/usr/share/gap/pkg/ctbllib/gap4/test.gi is in gap-character-tables 1r1p3-5.
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##
#W test.gi GAP character table library Thomas Breuer
##
#H @(#)$Id: test.gi,v 1.31 2004/03/30 09:01:40 gap Exp $
##
#Y Copyright (C) 2002, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
##
## This file contains the implementations of functions to test the data
## available in the {\GAP} Character Table Library.
##
## 1. General tools for checking character tables
## 2. Check ``construction tables''
##
Revision.( "ctbllib/gap4/test_gi" ) :=
"@(#)$Id: test.gi,v 1.31 2004/03/30 09:01:40 gap Exp $";
#############################################################################
##
## 1. General tools for checking character tables
##
#############################################################################
##
#F CTblLibTestMax( )
#F CTblLibTestMax( <tblname> )
##
InstallGlobalFunction( CTblLibTestMax, function( arg )
local result, name, parse, sub, tbl, fus;
# Initialize the result.
result:= true;
if Length( arg ) = 1 then
name:= arg[1];
parse:= PParseBackwards( arg[1], [ IsChar, "m", IsDigitChar ] );
if parse <> fail and parse[3] <> 0
and ForAny( parse[1], IsAlphaChar ) then
#T and different from 'x'!
sub:= CharacterTable( name );
tbl:= CharacterTable( parse[1] );
if sub = fail then
Print( "#I no character table with name `", name, "'\n" );
result:= false;
elif tbl = fail then
Print( "#I no character table with name `", parse[1], "'\n" );
result:= false;
elif GetFusionMap( sub, tbl ) = fail then
Print( "#I no fusion `", name, "' -> `", parse[1],
"' stored\n" );
fus:= CTblLibTestSubgroupFusion( sub, tbl, false );
if IsRecord( fus ) and GetFusionMap( sub, tbl ) = fail then
Print( "#I store the following fusion `", name, "' -> `",
Identifier( tbl ), "':\n",
LibraryFusion( Identifier( sub ), fus ) );
fi;
result:= false;
fi;
fi;
return result;
elif Length( arg ) = 0 then
for name in LIBLIST.firstnames do
result:= CTblLibTestMax( name ) and result;
od;
fi;
# Return the result.
return result;
end );
#############################################################################
##
#F CTblLibTestSylowNormalizers( )
#F CTblLibTestSylowNormalizers( <tblname> )
##
InstallGlobalFunction( CTblLibTestSylowNormalizers, function( arg )
local result, name, sub, size, nsg, classes, tbl, sizetbl, orders, fus;
# Initialize the result.
result:= true;
if Length( arg ) = 1 then
name:= PParseBackwards( arg[1], [ IsChar, "n", IsDigitChar ] );
if name <> fail and name[3] <> 0 then
sub:= CharacterTable( arg[1] );
if sub = fail then
Print( "#I no character table with name `", arg[1], "'\n" );
result:= false;
elif not IsPrimeInt( name[3] ) then
Print( "#E `", name[3], "' is not a prime for `", arg[1], "'\n" );
result:= false;
else
size:= First( Collected( Factors( Size( sub ) ) ),
pair -> pair[1] = name[3] );
size:= size[1]^size[2];
# Check that the Sylow `p' subgroup is normal.
nsg:= ClassPositionsOfNormalSubgroups( sub );
classes:= SizesConjugacyClasses( sub );
tbl:= CharacterTable( name[1] );
if tbl = fail then
Print( "#E the name `", arg[1], "' is admissible but `",
name[1], "' is not\n" );
else
sizetbl:= First( Collected( Factors( Size( tbl ) ) ),
pair -> pair[1] = name[3] );
if ForAll( nsg, l -> Sum( classes{ l } ) <> size ) then
Print( "#E `", arg[1], "': Sylow ", name[3],
" subgroup is not normal\n" );
result:= false;
elif size <> sizetbl[1]^sizetbl[2] then
Print( "#E `", arg[1],
"': does not have the same Sylow subgroup as `",
name[1], "'\n" );
result:= false;
fi;
# If the Sylow `p' subgroup is cyclic then check the order of the
# normalizer.
orders:= OrdersClassRepresentatives( tbl );
if size in orders and
Size( sub ) <> SizesCentralizers( tbl )[ Position( orders,
size ) ] * Phi( size ) / Number( orders, x -> x = size ) then
Print( "#E `", arg[1], "': has not the order of the Sylow ",
name[3], " subgroup of `", name[1], "'\n" );
result:= false;
fi;
# Check that a reasonable subgroup fusion is stored.
fus:= CTblLibTestSubgroupFusion( sub, tbl, false );
if IsRecord( fus ) and GetFusionMap( sub, tbl ) = fail then
Print( "#I store the following fusion `", arg[1],
"' -> `", Identifier( tbl ), "':\n",
LibraryFusion( Identifier( sub ), fus ) );
fi;
fi;
fi;
fi;
elif Length( arg ) = 0 then
for name in LIBLIST.firstnames do
result:= CTblLibTestSylowNormalizers( name ) and result;
od;
fi;
# Return the result.
return result;
end );
#############################################################################
##
#F CTblLibTestElementCentralizers( )
#F CTblLibTestElementCentralizers( <tblname> )
##
InstallGlobalFunction( CTblLibTestElementCentralizers, function( arg )
local result, sub, cen, record, fus, tbl, centralizers, i, name, info,
cand, orders, cname;
# Initialize the result.
result:= true;
if Length( arg ) = 1 then
sub:= CharacterTable( arg[1] );
if sub = fail then
Print( "#I no character table with name `", arg[1], "'\n" );
return false;
fi;
cen:= ClassPositionsOfCentre( sub );
# Deal with the possibility that `sub' is an element centralizer
# in a bigger table.
# (We do this only once for each table not for each name.)
if arg[1] = LowercaseString( Identifier( sub ) ) then
if 1 < Length( cen ) then
for record in ComputedClassFusions( sub ) do
fus:= record.map;
if Length( ClassPositionsOfKernel( fus ) ) = 1 then
tbl:= CharacterTable( record.name );
if tbl <> fail then
centralizers:= SizesCentralizers( tbl );
for i in [ 2 .. Length( cen ) ] do
if centralizers[ fus[ cen[i] ] ] = Size( sub ) then
name:= Concatenation( Identifier( tbl ), "C",
ClassNames( tbl )[ fus[ cen[i] ] ] );
info:= LibInfoCharacterTable( name );
if info = fail then
Print( "#I add the name `", name, "' for `",
Identifier( sub ), "'\n" );
elif info.firstName <> Identifier( sub ) then
Print( "#E `", name, "' should be a name for `",
Identifier( sub ), "' not `", info.firstName,
"'\n" );
result:= false;
fi;
fi;
od;
fi;
fi;
od;
fi;
fi;
# Deal with the case that the name has the form `<grpname>C<nam>'.
name:= PParseBackwards( arg[1],
[ IsChar, "c", IsDigitChar, IsAlphaChar ] );
if name <> fail and name[3] <> 0 and not IsEmpty( name[4] ) then
tbl:= CharacterTable( name[1] );
if tbl = fail then
Print( "#E the name `", arg[1], "' is admissible but `",
name[1], "' is not\n" );
return false;
fi;
# Check that a reasonable subgroup fusion is stored.
fus:= CTblLibTestSubgroupFusion( sub, tbl, false );
if IsRecord( fus ) and GetFusionMap( sub, tbl ) = fail then
Print( "#I store the following fusion `", name, "' -> `",
Identifier( tbl ), "':\n",
LibraryFusion( Identifier( sub ), fus ) );
fi;
# Check that there is a class whose centralizer order equals
# the size of `subtbl', and that the fusion is compatible with
# the centralizer condition.
centralizers:= SizesCentralizers( tbl );
cand:= Filtered( [ 1 .. Length( centralizers ) ],
i -> centralizers[i] = Size( sub ) );
if IsEmpty( cand ) then
Print( "#E `", arg[1], "' is not an element centralizer in `",
Identifier( tbl ), "'\n" );
return false;
fi;
orders:= OrdersClassRepresentatives( tbl );
cand:= Filtered( cand, i -> orders[i] = name[3] );
orders:= OrdersClassRepresentatives( sub );
cen:= Filtered( cen, i -> orders[i] = name[3] );
if IsRecord( fus ) then
cen:= Filtered( cen, i -> fus.map[i] in cand );
fi;
if IsEmpty( cen ) then
Print( "#E `", arg[1], "' is not the centralizer in `",
Identifier( tbl ), "'\n",
"#E of an element of order ", name[3], "\n" );
return false;
fi;
name:= LowercaseString( Concatenation( String( name[3] ), name[4] ) );
cname:= List( cen,
c -> LowercaseString( ClassNames( tbl )[ fus.map[c] ] ) );
if not name in cname then
Print( "#E `", arg[1], "' is the centralizer in `",
Identifier( tbl ), "'\n",
"#E of a class in `", cname, "' not of `", name, "'\n" );
return false;
fi;
fi;
elif Length( arg ) = 0 then
# Note that it is not sufficient to check the `Identifier' values.
for name in LIBLIST.allnames do
result:= CTblLibTestElementCentralizers( name ) and result;
od;
fi;
# Return the result.
return result;
end );
#############################################################################
##
#F CTblLibTestElementNormalizers( )
#F CTblLibTestElementNormalizers( <tblname> )
##
InstallGlobalFunction( CTblLibTestElementNormalizers, function( arg )
local result, sub, orders, classes, cen, record, fus, tbl, centralizers,
orbits, i, j, name, info, cand, cname;
# Initialize the result.
result:= true;
if Length( arg ) = 1 then
sub:= CharacterTable( arg[1] );
if sub = fail then
Print( "#I no character table with name `", arg[1], "'\n" );
return false;
fi;
orders:= OrdersClassRepresentatives( sub );
classes:= SizesConjugacyClasses( sub );
cen:= Filtered( [ 1 .. NrConjugacyClasses( sub ) ],
i -> orders[i] = Sum( classes{
ClassPositionsOfNormalClosure( sub, [ i ] ) } ) );
# Deal with the possibility that `sub' is an element normalizer
# in a bigger table.
# (We do this only once for each table not for each name.)
if arg[1] = LowercaseString( Identifier( sub ) ) then
if 1 < Length( cen ) then
for record in ComputedClassFusions( sub ) do
fus:= record.map;
if Length( ClassPositionsOfKernel( fus ) ) = 1 then
tbl:= CharacterTable( record.name );
centralizers:= SizesCentralizers( tbl );
orders:= OrdersClassRepresentatives( tbl );
orbits:= List( [ 1 .. NrConjugacyClasses( tbl ) ],
i -> Length( ClassOrbit( tbl, i ) ) );
for i in [ 2 .. Length( cen ) ] do
j:= fus[ cen[i] ];
if centralizers[j] * Phi( orders[j] ) / orbits[j]
= Size( sub ) then
name:= Concatenation( Identifier( tbl ), "N",
ClassNames( tbl )[ fus[ cen[i] ] ] );
info:= LibInfoCharacterTable( name );
if info = fail then
Print( "#I add the name `", name, "' for `",
Identifier( sub ), "'\n" );
elif info.firstName <> Identifier( sub ) then
Print( "#E `", name, "' should be a name for `",
Identifier( sub ), "' not `", info.firstName, "'\n" );
result:= false;
fi;
fi;
od;
fi;
od;
fi;
fi;
# Deal with the case that the name has the form `<grpname>N<nam>'.
name:= PParseBackwards( arg[1],
[ IsChar, "n", IsDigitChar, IsAlphaChar ] );
if name <> fail and name[3] <> 0 and not IsEmpty( name[4] ) then
tbl:= CharacterTable( name[1] );
if tbl = fail then
Print( "#E the name `", arg[1], "' is admissible but `",
name[1], "' is not\n" );
return false;
fi;
# Check that a reasonable subgroup fusion is stored.
fus:= CTblLibTestSubgroupFusion( sub, tbl, false );
if IsRecord( fus ) and GetFusionMap( sub, tbl ) = fail then
Print( "#I store the following fusion `", name, "' -> `",
Identifier( tbl ), "':\n",
LibraryFusion( Identifier( sub ), fus ) );
fi;
# Check that there is a class whose normalizer order equals
# the size of `subtbl', and that the fusion is compatible with
# the normalizer condition.
centralizers:= SizesCentralizers( tbl );
orders:= OrdersClassRepresentatives( tbl );
orbits:= List( [ 1 .. NrConjugacyClasses( tbl ) ],
i -> Length( ClassOrbit( tbl, i ) ) );
cand:= Filtered( [ 1 .. Length( centralizers ) ],
i -> centralizers[i] * Phi( orders[i] ) / orbits[i]
= Size( sub ) );
if IsEmpty( cand ) then
Print( "#E `", arg[1], "' is not an element normalizer in `",
Identifier( tbl ), "'\n" );
return false;
fi;
cand:= Filtered( cand, i -> orders[i] = name[3] );
orders:= OrdersClassRepresentatives( sub );
cen:= Filtered( cen, i -> orders[i] = name[3] );
if IsRecord( fus ) then
cen:= Filtered( cen, i -> fus.map[i] in cand );
fi;
if IsEmpty( cen ) then
Print( "#E `", arg[1], "' is not the normalizer in `",
Identifier( tbl ), "'\n",
"#E of an element of order ", name[3], "\n" );
return false;
fi;
name:= LowercaseString( Concatenation( String( name[3] ), name[4] ) );
cname:= List( cen,
c -> LowercaseString( ClassNames( tbl )[ fus.map[c] ] ) );
if not name in cname then
Print( "#E `", arg[1], "' is the centralizer in `",
Identifier( tbl ), "'\n",
"#E of a class in `", cname, "' not of `", name, "'\n" );
return false;
fi;
fi;
elif Length( arg ) = 0 then
# Note that it is not sufficient to check the `Identifier' values.
for name in LIBLIST.allnames do
result:= CTblLibTestElementNormalizers( name ) and result;
od;
fi;
# Return the result.
return result;
end );
#############################################################################
##
#V CTblLibHardFusions
##
## `CTblLibHardFusions' is a list of pairs `[ <subname>, <tblname> ]'
## where <subname> and <tblname> are `Identifier' values of character
## tables such that `CTblLibTestSubgroupFusion' shall omit the compatibility
## check for the class fusion between these tables.
##
InstallValue( CTblLibHardFusions, [] );
Add( CTblLibHardFusions, [ "Co1N3", "Co1" ] );
Add( CTblLibHardFusions, [ "Co1N2", "Co1" ] );
Add( CTblLibHardFusions, [ "Co2N2", "Co2" ] );
Add( CTblLibHardFusions, [ "Fi22N3", "Fi22" ] );
# computed via factorization through 3^(1+6):2^(3+4):3^2:2
Add( CTblLibHardFusions, [ "M24N2", "M24" ] );
# computed from the groups, time 227180 msec, incl. tables comput.
# (25-11-2002)
Add( CTblLibHardFusions, [ "M24N2", "He" ] );
# computed from the groups, time 12451360 msec, incl. tables comput.
# (26-11-2002)
Add( CTblLibHardFusions, [ "O8+(3)M14", "O8+(3)" ] );
# 1 orbit, 648 sol., time 154539590 msec on regulus (22-11-2002)
Add( CTblLibHardFusions, [ "L3(3)", "B" ] );
# 1 orbit, 36 sol., harmless if one forbids decomposition
Add( CTblLibHardFusions, [ "2^2xF4(2)", "2.2E6(2).2" ] );
#############################################################################
##
#F CTblLibTestDecompositions( <sub>, <fuslist>, <tbl> )
##
InstallGlobalFunction( CTblLibTestDecompositions,
function( sub, fuslist, tbl )
local bad, p, modtbl, modsub, fus, modfus;
bad:= [];
for p in Set( Factors( Size( tbl ) ) ) do
modtbl:= tbl mod p;
if modtbl <> fail then
modsub:= sub mod p;
if modsub <> fail then
for fus in fuslist do
modfus:= CompositionMaps( InverseMap(
GetFusionMap( modtbl, tbl ) ),
CompositionMaps( fus,
GetFusionMap( modsub, sub ) ) );
if fail in Decomposition( Irr( modsub ),
List( Irr( modtbl ), chi -> chi{ modfus } ),
"nonnegative" ) then
AddSet( bad, fus );
fi;
od;
fi;
fi;
od;
return Difference( fuslist, bad );
end );
#############################################################################
##
#F InitFusionsStatistics( <statfile> )
#F AmendFusionsStatistics( <statinfo>, <entry> )
#F FinalizeFusionsStatistics( <statinfo> )
##
## Create a file with information about all subgroup fusions stored in the
## {\GAP} Character Table Library.
## For the fusion from the table with identifier <subtbl> into that with
## identifier <tbl>, a list entry of the following form is printed.
##
## `[<subtbl>,<tbl>,<nrfus>,<nrorbs>,<nrcomp>,<nrcorbs>,<normtime>],'
##
## Here <nrfus> is the number of fusions,
## <nrorbs> is the number of orbits on the maps under table automorphisms,
## <nrcomp> is the number of those fusions that are compatible with the
## Brauer tables available for <subtbl> and <tbl>,
## <nrcorbs> is the number of orbits on the compatible maps under table
## automorphisms, and
## <normtime> is the time needed to compute the fusions,
## divided by ... (so this value is expected to be more or less independent
## of the machine used).
##
## Thus the fusion is unique if <nrfus> is $1$,
## it is unique up to table automorphisms if <nrorbs> is $1$;
## otherwise the fusion is ambiguous.
## If <nrcomp> is smaller than <nrfus> then the Brauer tables impose
## extra conditions on the fusions, and if <nrcorbs> is smaller than
## <nrorbs> then the Brauer tables reduce the ambiguity.
##
BindGlobal( "InitFusionsStatistics", function( statfile )
local time, l, i, j;
# Measure the time for some typical computations.
time:= Runtime();
l:= [];
for i in [ 1 .. 1000 ] do
for j in [ 1 .. 1000 ] do
l[j]:= j;
od;
od;
time:= ( Runtime() - time );
# Create the file.
PrintTo( statfile, "[\n" );
return rec( statfile:= statfile, time:= time );
end );
BindGlobal( "AmendFusionsStatistics", function( statinfo, entry )
AppendTo( statinfo.statfile, Concatenation( "[\"",
entry[1],
"\",\"",
entry[2],
"\",",
String( entry[3] ),
",",
String( entry[4] ),
",",
String( entry[5] ),
",",
String( entry[6] ),
",",
String( Int( entry[7] / statinfo.time ) ),
"],\n" ) );
end );
BindGlobal( "FinalizeFusionsStatistics", function( statinfo )
AppendTo( statinfo.statfile, "\n];\n" );
end );
#############################################################################
##
#F CTblLibTestSubgroupFusion( <sub>, <tbl>, <statfile> )
##
InstallGlobalFunction( CTblLibTestSubgroupFusion,
function( sub, tbl, statfile )
local fusrec, storedfus, time, fus, filt, fusreps, filtreps, result, bad;
fusrec:= First( ComputedClassFusions( sub ),
record -> record.name = Identifier( tbl ) );
# Shall the test be omitted?
if [ Identifier( sub ), Identifier( tbl ) ] in CTblLibHardFusions then
if fusrec = fail then
Print( "#E omitting fusion check for ", Identifier( sub ), " -> ",
Identifier( tbl ), " (no map stored)\n" );
else
# At least test the existing map for consistency.
fus:= PossibleClassFusions( sub, tbl,
rec( fusionmap:= fusrec.map ) );
if IsEmpty( fus ) then
Print( "#E stored fusion `", Identifier( sub ), "' -> `",
Identifier( tbl ), "' is wrong\n" );
else
Print( "#I omitting fusion check for ", Identifier( sub ), " -> ",
Identifier( tbl ), "\n" );
fi;
fi;
return fusrec;
fi;
if fusrec = fail then
fusrec:= rec();
storedfus:= fail;
else
storedfus:= fusrec.map;
fi;
time:= Runtime();
fus:= PossibleClassFusions( sub, tbl );
time:= Runtime() - time;
fusreps:= RepresentativesFusions( sub, fus, tbl );
filt:= CTblLibTestDecompositions( sub, fus, tbl );
filtreps:= RepresentativesFusions( sub, filt, tbl );
# Amend the statistics if wanted.
if IsRecord( statfile ) then
AmendFusionsStatistics( statfile, [
Identifier( sub ),
Identifier( tbl ),
Length( fus ),
Length( fusreps ),
Length( filt ),
Length( filtreps ),
time ] );
fi;
# Do the tables and fusions fit together?
if IsEmpty( fus ) then
Print( "#E no fusion `", Identifier( sub ), "' -> `",
Identifier( tbl ), "' possible\n" );
result:= false;
elif storedfus <> fail and not storedfus in fus then
Print( "#E stored fusion `", Identifier( sub ), "' -> `",
Identifier( tbl ), "' is wrong,\n" );
result:= false;
elif Length( fus ) = 1 then
# The fusion is unique.
if IsEmpty( filt ) then
Print( "#E unique fusion `", Identifier( sub ), "' -> `",
Identifier( tbl ), "' contradicts Brauer tables\n" );
result:= false;
else
if IsBound( fusrec.text )
and fusrec.text <> "fusion map is unique"
and ( Length( fusrec.text ) < 21 or
fusrec.text{ [ 1 .. 21 ] } <> "fusion map is unique," )
and ( Length( fusrec.text ) < 22 or
fusrec.text{ [ 1 .. 22 ] } <> "fusion map is unique (" )
then
Print( "#E text for stored fusion `", Identifier( sub ),
"' -> `", Identifier( tbl ),
"' is wrong (map is unique!)\n" );
fi;
result:= rec( name := Identifier( tbl ),
map := fus[1],
text := "fusion map is unique" );
fi;
elif 1 < Length( filtreps ) then
# The fusion is ambiguous.
if storedfus = fail then
Print( "#E ambiguous fusion `", Identifier( sub ), "' -> `",
Identifier( tbl ), "', no map stored\n" );
result:= fail;
elif not IsBound( fusrec.text ) then
# The ambiguity of the fusion is not mentioned in the stored fusion.
Print( "#E ambiguous fusion `", Identifier( sub ), "' -> `",
Identifier( tbl ), "', no text stored\n" );
result:= fail;
elif PositionSublist( fusrec.text, "together" ) = fail
and PositionSublist( fusrec.text, "determined" ) = fail then
# The ambiguity of the fusion is not mentioned in the stored fusion.
Print( "#E ambiguous fusion `", Identifier( sub ), "' -> `",
Identifier( tbl ),
"',\n#E without \"together\" or \"determined\" in text\n" );
result:= fail;
else
result:= fusrec;
fi;
# If the ambiguity is caused by the consideration of the Brauer tables
# then this effect of a generality problem should be mentioned in the
# text.
if Length( fusreps ) = 1
and not ( IsBound( fusrec.text ) and
PositionSublist( fusrec.text, "Brauer" ) <> fail ) then
Print( "#E ambiguity in fusion `", Identifier( sub ), "' -> `",
Identifier( tbl ), "'\n",
"#E caused by Brauer tables, this should be mentioned\n" );
result:= fail;
fi;
elif Length( filt ) = Length( fus ) then
# The fusion is unique up to table automorphisms.
# (Keep the stored fusion if exists.)
if IsBound( fusrec.text ) and
PositionSublist( fusrec.text, "unique up to table " ) = fail then
Print( "#E text for stored fusion `", Identifier( sub ),
"' -> `", Identifier( tbl ),
"' is wrong (map is unique up to table automorphisms!)\n" );
fi;
if storedfus = fail then
fus:= fus[1];
else
fus:= storedfus;
fi;
result:= rec( name := Identifier( tbl ),
map := fus,
text := "fusion map is unique up to table autom." );
else
# The Brauer tables impose additional conditions; together with this,
# the fusion is unique at least up to table automorphisms.
if IsEmpty( filt ) then
Print( "#E no fusion `", Identifier( sub ), "' -> `",
Identifier( tbl ), "' compatible with mod. tables\n" );
result:= false;
else
if storedfus = fail then
fus:= filt[1];
elif not storedfus in filt then
Print( "#E stored fusion `", Identifier( sub ), "' -> `",
Identifier( tbl ), "' not compatible with mod. tables\n" );
fus:= filt[1];
else
fus:= storedfus;
fi;
result:= rec( name := Identifier( tbl ),
map := fus );
if Length( fusreps ) = 1 then
if Length( filt ) = 1 then
# The fusion is unique.
result.text := Concatenation(
"fusion map is unique up to table autom.,\n",
"unique map that is compatible with Brauer tables" );
else
# The fusion is unique up to table automorphisms.
result.text := Concatenation(
"fusion map is unique up to table autom.,\n",
"compatible with Brauer tables" );
fi;
elif Length( filt ) = 1 then
result.text:= "fusion map uniquely determined by Brauer tables";
else
result.text:=
"fusion map determined up to table autom. by Brauer tables";
fi;
#T for all cases, check the status in the `InfoText', perhaps print something!
fi;
fi;
# Return the result.
return result;
end );
#############################################################################
##
#F CTblLibTestFactorFusion( <tbl>, <fact> )
##
InstallGlobalFunction( CTblLibTestFactorFusion, function( tbl, fact )
local storedfus, quot, classes, kernels, fus, ker, f, trans, map, reps,
record;
storedfus:= GetFusionMap( tbl, fact );
quot:= Size( tbl ) / Size( fact );
classes:= SizesConjugacyClasses( tbl );
kernels:= Filtered( ClassPositionsOfNormalSubgroups( tbl ),
list -> Sum( classes{ list } ) = quot );
fus:= [];
for ker in kernels do
f:= CharacterTableFactorGroup( tbl, ker );
trans:= TransformingPermutationsCharacterTables( f, fact );
if IsRecord( trans ) then
map:= OnTuples( GetFusionMap( tbl, f ), trans.columns );
Append( fus, Orbit( trans.group, map, OnTuples ) );
fi;
od;
# Do the tables and fusions fit together?
if IsEmpty( fus ) then
Print( "#E no fusion `", Identifier( tbl ), "' ->> `",
Identifier( fact ), "' possible\n" );
return false;
elif storedfus <> fail and not storedfus in fus then
Print( "#E stored fusion `", Identifier( tbl ), "' ->> `",
Identifier( fact ), "' is wrong\n" );
return false;
fi;
# Is the fusion perhaps unique (with given kernel)?
if Length( fus ) = 1 then
return rec( name := Identifier( fact ),
map := fus[1],
text := "fusion map is unique" );
fi;
# Is the fusion perhaps unique up to table automorphisms
# (with given kernel, of course)?
reps:= RepresentativesFusions( Group( () ), fus,
AutomorphismsOfTable( fact ) );
if Length( reps ) = 1 then
return rec( name := Identifier( fact ),
map := fus[1],
text := "fusion map is unique up to table autom." );
fi;
# Is the ambiguity of the fusion perhaps mentioned in the stored fusion?
if storedfus = fail then
Print( "#E ambiguous fusion `", Identifier( tbl ), "' ->> `",
Identifier( fact ), "', no map stored\n" );
return fail;
fi;
record:= First( ComputedClassFusions( tbl ),
x -> x.name = Identifier( fact ) and x.map = storedfus );
#T more possibilities?
return record;
end );
#############################################################################
##
#F CTblLibTestFusions( <statistics> )
#F CTblLibTestFusions( <tblname>, <statistics> )
##
InstallGlobalFunction( CTblLibTestFusions, function( arg )
local result, sub, record, tbl, fus, name;
# Initialize the result.
result:= true;
if Length( arg ) = 2 then
sub:= CharacterTable( arg[1] );
if sub = fail then
Print( "#I no character table with name `", arg[1], "'\n" );
result:= false;
else
for record in ShallowCopy( ComputedClassFusions( sub ) ) do
tbl:= CharacterTable( record.name );
# Do not report a problem if `fail' is returned,
# since direct products involving `sub' may have been
# constructed before this test started.
if tbl <> fail then
if Size( sub ) <= Size( tbl ) then
fus:= CTblLibTestSubgroupFusion( sub, tbl, arg[2] );
if not IsRecord( fus ) then
result:= false;
elif record.map <> fus.map then
Print( "#E replace the stored fusion `", arg[1], "' -> `",
record.name, "' by the following one.\n",
LibraryFusion( arg[1], fus ) );
fi;
else
fus:= CTblLibTestFactorFusion( sub, tbl );
if not IsRecord( fus ) then
result:= false;
fi;
fi;
fi;
od;
fi;
elif Length( arg ) = 1 then
for name in LIBLIST.firstnames do
result:= CTblLibTestFusions( name, arg[1] ) and result;
od;
fi;
# Return the result.
return result;
end );
#############################################################################
##
#F CTblLibTestPowerMaps( )
#F CTblLibTestPowerMaps( <tblname> )
##
## First suppose that `CTblLibTestPowerMaps' is called with one
## argument <tblname>, which is an admissible name of a character table.
## Then it is checked whether all power maps of prime divisors of the group
## order are stored on the table, and whether they are correct.
## (This includes the information about ambiguities of the power maps in the
## `InfoText' values of the tables.)
##
## If no argument is given then all standard character table names are
## checked with the one argument version.
##
## In all cases, the return value is `false' if an error occurred,
## and `true' otherwise.
##
InstallGlobalFunction( CTblLibTestPowerMaps, function( arg )
local result, tbl, powermaps, info, p, pow, reps, storedmap, name;
# Initialize the result.
result:= true;
if Length( arg ) = 1 then
tbl:= CharacterTable( arg[1] );
if tbl = fail then
Print( "#I no character table with name `", arg[1], "'\n" );
result:= false;
else
powermaps:= ComputedPowerMaps( tbl );
if HasInfoText( tbl ) then
info:= InfoText( tbl );
else
info:= "";
fi;
for p in Set( Factors( Size( tbl ) ) ) do
pow:= PossiblePowerMaps( tbl, p );
reps:= RepresentativesPowerMaps( pow,
MatrixAutomorphisms( Irr( tbl ) ) );
if not IsBound( powermaps[p] ) then
Print( "#I no ", Ordinal( p ), " power map stored on table `",
arg[1], "'\n" );
storedmap:= fail;
else
storedmap:= powermaps[p];
fi;
if IsEmpty( pow ) then
Print( "#E no ", Ordinal( p ), " power map possible for `",
Identifier( tbl ), "'\n" );
result:= false;
elif storedmap <> fail and not storedmap in pow then
Print( "#E stored ", Ordinal( p ), " power map for `",
Identifier( tbl ), "' is wrong\n" );
result:= false;
elif Length( reps ) <> 1 then
if PositionSublist( info, Concatenation( Ordinal( p ),
" power map determined" ) ) = fail then
Print( "#E ambiguous ", Ordinal( p ), " power map for `",
Identifier( tbl ), "'\n" );
result:= false;
fi;
elif Length( pow ) <> 1 then
if PositionSublist( info, Concatenation( Ordinal( p ),
" power map determined only up to matrix automorphism" ) )
<> fail then
Print( "#E ", Ordinal( p ), " power map for `",
Identifier( tbl ), "' is det. only up to mat. aut.\n" );
result:= false;
fi;
else
Print( "#E unnecessary statement about ", Ordinal( p ),
" power map for `", Identifier( tbl ), "'\n" );
result:= false;
fi;
if storedmap = fail then
if Length( pow ) = 1 then
Print( "#I store the following unique ", Ordinal( p ),
" power map on `", Identifier( tbl ), "':\n", pow[1],
"\n" );
elif Length( reps ) = 1 then
Print( "#I store the following ", Ordinal( p ),
" power map on `", Identifier( tbl ), "':\n", reps[1],
"\n#I (unique up to matrix automorphisms)\n" );
fi;
fi;
od;
fi;
elif Length( arg ) = 0 then
for name in LIBLIST.firstnames do
result:= CTblLibTestPowerMaps( name ) and result;
od;
fi;
# Return the result.
return result;
end );
#############################################################################
##
#F CTblLibTestTensorDecomposition( <ordname>, <p> )
##
InstallGlobalFunction( CTblLibTestTensorDecomposition, function( ordname, p )
local tbl, ibr, i, tens;
tbl:= CharacterTable( ordname ) mod p;
if tbl <> fail then
ibr:= IBr( tbl );
for i in [ 1 .. Length( ibr ) ] do
tens:= Set( Tensored( [ ibr[i] ], ibr{ [ 1 .. i ] } ) );
if not ForAll( Decomposition( ibr, tens, "nonnegative" ),
IsList ) then
Print( "#I tensor decomposition for ", Identifier( tbl ),
" failed for at least some products with X[", i, "]\n" );
return false;
fi;
od;
if HasInfoText( tbl ) and
PositionSublist( InfoText( tbl ), "TENS" ) = fail then
Print( "#I add \"TENS\" to `InfoText' for ", Identifier( tbl ),
"\n" );
fi;
fi;
return true;
end );
#############################################################################
##
#F CheckIndicators( <modtbl> )
##
## Test if the $2$nd indicators of the Brauer table <modtbl> are correct.
##
BindGlobal( "CheckIndicators", function( modtbl )
local ind, modind, irr, result, i, info, decmat, j, chi, odd;
if not IsBrauerTable( modtbl ) then
Error( "<modtbl> must be a Brauer table" );
fi;
ind:= Indicator( OrdinaryCharacterTable( modtbl ), 2 );
modind:= Indicator( modtbl, 2 );
irr:= Irr( modtbl );
result:= true;
for i in [ 1 .. Length( BlocksInfo( modtbl ) ) ] do
info:= BlocksInfo( modtbl )[i];
decmat:= DecompositionMatrix( modtbl, i );
for j in [ 1 .. Length( info.modchars ) ] do
chi:= irr[ info.modchars[j] ];
if ForAny( chi, x -> GaloisCyc( x, -1 ) <> x ) then
# The indicator of a Brauer character is zero iff it has
# at least one nonreal value.
if modind[ info.modchars[j] ] <> 0 then
Print( "#X ", Identifier( modtbl ), ": indicator of X[",
info.modchars[j], "] (degree ", chi[1],
") must be 0, not ", modind[ info.modchars[j] ], "\n" );
result:= false;
fi;
elif UnderlyingCharacteristic( modtbl ) <> 2 then
# The indicator is equal to the indicator of an ordinary character
# that contains it as a constituent, with odd multiplicity.
odd:= Filtered( [ 1 .. Length( decmat ) ],
x -> decmat[x][j] mod 2 <> 0 );
if IsEmpty( odd ) then
Print( "#X ", Identifier( modtbl ),
": no odd constituent for X[", info.modchars[j],
"] (degree ", chi[1], ")\n" );
result:= false;
else
odd:= List( odd, x -> ind[ info.ordchars[x] ] );
if ForAny( odd,
x -> x <> 0 and x <> modind[ info.modchars[j] ] ) then
if 1 < Length( Set( odd ) ) then
Print( "#X ", Identifier( modtbl ),
": ind. of odd const. not unique for X[",
info.modchars[j], "] (degree ", chi[1], ")\n" );
else
Print( "#X ", Identifier( modtbl ),
": indicator of X[", i.modchars[j],
"] (degree ", chi[1], ") must be ", odd[1], ", not ",
modind[ info.modchars[j] ], "\n" );
fi;
result:= false;
fi;
fi;
else
# Test that all nontrivial character degrees are even.
if not ForAll( chi, x -> x = 1 ) and chi[1] mod 2 <> 0 then
Print( "#X ", Identifier( modtbl ), ": degree X[",
info.modchars[j], "][1] = ", chi[1],
" but should be even\n" );
result:= false;
fi;
fi;
od;
od;
# Return the result.
return result;
end );
#############################################################################
##
#F CTblLibTestInfoText()
#F CTblLibTestInfoText( <tblname> )
##
InstallGlobalFunction( "CTblLibTestInfoText", function( arg )
local result, name, tbl, info, pos, pos2, int, pos3, groupname, suptbl,
maxes;
result:= true;
if Length( arg ) = 0 then
for name in LIBLIST.firstnames do
result:= CTblLibTestInfoText( name ) and result;
od;
elif Length( arg ) = 1 and IsString( arg[1] ) then
tbl:= CharacterTable( arg[1] );
name:= Identifier( tbl );
# If there is no `InfoText' value then nothing is to do.
if not HasInfoText( tbl ) then
return true;
fi;
for info in SplitString( InfoText( tbl ), "\n" ) do
# Filter out phrases of the form `maximal subgroup of <grpname>'.
pos:= PositionSublist( info, "maximal subgroup of " );
if pos <> fail then
# Get the character table of the overgroup.
pos3:= PositionSublist( info, ",", pos + 20 );
if pos3 = fail then
pos3:= Length( info ) + 1;
fi;
groupname:= info{ [ pos + 20 .. pos3 - 1 ] };
suptbl:= CharacterTable( groupname );
if suptbl = fail then
Print( "#E InfoText for `", name, "' contains `",
info{ [ pos .. pos3 - 1 ] }, "'\n",
"#E but there is no table of `", groupname, "'\n" );
return false;
fi;
# Check whether there is a class fusion into `suptbl'.
if GetFusionMap( tbl, suptbl ) = fail then
Print( "#E missing fusion ", name, " -> ",
Identifier( suptbl ), " in spite of `InfoText'\n" );
fi;
if 3 < pos and
info{ [ pos-2 .. pos-1 ] } in [ "st ", "nd ", "rd ", "th " ] then
# Get the position in the `Maxes' list.
#T there may be several, separated with ` and '!
pos2:= pos - 3;
while 0 < pos2 and IsDigitChar( info[ pos2 ] ) do
pos2:= pos2-1;
od;
int:= Int( info{ [ pos2 + 1 .. pos-3 ] } );
if int = 0 then
Print( "#E InfoText for `", name, "' contains `",
info{ [ pos2 + 1 .. pos3 - 1 ] }, "'\n" );
elif not HasMaxes( suptbl ) then
Print( "#I InfoText for `", name, "' contains `",
info{ [ pos2 + 1 .. pos3 - 1 ] }, "'\n",
"#I but `", groupname, "' has no `Maxes'\n" );
return false;
else
# Get the `Maxes' value of the character table of the overgroup.
maxes:= Maxes( suptbl );
# Compare the two values.
if Length( maxes ) < int or maxes[ int ] <> name then
Print( "#E InfoText for `", name, "' contains `",
info{ [ pos2 + 1 .. pos3 - 1 ] }, "'\n",
"#E but the position in `Maxes' for `", groupname,
"' is `", Position( maxes, name ), "'\n" );
return false;
fi;
fi;
fi;
fi;
od;
fi;
return result;
end );
#############################################################################
##
#V CTblLibHardTableAutomorphisms
##
## `CTblLibHardTableAutomorphisms' is a list of `Identifier' values of
## character tables such that `CTblLibTestTableAutomorphisms' shall omit the
## check for these tables.
##
InstallValue( CTblLibHardTableAutomorphisms, [] );
Add( CTblLibHardTableAutomorphisms, "O8+(3)M14" );
Add( CTblLibHardTableAutomorphisms, "3.U6(2).3" );
#############################################################################
##
#F CTblLibTestTableAutomorphisms( [<tblname>] )
##
InstallGlobalFunction( CTblLibTestTableAutomorphisms, function( arg )
local result, tbl, aut, irr, irrset, stored, name;
# Initialize the result.
result:= true;
if Length( arg ) = 1 then
# Shall the test be omitted?
if arg[1] in CTblLibHardTableAutomorphisms then
Print( "#I omitting table automorphisms check for ", arg[1], "\n" );
return true;
fi;
tbl:= CharacterTable( arg[1] );
if tbl = fail then
Print( "#I no character table with name `", arg[1], "'\n" );
result:= false;
elif not HasAutomorphismsOfTable( tbl ) then
aut:= TableAutomorphisms( tbl, Irr( tbl ), "closed" );
Print( "#I table automorphisms missing for `", arg[1], "', add\n",
GeneratorsOfGroup( aut ), "\n" );
result:= false;
else
# Check that the stored automorphisms are automorphisms,
# and that there are not more automorphisms than the stored ones.
irr:= Irr( tbl );
irrset:= Set( irr );
stored:= AutomorphismsOfTable( tbl );
if ForAll( GeneratorsOfGroup( stored ),
gen -> Set( List( irr,
chi -> Permuted( chi, gen ) ) ) = irrset ) then
aut:= TableAutomorphisms( tbl, Irr( tbl ), stored );
else
Print( "#E wrong automorphism(s) stored for `", arg[1], "'!\n" );
aut:= TableAutomorphisms( tbl, Irr( tbl ), "closed" );
fi;
if aut <> stored then
Print( "#E replace wrong automorphisms for `", arg[1], "' by\n",
GeneratorsOfGroup( aut ), "\n" );
result:= false;
fi;
fi;
elif Length( arg ) = 0 then
for name in LIBLIST.firstnames do
result:= CTblLibTestTableAutomorphisms( name ) and result;
od;
fi;
# Return the result.
return result;
end );
#############################################################################
##
## 2. Check ``construction tables''
##
#############################################################################
##
#F CTblLibTestDirectProductConstruction( <tbl> )
##
## Check that there are at least two factors.
##
BindGlobal( "CTblLibTestDirectProductConstruction", function( tbl )
local result, info;
result:= true;
info:= ConstructionInfoCharacterTable( tbl );
if Length( info[2] ) < 2 then
Print( "#E for `", Identifier( tbl ),
"', use `ConstructPermuted' not `ConstructDirectProduct'\n" );
result:= false;
fi;
return result;
end );
#############################################################################
##
#F CTblLibTestGS3Construction( <tbl> )
##
## Assume that <tbl> is an ordinary character table such that the first
## entry of `ConstructionInfoCharacterTable( <tbl> )' is `"ConstructGS3"'.
## `CTblLibTestGS3Construction' checks whether the action on the classes
## of the index two subgroup is correct, that the construction with
## `CharacterTableOfTypeGS3' yields the same irreducibles as those of <tbl>,
## and that the Brauer tables (if available
##
BindGlobal( "CTblLibTestGS3Construction", function( tbl )
local result, info, t2, t3, tnames, t, t3fustbl, aut, poss, ts3,
p, tmodp, t2modp, t3modp, tblmodp, ts3modp, nsg;
result:= true;
info:= ConstructionInfoCharacterTable( tbl );
t2:= CharacterTable( info[2] );
t3:= CharacterTable( info[3] );
tnames:= Intersection( NamesOfFusionSources( t2 ),
NamesOfFusionSources( t3 ) );
t:= Filtered( List( tnames, CharacterTable ),
ttbl -> 6 * Size( ttbl ) = Size( tbl ) );
if Length( t ) <> 1 then
Print( "#E table of the kernel of S3 not identified\n" );
return false;
fi;
t:= t[1];
# Get the action of `tbl' on the classes of `t3'.
t3fustbl:= GetFusionMap( t3, tbl );
aut:= Product( List( Filtered( InverseMap( t3fustbl ), IsList ),
x -> ( x[1], x[2] ) ), () );
poss:= PossibleActionsForTypeGS3( t, t2, t3 );
if not aut in poss then
Print( "#E for `", Identifier( tbl ),
"' the action of G.S3 on G.3 is not possible\n" );
result:= false;
elif Length( poss ) <> 1 then
Print( "#I for `", Identifier( tbl ),
"' the action of G.S3 on G.3 is not unique\n" );
result:= false;
fi;
# Check that the two constructions (from the tables of subgroups
# and from the info stored on `tbl') yield the same result.
ts3:= CharacterTableOfTypeGS3( t, t2, t3, aut, "test" );
if Irr( ts3.table ) <> Irr( tbl ) then
Print( "#E constructed and library table for `",
Identifier( tbl ), "' sorted incompatibly\n" );
result:= false;
fi;
# Check that also the Brauer tables are available.
for p in Set( Factors( Size( tbl ) ) ) do
tmodp:= t mod p;
t2modp:= t2 mod p;
t3modp:= t3 mod p;
if tmodp <> fail and t2modp <> fail and t3modp <> fail then
tblmodp:= tbl mod p;
ts3modp:= CharacterTableOfTypeGS3( tmodp, t2modp, t3modp, tbl,
Concatenation( Identifier( tbl ), "mod", String( p ) ) );
if tblmodp = fail then
# Add the table to the library if it has trivial $O_p(G)$.
nsg:= List( ClassPositionsOfNormalSubgroups( tbl ),
x -> Sum( SizesConjugacyClasses( tbl ){ x } ) );
if not ForAny( nsg, n -> IsPrimePowerInt( n ) and n mod p = 0 ) then
AutomorphismsOfTable( ts3modp.table );
#T better call a function that performs all checks for new Brauer tables!
Print( "#I add the following ", p, "-modular table of `",
Identifier( tbl ), "':\n",
CTblLibStringBrauer( ts3modp.table) );
fi;
elif Irr( ts3modp.table ) <> Irr( tblmodp ) then
Print( "#E constructed and library table for `",
Identifier( tbl ), "' sorted incompatibly\n" );
result:= false;
fi;
fi;
od;
return result;
end );
#############################################################################
##
#V CTblLibTestConstructionsFunctions
##
InstallValue( CTblLibTestConstructionsFunctions, [
"ConstructGS3", CTblLibTestGS3Construction,
"ConstructDirectProduct", CTblLibTestDirectProductConstruction,
] );
#############################################################################
##
#F CTblLibTestConstructions()
#F CTblLibTestConstructions( <tblname> )
##
InstallGlobalFunction( CTblLibTestConstructions, function( arg )
local result, name, tbl, constr, pos;
result:= true;
if Length( arg ) = 0 then
for name in LIBLIST.firstnames do
result:= CTblLibTestConstructions( name ) and result;
od;
elif Length( arg ) = 1 and IsString( arg[1] ) then
tbl:= CharacterTable( arg[1] );
if HasConstructionInfoCharacterTable( tbl ) then
# Apply tests depending on the construction type of the table.
constr:= ConstructionInfoCharacterTable( tbl );
if IsList( constr ) then
pos:= Position( CTblLibTestConstructionsFunctions,
ConstructionInfoCharacterTable( tbl )[1] );
if pos <> fail then
return CTblLibTestConstructionsFunctions[ pos + 1 ]( tbl );
fi;
fi;
fi;
fi;
return result;
end );
#############################################################################
##
#E
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