharacterTable( "J1" ) );
[ "L2(11)", "2^3.7.3", "2xA5", "19:6", "11:10", "D6xD10", "7:6" ]
gap> List( mx, name -> IsNontrivialDirectProduct(
> CharacterTable( name ) ) );
[ false, false, true, false, false, true, false ]
## doc/../dlnames/dlnames.gd (138-144)
gap> tbl:= CharacterTable( "U4(2).2" );;
gap> UnipotentCharacter( tbl, [ [ 0, 1 ], [ 2 ] ] );
Character( CharacterTable( "U4(2).2" ),
[ 15, 7, 3, -3, 0, 3, -1, 1, 0, 1, -2, 1, 0, 0, -1, 5, 1, 3, -1, 2,
-1, 1, -1, 0, 0 ] )
## doc/../dlnames/dlnames.gd (62-72)
gap> DeligneLusztigNames( "L2(7)" );
[ [ 2 ],,,, [ 1, 1 ] ]
gap> tbl:= CharacterTable( "L2(7)" );
CharacterTable( "L3(2)" )
gap> HasDeligneLusztigNames( tbl );
true
gap> DeligneLusztigNames( rec( isoc:= "A", isot:= "simple",
> l:= 2, q:= 2 ) );
[ [ 3 ],,, [ 2, 1 ],, [ 1, 1, 1 ] ]
## doc/../dlnames/dlnames.gd (96-105)
gap> tbl:= CharacterTable( "F4(2)" );;
gap> DeligneLusztigName( Irr( tbl )[9] );
fail
gap> HasDeligneLusztigNames( tbl );
true
gap> List( [ 1 .. 8 ], i -> DeligneLusztigName( Irr( tbl )[i] ) );
[ "phi{1,0}", "[ [ 2 ], [ ] ]", "phi{2,4}''", "phi{2,4}'",
"F4^II[1]", "phi{4,1}", "F4^I[1]", "phi{9,2}" ]
## doc/../gap4/ctadmin.gd (2260-2265)
gap> KnowsDeligneLusztigNames( CharacterTable( "A5" ) );
true
gap> KnowsDeligneLusztigNames( CharacterTable( "M" ) );
false
## doc/../gap4/ctbltoct.g (807-812)
gap> StringCTblLibInfo( CharacterTable( "A5" ) );;
gap> StringCTblLibInfo( CharacterTable( "A5" ) mod 2 );;
gap> StringCTblLibInfo( "A5" );;
gap> StringCTblLibInfo( "A5", 2 );;
## doc/../gap4/atlasstr.g (1854-1868)
gap> str:= StringAtlasContents();;
gap> pos:= PositionNthOccurrence( str, '\n', 10 );;
gap> Print( str{ [ 1 .. pos ] } );
A5 = L2(4) = L2(5) 2 2:2, 3:2, 5:2
L3(2) = L2(7) 3 2:3, 3:3, 7:3
A6 = L2(9) = S4(2)' 4 2:4, 3:4, 5:5
L2(8) = R(3)' 6 2:6, 3:6, 7:6
L2(11) 7 2:7, 3:7, 5:8, 11:8
L2(13) 8 2:9, 3:9, 7:10, 13:10
L2(17) 9 2:11, 3:11, 17:12
A7 10 2:13, 3:13, 5:14, 7:15
L2(19) 11 2:16, 3:16, 5:17, 19:18
L2(16) 12 2:19, 3:20, 5:20, 17:21
## doc/../gap4/atlasstr.g (1235-1258)
gap> DisplayAtlasMap( "M12" );
--------- ---------
| | | |
| G | | G.2 | 15
| | | |
--------- ---------
--------- ---------
| | | |
| 2.G | | 2.G.2 | 11
| | | |
--------- ---------
15 9
gap> DisplayAtlasMap( "M12", 2 );
--------- ---------
| | | |
| G | | G.2 | 6
| | | |
--------- ---------
6 0
gap> StringsAtlasMap( "M11" );
[ "--------- ", "| | ", "| G | 10", "| | ",
"--------- ", " 10 " ]
## doc/../gap4/atlasstr.g (1271-1282)
gap> DisplayAtlasMap( "S10(2)" );
---------
| |
| G | 198
| |
---------
198
gap> DisplayAtlasMap( "L12(27)" );
gap> StringsAtlasMap( "L12(27)" );
fail
## doc/../gap4/atlasstr.g (1292-1322)
gap> DisplayAtlasMap( rec(
> labels:= [ [ "G", "G.3" ],
> [ "2.G", "" ],
> [ "2'.G", "" ],
> [ "2''.G", "" ] ],
> shapes:= [ [ "closed", "closed" ],
> [ "closed", "empty" ],
> [ "closed", "empty" ],
> [ "closed", "empty" ] ],
> labelscol:= [ "1", "1" ],
> labelsrow:= [ "1", "1", "1", "1" ],
> dashedhorz:= [ false, false, true, true ],
> dashedvert:= [ false, false ],
> showdashedrows:= true ) );
--------- ---------
| | | |
| G | | G.3 | 1
| | | |
--------- ---------
---------
| |
| 2.G | 1
| |
---------
2'.G ---------
---------
2''.G ---------
---------
1 1
## doc/../gap4/atlasstr.g (1328-1350)
gap> DisplayAtlasMap( rec(
> labels:= [ [ "G", "G.2" ],
> [ "3.G", "3.G.2" ] ],
> shapes:= [ [ "closed", "closed" ],
> [ "closed", "open" ] ],
> labelscol:= [ "1", "1" ],
> labelsrow:= [ "1", "1" ],
> dashedhorz:= [ false, false ],
> dashedvert:= [ false, false ],
> showdashedrows:= true ) );
--------- ---------
| | | |
| G | | G.2 | 1
| | | |
--------- ---------
--------- --------
| | |
| 3.G | | 3.G.2 1
| | |
---------
1 1
## doc/../gap4/ctadmin.gd (2365-2372)
gap> Maxes( CharacterTable( "A6" ) );
[ "A5", "A6M2", "3^2:4", "s4", "A6M5" ]
gap> IsDuplicateTable( CharacterTable( "A5" ) );
false
gap> IsDuplicateTable( CharacterTable( "A6M2" ) );
true
## doc/../gap4/ctadmin.gd (2402-2409)
gap> Maxes( CharacterTable( "A6" ) );
[ "A5", "A6M2", "3^2:4", "s4", "A6M5" ]
gap> IdentifierOfMainTable( CharacterTable( "A5" ) );
fail
gap> IdentifierOfMainTable( CharacterTable( "A6M2" ) );
"A5"
## doc/../gap4/ctadmin.gd (2439-2446)
gap> Maxes( CharacterTable( "A6" ) );
[ "A5", "A6M2", "3^2:4", "s4", "A6M5" ]
gap> IdentifiersOfDuplicateTables( CharacterTable( "A5" ) );
[ "A6M2", "Alt(5)" ]
gap> IdentifiersOfDuplicateTables( CharacterTable( "A6M2" ) );
[ ]
## doc/../gap4/ctadmin.gd (1747-1755)
gap> tbl:= CharacterTable( "M11" );;
gap> HasMaxes( tbl );
true
gap> maxes:= Maxes( tbl );
[ "A6.2_3", "L2(11)", "3^2:Q8.2", "A5.2", "2.S4" ]
gap> CharacterTable( maxes[1] );
CharacterTable( "A6.2_3" )
## doc/../gap4/ctadmin.gd (1876-1882)
gap> ProjectivesInfo( CharacterTable( "A5" ) );
[ rec(
chars := [ [ 2, 0, -1, E(5)+E(5)^4, E(5)^2+E(5)^3 ],
[ 2, 0, -1, E(5)^2+E(5)^3, E(5)+E(5)^4 ],
[ 4, 0, 1, -1, -1 ], [ 6, 0, 0, 1, 1 ] ], name := "2.A5" ) ]
## doc/../gap4/ctadmin.gd (1924-1927)
gap> ExtensionInfoCharacterTable( CharacterTable( "A5" ) );
[ "2", "2" ]
## doc/../gap4/ctadmin.gi (38-41)
gap> AllCharacterTableNames( InfoText,
> s -> PositionSublist( s, "tests:" ) <> fail );;
## doc/../gap4/ctadmin.gd (960-969)
gap> Maxes( CharacterTable( "A6" ) );
[ "A5", "A6M2", "3^2:4", "s4", "A6M5" ]
gap> IsAtlasCharacterTable( CharacterTable( "A5" ) );
true
gap> IsAtlasCharacterTable( CharacterTable( "A5" ) mod 2 );
true
gap> IsAtlasCharacterTable( CharacterTable( "A6M2" ) );
false
## doc/../gap4/ctadmin.gd (788-807)
gap> c5:= CharacterTableSpecialized( CharacterTable( "Cyclic" ), 5 );
CharacterTable( "C5" )
gap> Display( c5 );
C5
5 1 1 1 1 1
1a 5a 5b 5c 5d
5P 1a 1a 1a 1a 1a
X.1 1 1 1 1 1
X.2 1 A B /B /A
X.3 1 B /A A /B
X.4 1 /B A /A B
X.5 1 /A /B B A
A = E(5)
B = E(5)^2
## doc/../gap4/ctadmin.gd (812-821)
gap> HasClassParameters( c5 ); HasCharacterParameters( c5 );
true
true
gap> ClassParameters( c5 ); CharacterParameters( c5 );
[ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ]
[ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ]
gap> ClassParameters( CharacterTable( "Symmetric", 3 ) );
[ [ 1, [ 1, 1, 1 ] ], [ 1, [ 2, 1 ] ], [ 1, [ 3 ] ] ]
## doc/../gap4/ctadmin.gd (828-836)
gap> CharacterTable( "Cyclic" ).irreducibles[1][1]( 5, 2, 3 );
E(5)
gap> tbl:= CharacterTable( "Symmetric" );;
gap> tbl.irreducibles[1][1]( 5, [ 3, 2 ], [ 2, 2, 1 ] );
1
gap> tbl.orders[1]( 5, [ 2, 1, 1, 1 ] );
2
## doc/ctbllibr.xml (596-630)
gap> Print( CharacterTable( "Cyclic" ), "\n" );
rec(
centralizers := [ function ( n, k )
return n;
end ],
charparam := [ function ( n )
return [ 0 .. n - 1 ];
end ],
classparam := [ function ( n )
return [ 0 .. n - 1 ];
end ],
domain := <Category "(IsInt and IsPosRat)">,
identifier := "Cyclic",
irreducibles := [ [ function ( n, k, l )
return E( n ) ^ (k * l);
end ] ],
isGenericTable := true,
libinfo := rec(
firstname := "Cyclic",
othernames := [ ] ),
orders := [ function ( n, k )
return n / Gcd( n, k );
end ],
powermap := [ function ( n, k, pow )
return [ 1, k * pow mod n ];
end ],
size := function ( n )
return n;
end,
specializedname := function ( q )
return Concatenation( "C", String( q ) );
end,
text := "generic character table for cyclic groups" )
## doc/../gap4/ctadmin.gd (2579-2595)
gap> AtlasLabelsOfIrreducibles( CharacterTable( "3.A7.2" ) );
[ "\\chi_{1,0}", "\\chi_{1,1}", "\\chi_{2,0}", "\\chi_{2,1}",
"\\chi_{3+4}", "\\chi_{5,0}", "\\chi_{5,1}", "\\chi_{6,0}",
"\\chi_{6,1}", "\\chi_{7,0}", "\\chi_{7,1}", "\\chi_{8,0}",
"\\chi_{8,1}", "\\chi_{9,0}", "\\chi_{9,1}", "\\chi_{17+17\\ast 2}",
"\\chi_{18+18\\ast 2}", "\\chi_{19+19\\ast 2}",
"\\chi_{20+20\\ast 2}", "\\chi_{21+21\\ast 2}",
"\\chi_{22+23\\ast 8}", "\\chi_{22\\ast 8+23}" ]
gap> AtlasLabelsOfIrreducibles( CharacterTable( "3.A7.2" ), "short" );
[ "\\chi_{1,0}", "\\chi_{1,1}", "\\chi_{2,0}", "\\chi_{2,1}",
"\\chi_{3+}", "\\chi_{5,0}", "\\chi_{5,1}", "\\chi_{6,0}",
"\\chi_{6,1}", "\\chi_{7,0}", "\\chi_{7,1}", "\\chi_{8,0}",
"\\chi_{8,1}", "\\chi_{9,0}", "\\chi_{9,1}", "\\chi_{17+}",
"\\chi_{18+}", "\\chi_{19+}", "\\chi_{20+}", "\\chi_{21+}",
"\\chi_{22+}", "\\chi_{23+}" ]
## doc/../gap4/ctadmin.gd (1692-1702)
gap> tbl:= CharacterTable( "m10" );
CharacterTable( "A6.2_3" )
gap> HasCASInfo( tbl );
true
gap> CASInfo( tbl );
[ rec( name := "m10", permchars := (3,5)(4,8,7,6), permclasses := (),
text := "names: m10\norder: 2^4.3^2.5 = 720\nnumber of c\
lasses: 8\nsource: cambridge atlas\ncomments: point stabilizer of \
mathieu-group m11\ntest: orth, min, sym[3]\n" ) ]
## doc/../gap4/ctadmin.gd (1711-1716)
gap> First( ComputedClassFusions( tbl ), x -> x.name = "M11" );
rec( map := [ 1, 2, 3, 4, 5, 4, 7, 8 ], name := "M11",
text := "fusion is unique up to table automorphisms,\nthe representa\
tive is equal to the fusion map on the CAS table" )
## doc/../gap4/ctadmin.gd (570-577)
gap> LibInfoCharacterTable( "S5" );
rec( fileName := "ctoalter", firstName := "A5.2" )
gap> LibInfoCharacterTable( "S5mod2" );
rec( fileName := "ctbalter", firstName := "A5.2mod2" )
gap> LibInfoCharacterTable( "J5" );
fail
## doc/../gap4/ctadmin.gd (344-350)
gap> CharacterTable( "private" );
fail
gap> NotifyNameOfCharacterTable( "A5", [ "private" ] );
gap> a5:= CharacterTable( "private" );
CharacterTable( "A5" )
## doc/../gap4/ctadmin.gd (1465-1475)
gap> s5:= CharacterTable( "S5" );
CharacterTable( "A5.2" )
gap> fus:= PossibleClassFusions( a5, s5 );
[ [ 1, 2, 3, 4, 4 ] ]
gap> fusion:= rec( name:= s5, map:= fus[1], text:= "unique" );;
gap> Print( LibraryFusion( "A5", fusion ) );
ALF("A5","A5.2",[1,2,3,4,4],[
"unique"
]);
## doc/../gap4/ctadmin.gd (1536-1548)
gap> tbl:= CharacterTable( "A5" );
CharacterTable( "A5" )
gap> tom:= TableOfMarks( "A5" );
TableOfMarks( "A5" )
gap> fus:= PossibleFusionsCharTableTom( tbl, tom );
[ [ 1, 2, 3, 5, 5 ] ]
gap> fusion:= rec( name:= tom, map:= fus[1], text:= "unique" );;
gap> Print( LibraryFusionTblToTom( "A5", fusion ) );
ARC("A5","tomfusion",rec(name:="A5",map:=[1,2,3,5,5],text:=[
"unique"
]));
## doc/../gap4/ctblothe.gd (99-122)
gap> Print( CASString( CharacterTable( "Cyclic", 2 ) ), "\n" );
'C2'
00/00/00. 00.00.00.
(2,2,0,2,-1,0)
text:
(#computed using generic character table for cyclic groups#),
order=2,
centralizers:(
2,2
),
reps:(
1,2
),
powermap:2(
1,1
),
characters:
(1,1
,0:0)
(1,-1
,0:0);
/// converted from GAP
## doc/../gap4/ctblothe.gd (200-204)
gap> MAKElb11( [ 3, 4 ] );
3 2 0 1 0
4 2 0 1 0
## doc/../gap4/ctblothe.gd (379-412)
gap> moca5:= MOCTable( CharacterTable( "A5" ) );
rec( 30170 := [ [ ], [ 2, 2, 1, 1 ], [ 3, 3, 1, 1 ], [ 4, 5, 1, 1 ] ]
,
30900 := [ [ 1, 1, 1, 1, 0 ], [ 3, -1, 0, 0, -1 ],
[ 3, -1, 0, 1, 1 ], [ 4, 0, 1, -1, 0 ], [ 5, 1, -1, 0, 0 ] ],
GAPtbl := CharacterTable( "A5" ), centralizers := [ 60, 4, 3, 5 ],
cycsubgps := [ 1, 2, 3, 4, 4 ],
fieldbases :=
[ CanonicalBasis( Rationals ), CanonicalBasis( Rationals ),
CanonicalBasis( Rationals ),
Basis( NF(5,[ 1, 4 ]), [ 1, E(5)+E(5)^4 ] ) ], fields := [ ],
galconjinfo := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ],
identifier := "MOCTable(A5)",
invmap := [ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ],
[ 1, 4, 0, 1, 5, 0 ] ], orders := [ 1, 2, 3, 5 ],
powerinfo :=
[ ,
[ [ 1, 1, 0 ], [ 1, 1, 0 ], [ 1, 3, 0 ],
[ 1, 4, -1, 5, 0, -1, 5, 0 ] ],
[ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 1, 0 ],
[ 1, 4, -1, 5, 0, -1, 5, 0 ] ],,
[ [ 1, 1, 0 ], [ 1, 2, 0 ], [ 1, 3, 0 ], [ 1, 1, 0, 0 ] ] ],
prime := 0, repcycsub := [ 1, 2, 3, 4 ],
tensinfo :=
[ [ 1 ], [ 1 ], [ 1 ],
[ 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2, 2, 0 ] ] )
gap> str:= MOCString( moca5 );;
gap> str{[1..68]};
"y100y105ay110fey130t60edfy140bcdfy150bbbfcabbey160bbcbdbebecy170ccbb"
gap> moca5mod3:= MOCTable( CharacterTable( "A5" ) mod 3, [ 1 .. 4 ] );;
gap> MOCString( moca5mod3 ){ [ 1 .. 68 ] };
"y100y105dy110edy130t60efy140bcfy150bbfcabbey160bbcbdbdcy170ccbbdfbby"
## doc/../gap4/ctblothe.gd (483-504)
gap> scan:= ScanMOC( str );
rec( y050 := [ 5, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 1, 0, 0 ],
y105 := [ 0 ], y110 := [ 5, 4 ], y130 := [ 60, 4, 3, 5 ],
y140 := [ 1, 2, 3, 5 ], y150 := [ 1, 1, 1, 5, 2, 0, 1, 1, 4 ],
y160 := [ 1, 1, 2, 1, 3, 1, 4, 1, 4, 2 ],
y170 := [ 2, 2, 1, 1, 3, 3, 1, 1, 4, 5, 1, 1 ],
y210 := [ 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 1, 2, 1, -1, 2,
2, 0 ], y220 := [ 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0 ],
y230 := [ 2, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 4, -1, 5, 0, -1, 5, 0 ],
y900 := [ 1, 1, 1, 1, 0, 3, -1, 0, 0, -1, 3, -1, 0, 1, 1, 4, 0, 1,
-1, 0, 5, 1, -1, 0, 0 ] )
gap> gapchars:= GAPChars( moca5, scan.y900 );
[ [ 1, 1, 1, 1, 1 ], [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ],
[ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ], [ 4, 0, 1, -1, -1 ],
[ 5, 1, -1, 0, 0 ] ]
gap> mocchars:= MOCChars( moca5, gapchars );
[ [ 1, 1, 1, 1, 0 ], [ 3, -1, 0, 0, -1 ], [ 3, -1, 0, 1, 1 ],
[ 4, 0, 1, -1, 0 ], [ 5, 1, -1, 0, 0 ] ]
gap> Concatenation( mocchars ) = scan.y900;
true
## doc/../gap4/ctblothe.gd (590-616)
gap> tbl:= CharacterTable( "Alternating", 5 );;
gap> str:= GAP3CharacterTableString( tbl );;
gap> Print( str );
rec(
centralizers := [ 60, 4, 3, 5, 5 ],
fusions := [ rec( map := [ 1, 3, 4, 7, 7 ], name := "Sym(5)" ) ],
identifier := "Alt(5)",
irreducibles := [
[ 1, 1, 1, 1, 1 ],
[ 4, 0, 1, -1, -1 ],
[ 5, 1, -1, 0, 0 ],
[ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ],
[ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ]
],
orders := [ 1, 2, 3, 5, 5 ],
powermap := [ , [ 1, 1, 3, 5, 4 ], [ 1, 2, 1, 5, 4 ], , [ 1, 2, 3, 1, \
1 ] ],
size := 60,
text := "computed using generic character table for alternating groups\
",
operations := CharTableOps )
gap> scan:= GAP3CharacterTableScan( str );
CharacterTable( "Alt(5)" )
gap> TransformingPermutationsCharacterTables( tbl, scan );
rec( columns := (), group := Group([ (4,5) ]), rows := () )
## doc/../gap4/ctblothe.gd (748-756)
gap> CambridgeMaps( CharacterTable( "A5" ) );
rec( names := [ "1A", "2A", "3A", "5A", "B*" ],
power := [ "", "A", "A", "A", "A" ],
prime := [ "", "A", "A", "A", "A" ] )
gap> CambridgeMaps( CharacterTable( "A5" ) mod 2 );
rec( names := [ "1A", "3A", "5A", "B*" ],
power := [ "", "A", "A", "A" ], prime := [ "", "A", "A", "A" ] )
## doc/../gap4/ctblothe.gd (798-843)
gap> str:= StringOfCambridgeFormat( [ [ "A5", "A5.2" ],
> [ "2.A5", "2.A5.2" ] ] );;
gap> Print( str );
#23 ? A5
#7 4 4 4 4 4 4 4 4 4 4 4
#9 ; @ @ @ @ @ ; ; @ @ @
#1 | 60 4 3 5 5 | | 6 2 3
#2 p power A A A A | | A A AB
#3 p' part A A A A | | A A AB
#4 ind 1A 2A 3A 5A B* fus ind 2B 4A 6A
#5 + 1 1 1 1 1 : ++ 1 1 1
#5 + 3 -1 0 -b5 * . + 0 0 0
#5 + 3 -1 0 * -b5 . | | | |
#5 + 4 0 1 -1 -1 : ++ 2 0 -1
#5 + 5 1 -1 0 0 : ++ 1 -1 1
#6 ind 1 4 3 5 5 fus ind 2 8 6
#6 | 2 | 6 10 10 | | | 8 6
#5 - 2 0 -1 b5 * . - 0 0 0
#5 - 2 0 -1 * b5 . | | | |
#5 - 4 0 1 -1 -1 : oo 0 0 i3
#5 - 6 0 0 1 1 : oo 0 i2 0
#8
gap> str:= StringOfCambridgeFormat( [ [ "A5", "A5.2" ],
> [ "2.A5", "2.A5.2" ] ], 3 );;
gap> Print( str );
#23 A5 (Mod 3)
#7 4 4 4 4 4 4 4 4 4
#9 ; @ @ @ @ ; ; @ @
#1 | 60 4 5 5 | | 6 2
#2 p power A A A | | A A
#3 p' part A A A | | A A
#4 ind 1A 2A 5A B* fus ind 2B 4A
#5 + 1 1 1 1 : ++ 1 1
#5 + 3 -1 -b5 * . + 0 0
#5 + 3 -1 * -b5 . | | |
#5 + 4 0 -1 -1 : ++ 2 0
#6 ind 1 4 5 5 fus ind 2 8
#6 | 2 | 10 10 | | | 8
#5 - 2 0 b5 * . - 0 0
#5 - 2 0 * b5 . | | |
#5 - 6 0 1 1 : oo 0 i2
#8
gap> StringOfCambridgeFormat( [ [ "L10(11)" ] ], 0 );
fail
## doc/../gap4/ctblothe.gd (923-940)
gap> b:= BosmaBase( 8 );
[ 0, 1, 2, 3 ]
gap> b:= Basis( CF(8), List( b, i -> E(8)^i ) );
Basis( CF(8), [ 1, E(8), E(4), E(8)^3 ] )
gap> Coefficients( b, Sqrt(2) );
[ 0, 1, 0, -1 ]
gap> Coefficients( b, Sqrt(-2) );
[ 0, 1, 0, 1 ]
gap> b:= BosmaBase( 15 );
[ 0, 5, 3, 8, 6, 11, 9, 14 ]
gap> b:= List( b, i -> E(15)^i );
[ 1, E(3), E(5), E(15)^8, E(5)^2, E(15)^11, E(5)^3, E(15)^14 ]
gap> Coefficients( Basis( CF(15), b ), EB(15) );
[ -1, -1, 0, 0, -1, -2, -1, -2 ]
gap> BosmaBase( 48 );
[ 0, 3, 6, 9, 12, 15, 18, 21, 16, 19, 22, 25, 28, 31, 34, 37 ]
## doc/../gap4/ctblothe.gd (974-1076)
gap> tmpdir:= DirectoryTemporary();;
gap> file:= Filename( tmpdir, "magmatable" );;
gap> str:= "\
> Character Table of Group G\n\
> --------------------------\n\
> \n\
> ---------------------------\n\
> Class | 1 2 3 4 5\n\
> Size | 1 15 20 12 12\n\
> Order | 1 2 3 5 5\n\
> ---------------------------\n\
> p = 2 1 1 3 5 4\n\
> p = 3 1 2 1 5 4\n\
> p = 5 1 2 3 1 1\n\
> ---------------------------\n\
> X.1 + 1 1 1 1 1\n\
> X.2 + 3 -1 0 Z1 Z1#2\n\
> X.3 + 3 -1 0 Z1#2 Z1\n\
> X.4 + 4 0 1 -1 -1\n\
> X.5 + 5 1 -1 0 0\n\
> \n\
> Explanation of Character Value Symbols\n\
> --------------------------------------\n\
> \n\
> # denotes algebraic conjugation, that is,\n\
> #k indicates replacing the root of unity w by w^k\n\
> \n\
> Z1 = (CyclotomicField(5: Sparse := true)) ! [\n\
> RationalField() | 1, 0, 1, 1 ]\n\
> ";;
gap> FileString( file, str );;
gap> tbl:= GAPTableOfMagmaFile( file, "MagmaA5" );;
gap> Display( tbl );
MagmaA5
2 2 2 . . .
3 1 . 1 . .
5 1 . . 1 1
1a 2a 3a 5a 5b
2P 1a 1a 3a 5b 5a
3P 1a 2a 1a 5b 5a
5P 1a 2a 3a 1a 1a
X.1 1 1 1 1 1
X.2 3 -1 . A *A
X.3 3 -1 . *A A
X.4 4 . 1 -1 -1
X.5 5 1 -1 . .
A = -E(5)-E(5)^4
= (1-Sqrt(5))/2 = -b5
gap> tbl2:= GAPTableOfMagmaFile( str, "MagmaA5", "string" );;
gap> Irr( tbl ) = Irr( tbl2 );
true
gap> str:= "\
> Character Table of Group G\n\
> --------------------------\n\
> \n\
> ------------------------------\n\
> Class | 1 2 3 4 5 6\n\
> Size | 1 1 1 1 1 1\n\
> Order | 1 2 3 3 6 6\n\
> ------------------------------\n\
> p = 2 1 1 4 3 3 4\n\
> p = 3 1 2 1 1 2 2\n\
> ------------------------------\n\
> X.1 + 1 1 1 1 1 1\n\
> X.2 + 1 -1 1 1 -1 -1\n\
> X.3 0 1 1 J-1-J-1-J J\n\
> X.4 0 1 -1 J-1-J 1+J -J\n\
> X.5 0 1 1-1-J J J-1-J\n\
> X.6 0 1 -1-1-J J -J 1+J\n\
> \n\
> \n\
> Explanation of Character Value Symbols\n\
> --------------------------------------\n\
> \n\
> J = RootOfUnity(3)\n\
> ";;
gap> FileString( file, str );;
gap> tbl:= GAPTableOfMagmaFile( file, "MagmaC6" );;
gap> Display( tbl );
MagmaC6
2 1 1 1 1 1 1
3 1 1 1 1 1 1
1a 2a 3a 3b 6a 6b
2P 1a 1a 3b 3a 3a 3b
3P 1a 2a 1a 1a 2a 2a
X.1 1 1 1 1 1 1
X.2 1 -1 1 1 -1 -1
X.3 1 1 A /A /A A
X.4 1 -1 A /A -/A -A
X.5 1 1 /A A A /A
X.6 1 -1 /A A -A -/A
A = E(3)
= (-1+Sqrt(-3))/2 = b3
## doc/../gap4/ctblothe.gd (1153-1168)
gap> if CTblLib.IsMagmaAvailable() then
> g:= MathieuGroup( 24 );
> ccl:= ConjugacyClasses( g );
> t:= CharacterTableComputedByMagma( g, "testM24" );
> if t = fail then
> Print( "#E Magma did not compute a character table.\n" );
> elif ( not HasConjugacyClasses( t ) ) or
> ( ConjugacyClasses( t ) <> ccl ) then
> Print( "#E The conjugacy classes do not fit.\n" );
> elif TransformingPermutationsCharacterTables( t,
> CharacterTable( "M24" ) ) = fail then
> Print( "#E Inconsistency of character tables?\n" );
> fi;
> fi;
## doc/../gap4/atlasirr.g (970-975)
gap> values:= List( [ "e31", "y'24+3", "r2+i", "r2+i2" ],
> AtlasIrrationality );;
gap> List( values, CTblLib.StringOfAtlasIrrationality );
[ "e31", "y'24+3", "z8-&3+i", "2z8" ]
## doc/../gap4/ctadmin.gd (2482-2485)
gap> StructureDescriptionCharacterTableName( "M12M2" );
"M11"
##
gap> STOP_TEST( "docxpl2.tst" );
gap> SizeScreen( save );;
#############################################################################
##
#E
