Spracherkennung für: .gi vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen]
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##
#W construc.gi GAP 4 package CTblLib Thomas Breuer
##
## 1. Character Tables of Groups of Structure $M.G.A$
## 2. Character Tables of Groups of Structure $G.S_3$
## 3. Character Tables of Groups of Structure $G.2^2$
## 4. Character Tables of Groups of Structure $2^2.G$
## 5. Character Tables of Subdirect Products of Index Two
## 6. Brauer Tables of Extensions by $p$-regular Automorphisms
## 7. Construction Functions used in the Character Table Library
## 8. Character Tables of Coprime Central Extensions
## 9. Miscellaneous
##
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##
## 1. Character Tables of Groups of Structure $M.G.A$
##
#############################################################################
##
#F IrreducibleCharactersOfTypeMGA( <tblMG>, <tblGA>, <Mclasses>, <MGAfusGA>,
#F <orbs>, <a> )
##
## <Mclasses> is assumed to be the set of classes in <tblMG> that are mapped
## to the identity in the factor group $G$
## that is a normal subgroup of <tblGA>.
## Note that the table of $G$ is not needed at all in the construction,
## only the quotient $|G.a| / |G|$ seems to be a necessary parameter.
##
BindGlobal( "IrreducibleCharactersOfTypeMGA",
function( tblMG, tblGA, Mclasses, MGAfusGA, orbs, a )
local irr, p, m, ordtbl, zero, chi, ind, i;
irr:= List( Irr( tblGA ), chi -> CompositionMaps( chi, MGAfusGA ) );
p:= UnderlyingCharacteristic( tblMG );
m:= Sum( SizesConjugacyClasses( tblMG ){ Mclasses }, 0 );
if p <> 0 then
# Consider the ordinary tables, and divide the group orders by the
# orders of the `p'-cores.
# Note that we may construct the 3-modular table of $6.A_6.2_1$ from
# the table of its derived subgroup and the table of one of the factor
# groups $3.A_6.2_1$ or $A_6.2_1$.
ordtbl:= OrdinaryCharacterTable( tblMG );
m:= m * Sum( SizesConjugacyClasses( ordtbl ){
ClassPositionsOfPCore( ordtbl, p ) }, 0 );
ordtbl:= OrdinaryCharacterTable( tblGA );
m:= m / Sum( SizesConjugacyClasses( ordtbl ){
ClassPositionsOfPCore( ordtbl, p ) }, 0 );
fi;
zero:= Zero( MGAfusGA );
for chi in Irr( tblMG ) do
if not IsSubset( ClassPositionsOfKernel( chi ), Mclasses ) then
ind:= ShallowCopy( zero );
for i in [ 1 .. Length( orbs ) ] do
if IsBound( orbs[i] ) then
ind[i]:= Sum( chi{ orbs[i] }, 0 ) * ( a / Length( orbs[i] ) );
fi;
od;
if not ind in irr then
Add( irr, ind );
fi;
fi;
od;
return irr;
end );
#############################################################################
##
#F PossibleCharacterTablesOfTypeMGA( <tblMG>, <tblG>, <tblGA>, <orbs>,
#F <identifier> )
##
InstallGlobalFunction( PossibleCharacterTablesOfTypeMGA,
function( tblMG, tblG, tblGA, orbs, identifier )
local MGfusG, # factor fusion map from `tblMG' onto `tblG'
GfusGA, # subgroup fusion map from `tblG' into `tblGA'
tblMGA, # record for the desired table
MGfusMGA, # subgroup fusion map from `tblMG' into `tblMGA'
factouter, # positions of classes of `tblGA' outside `tblG'
MGAfusGA, # factor fusion map from `tblMGA' onto `tblGA'
inner, # inner classes of `tblMGA'
outer, # outer classes of `tblMGA'
nccl, # class number of `tblMG'
classes, # class lengths of `tblMG'
i, # loop variable
primes, # prime divisors of the order of `tblMGA'
invMGAfusGA, # inverse of `MGAfusGA'
p, # loop variable
GAmapp, # `p'-th power map of `tblGA'
orders, # element orders of `tblMGA'
suborders, # element orders of `tblMG'
outerorders, # outer part of the orders
gcd, # g.c.d. of the orders of `M' and `A'
matautos, # matrix automorphisms of the irred. of `tblMGA'
tblrecord, # record of `tblMGA' (power maps perhaps ambiguous)
info, # list of possible tables
newinfo, # list of possible tables for the next step
pair, # loop variable
pow, # one possible power map
newmatautos, # automorphisms respecting one more power map
newtblMGA, # intermediate table with one more unique power map
oldfus;
# Check the arguments.
if not ForAll( [ tblMG, tblG, tblGA ], IsOrdinaryTable ) then
Error( "<tblG>, <tblMG>, <tblGA> must be ordinary character tables" );
fi;
# Fetch the stored fusions.
MGfusG:= GetFusionMap( tblMG, tblG );
GfusGA:= GetFusionMap( tblG, tblGA );
if MGfusG = fail or GfusGA = fail then
Error( "fusions <tblMG> -> <tblG>, <tblG> -> <tblGA> must be stored" );
fi;
# Initialize the table record `tblMGA' of $m.G.a$.
tblMGA:= rec( UnderlyingCharacteristic := 0,
Identifier := identifier,
Size := Size( tblMG ) * Size( tblGA ) / Size( tblG ),
ComputedPowerMaps := [] );
# The class fusion of `tblMG' into `tblMGA' is given by `orbs'.
MGfusMGA:= InverseMap( orbs );
# Determine the outer classes of `tblGA'.
factouter:= Difference( [ 1 .. NrConjugacyClasses( tblGA ) ], GfusGA );
# Compute the fusion of `tblMGA' onto `tblGA'.
MGAfusGA:= CompositionMaps( GfusGA, CompositionMaps( MGfusG, orbs ) );
Append( MGAfusGA, factouter );
# Distinguish inner and outer classes of `tblMGA'.
inner:= [ 1 .. Maximum( MGfusMGA ) ];
outer:= [ Maximum( MGfusMGA ) + 1 .. Length( MGAfusGA ) ];
nccl:= Length( inner ) + Length( outer );
# Compute the class lengths of `tblMGA'.
tblMGA.SizesConjugacyClasses:= Concatenation( Zero( inner ),
( Size( tblMG ) / Size( tblG ) )
* SizesConjugacyClasses( tblGA ){ factouter } );
classes:= SizesConjugacyClasses( tblMG );
for i in inner do
tblMGA.SizesConjugacyClasses[i]:= Sum( classes{ orbs[i] } );
od;
# Compute the centralizer orders of `tblMGA'.
tblMGA.SizesCentralizers:= List( tblMGA.SizesConjugacyClasses,
x -> tblMGA.Size / x );
# Compute the irreducible characters of `tblMGA'.
tblMGA.Irr:= IrreducibleCharactersOfTypeMGA( tblMG, tblGA,
ClassPositionsOfKernel( MGfusG ), MGAfusGA, orbs,
Size( tblGA ) / Size( tblG ) );
# Compute approximations for power maps of `tblMGA'.
# (All $p$-th power maps for $p$ coprime to $|A|$ are uniquely
# determined this way, since inner and outer part are kept separately.)
#T We know more:
#T If |A| is a prime and does not divide |M| then the action is
#T semiregular; we have a unique fixed point for any element in N
#T that has a p-th root outside N.
primes:= PrimeDivisors( tblMGA.Size );
invMGAfusGA:= InverseMap( MGAfusGA );
for p in primes do
# inner part: Transfer the map from `tblMG' to `tblMGA'.
tblMGA.ComputedPowerMaps[p]:= CompositionMaps( MGfusMGA,
CompositionMaps( PowerMap( tblMG, p ), orbs ) );
# outer part: Use the map of `tblGA' for an approximation.
GAmapp:= PowerMap( tblGA, p );
for i in outer do
tblMGA.ComputedPowerMaps[p][i]:=
invMGAfusGA[ GAmapp[ MGAfusGA[i] ] ];
od;
od;
# Enter the element orders.
# (If $|A|$ and $|M|$ are coprime then the orders of outer elements
# are uniquely determined; otherwise there may be ambiguities.)
orders:= [];
suborders:= OrdersClassRepresentatives( tblMG );
for i in [ 1 .. Length( MGfusMGA ) ] do
orders[ MGfusMGA[i] ]:= suborders[i];
od;
outerorders:= OrdersClassRepresentatives( tblGA ){ factouter };
gcd:= Gcd( Size( tblMG ), Size( tblGA ) ) / Size( tblG );
if gcd <> 1 then
gcd:= DivisorsInt( gcd );
outerorders:= List( outerorders, x -> gcd * x );
fi;
tblMGA.OrdersClassRepresentatives:= Concatenation( orders, outerorders );
# Compute the automorphisms of the matrix of characters.
if gcd = 1 then
matautos:= [ tblMGA.SizesCentralizers,
tblMGA.OrdersClassRepresentatives ];
else
matautos:= [ tblMGA.SizesCentralizers ];
fi;
matautos:= MatrixAutomorphisms( tblMGA.Irr, matautos,
GroupByGenerators( [], () ) );
# Convert the record to a character table object.
# (Keep a record for the case that we need copies later.)
tblrecord:= ShallowCopy( tblMGA );
Unbind( tblrecord.ComputedPowerMaps );
ConvertToCharacterTableNC( tblMGA );
# Test and improve the (perhaps ambiguous) power maps
# (and update the automorphisms if necessary) using characters.
# Whenever several $p$-th power maps are possible then we branch,
# so we end up with a list of possible character tables.
info:= [ [ tblMGA, matautos ] ];
for p in primes do
newinfo:= [];
for pair in info do
tblMGA:= pair[1];
matautos:= pair[2];
pow:= ComputedPowerMaps( tblMGA )[p];
pow:= PossiblePowerMaps( tblMGA, p, rec( powermap:= pow ) );
if not IsEmpty( pow ) then
# Consider representatives up to matrix automorphisms.
for pow in RepresentativesPowerMaps( pow, matautos ) do
newmatautos:= SubgroupProperty( matautos,
perm -> ForAll( [ 1 .. nccl ],
i -> pow[ i^perm ] = pow[i]^perm ),
TrivialSubgroup( matautos ),
TrivialSubgroup( matautos ) );
newtblMGA:= ConvertToLibraryCharacterTableNC(
ShallowCopy( tblrecord ) );
SetComputedPowerMaps( newtblMGA,
StructuralCopy( ComputedPowerMaps( tblMGA ) ) );
ComputedPowerMaps( newtblMGA )[p]:= pow;
Add( newinfo, [ newtblMGA, newmatautos ] );
od;
fi;
od;
# Hand over the list for the next step.
info:= newinfo;
od;
# Here we have the final list of tables.
for pair in info do
tblMGA:= pair[1];
SetAutomorphismsOfTable( tblMGA, pair[2] );
StoreFusion( tblMGA, MGAfusGA, tblGA );
oldfus:= ShallowCopy( ComputedClassFusions( tblMG ) );
StoreFusion( tblMG, MGfusMGA, tblMGA );
SetConstructionInfoCharacterTable( tblMGA,
ConstructMGAInfo( tblMGA, tblMG, tblGA ) );
if Length( oldfus ) < Length( ComputedClassFusions( tblMG ) ) then
Unbind( ComputedClassFusions( tblMG )[
Length( ComputedClassFusions( tblMG ) ) ] );
fi;
SetInfoText( tblMGA,
"constructed using `PossibleCharacterTablesOfTypeMGA'" );
# Store the unique element orders if necessary.
if gcd <> 1 then
ResetFilterObj( tblMGA, HasOrdersClassRepresentatives );
SetOrdersClassRepresentatives( tblMGA,
ElementOrdersPowerMap( ComputedPowerMaps( tblMGA ) ) );
fi;
od;
# Return the result list.
return List( info, pair -> rec( table := pair[1],
MGfusMGA := MGfusMGA ) );
end );
#############################################################################
##
#F BrauerTableOfTypeMGA( <modtblMG>, <modtblGA>, <ordtblMGA> )
##
InstallGlobalFunction( BrauerTableOfTypeMGA,
function( modtblMG, modtblGA, ordtblMGA )
local p, modtblMGA, MGfusMGA, MGAfusGA, orbs, i, kernel;
# Fetch the underlying characteristic, and check the arguments.
p:= UnderlyingCharacteristic( modtblMG );
if UnderlyingCharacteristic( modtblGA ) <> p then
Info( InfoCharacterTable, 1,
"BrauerTableOfTypeMGA: UnderlyingCharacteristic values differ\n",
"#I for <modtblMG>, <modtblGA>" );
return fail;
elif not IsOrdinaryTable( ordtblMGA ) then
Info( InfoCharacterTable, 1,
"BrauerTableOfTypeMGA: <ordtblMGA> must be the ordinary table\n",
"#I of M.G.A" );
return fail;
fi;
# We cannot assume that the ordinary table of `tblMGA' has the same
# ordering of classes as is guaranteed for the table to be constructed.
# (Consider the case of $M.G = 3.U.3$ and $G.A = U.6$, where the
# outer classes of $U.2$ precede the outer classes of $U.3$.)
modtblMGA:= CharacterTableRegular( ordtblMGA, p );
# Compute the restriction of the action to the `p'-regular classes.
MGfusMGA:= GetFusionMap( modtblMG, modtblMGA );
if MGfusMGA = fail then
Info( InfoCharacterTable, 1,
"BrauerTableOfTypeMGA: the class fusion\n",
"#I OrdinaryCharacterTable( <modtblMG> ) -> <ordtblMGA> ",
"must be stored" );
return fail;
fi;
# Compute the irreducibles.
MGAfusGA:= GetFusionMap( modtblMGA, modtblGA );
orbs:= InverseMap( MGfusMGA );
for i in [ 1 .. Length( orbs ) ] do
if IsBound( orbs[i] ) and IsInt( orbs[i] ) then
orbs[i]:= [ orbs[i] ];
fi;
od;
kernel:= Filtered( [ 1 .. NrConjugacyClasses( modtblMG ) ],
i -> MGAfusGA[ MGfusMGA[i] ] = 1 );
SetIrr( modtblMGA, List( IrreducibleCharactersOfTypeMGA( modtblMG,
modtblGA, kernel, MGAfusGA, orbs,
Size( ordtblMGA ) / Size( modtblMG ) ),
x -> Character( modtblMGA, x ) ) );
SetInfoText( modtblMGA, "constructed using `BrauerTableOfTypeMGA'" );
# Return the result.
return rec( table:= modtblMGA, MGfusMGA:= MGfusMGA );
end );
#############################################################################
##
#F PossibleActionsForTypeMGA( <tblMG>, <tblG>, <tblGA> )
##
InstallGlobalFunction( PossibleActionsForTypeMGA,
function( tblMG, tblG, tblGA )
local tfustA,
Mtfust,
ker,
index,
inner,
i,
elms,
cenMG,
cenG,
inv,
factorbits,
img,
newelms,
chars;
# Check that the function is applicable.
tfustA:= GetFusionMap( tblG, tblGA );
if tfustA = fail then
Error( "class fusion <tblG> -> <tblGA> must be stored on <tblG>" );
fi;
Mtfust:= GetFusionMap( tblMG, tblG );
if Mtfust = fail then
Error( "class fusion <tblMG> -> <tblG> must be stored on <tblMG>" );
fi;
index:= Size( tblGA ) / Size( tblG );
if not IsPrimeInt( index ) then
inner:= Set( tfustA );
for i in PrimeDivisors( index ) do
if ForAll( PowerMap( tblGA, index / i ), j -> j in inner ) then
Error( "factor of <tblGA> by <tblG> must be cyclic" );
fi;
od;
fi;
# The automorphism must have order equal to the order of $A$.
# We need to consider only one generator for each cyclic group of
# the right order.
elms:= Filtered( Elements( AutomorphismsOfTable( tblMG ) ),
#T better avoid computing all elements
x -> Order( x ) = index );
elms:= Set( List( elms, SmallestGeneratorPerm ) );
Info( InfoCharacterTable, 1,
Length( elms ), " automorphism(s) of order ", index );
# The automorphism respects the fusion of classes of $G$ into $G.A$.
inv:= InverseMap( Mtfust );
for i in [ 1 .. Length( inv ) ] do
if IsInt( inv[i] ) then
inv[i]:= [ inv[i] ];
fi;
od;
factorbits:= Filtered( InverseMap( tfustA ), IsList );
for i in [ 1 .. Length( inv ) ] do
img:= First( factorbits, orb -> i in orb );
if img = fail then
img:= inv[i];
newelms:= Filtered( elms, x -> OnSets( img, x ) = img );
else
img:= Union( inv{ Difference( img, [ i ] ) } );
newelms:= Filtered( elms, x -> IsSubset( img, OnSets( inv[i], x ) ) );
fi;
if newelms <> elms then
elms:= newelms;
Info( InfoCharacterTable, 1,
Length( elms ), " automorphism(s) mapping ",i," compatibly" );
fi;
od;
# The automorphism must act semiregularly on those characters of $M.G$
# that are not characters of $G$.
# (Think of the case that the centres of $G$ and $M.G$ have orders
# $2$ and $6$, respectively, and $A$ is of order $2$.)
ker:= ClassPositionsOfKernel( Mtfust );
chars:= Filtered( Irr( tblMG ),
chi -> not IsSubset( ClassPositionsOfKernel( chi ), ker ) );
elms:= Filtered( elms,
x -> Set( OrbitLengths( Group(x), chars, Permuted ) )
= [ index ] );
Info( InfoCharacterTable, 1,
Length( elms ), " automorphism(s) acting semiregularly" );
# Form the orbits on the class positions.
elms:= Set( List( elms, x -> Set( List( Orbits( Group(x),
[ 1 .. NrConjugacyClasses( tblMG ) ] ), Set ) ) ) );
# Return the result.
return elms;
end );
#############################################################################
##
#F ConstructMGA( <tbl>, <subname>, <factname>, <plan>, <perm> )
##
InstallGlobalFunction( ConstructMGA,
function( tbl, subname, factname, plan, perm )
local factfus, subfus, proj, irreds, zero, irr, newirr, entry, sum, chi,
i;
factfus := First( tbl.ComputedClassFusions,
fus -> fus.name = factname ).map;
factname := CharacterTableFromLibrary( factname );
subname := CharacterTableFromLibrary( subname );
subfus := First( ComputedClassFusions( subname ),
fus -> fus.name = tbl.Identifier ).map;
proj := ProjectionMap( subfus );
irreds := List( Irr( factname ),
x -> ValuesOfClassFunction( x ){ factfus } );
zero:= [ 1 .. Length( factfus ) ] * 0;
irr:= Irr( subname );
newirr:= [];
for entry in plan do
# Note that `proj' need not be dense.
sum:= Sum( irr{ entry } );
chi:= ShallowCopy( zero );
for i in [ 1 .. Length( chi ) ] do
if IsBound( proj[i] ) then
chi[i]:= sum[ proj[i] ];
fi;
od;
Add( newirr, chi );
od;
Append( irreds, newirr );
tbl.Irr:= Permuted( irreds, perm );
end );
#############################################################################
##
## 2. Character Tables of Groups of Structure $G.S_3$
##
#############################################################################
##
#F IrreducibleCharactersOfTypeGS3( <tbl>, <tblC>, <tblK>, <aut>,
#F <tblfustblC>, <tblfustblK>, <tblCfustblKC>, <tblKfustblKC>, <outerC> )
##
BindGlobal( "IrreducibleCharactersOfTypeGS3",
function( tbl, tblC, tblK, aut, tblfustblC, tblfustblK, tblCfustblKC,
tblKfustblKC, outerC )
local irreducibles, # list of irreducible characters, result
zero, # zero vector on the classes of `tblC' \ `tbl'
irrtbl, # irreducible of `tbl'
irrtblC, # irreducible of `tblC'
irrtblK, # irreducible of `tblK'
done, # Boolean list, `true' for all processed characters
outerKC, # position of classes outside `tblK'
k, # order of the factor `tblK / tbl'
c, # order of the factor `tblC / tbl'
p, # characteristic of `tbl'
r, # ramification index
i, # loop over the irreducibles of `tblK'
chi, # currently processed character of `tblK'
img, # image of `chi' under `aut'
rest, # restriction of `chi' to `tbl' (via `tblfustblK')
e, # current ramification
const, # irreducible constituents of `rest'
ext, # extensions of an extendible constituent to `tblC'
chitilde, # one extension
irr, # one irreducible character
j, # loop over the classes of `tblK'
sum; # an induced character
# Initializations.
irreducibles:= [];
zero:= 0 * outerC;
irrtbl:= Irr( tbl );
irrtblC:= Irr( tblC );
irrtblK:= Irr( tblK );
done:= BlistList( [ 1 .. Length( irrtblK ) ], [] );
outerKC:= tblCfustblKC{ outerC };
k:= Size( tblK ) / Size( tbl );
c:= Size( tblC ) / Size( tbl );
p:= UnderlyingCharacteristic( tbl );
r:= RootInt( k );
if r^2 <> k then
r:= 1;
fi;
# Loop over the irreducibles of `tblK'.
for i in [ 1 .. Length( irrtblK ) ] do
if not done[i] then
done[i]:= true;
chi:= irrtblK[i];
img:= Permuted( chi, aut );
if img = chi then
# `chi' extends.
rest:= chi{ tblfustblK };
e:= 1;
if rest in irrtbl then
# `rest' is invariant in `tblKC', so we take the values
# of its extensions to `tblC' on the outer classes.
const:= [ rest ];
elif r <> 1 and rest / r in irrtbl then
# `rest' is a multiple of an irreducible character of `tbl'.
const:= [ rest / r ];
e:= r;
else
# `rest' is a sum of `k' irreducibles of `tbl';
# exactly one of them is fixed under the action of `tblC',
# so we take the values of the extensions of this constituent
# on the outer classes.
const:= Filtered( irrtbl,
x -> x[1] = rest[1] / k and
Induced( tbl, tblK, [ x ], tblfustblK )[1] = chi );
Assert( 1, Length( const ) = k,
"Strange number of constituents.\n" );
fi;
ext:= Filtered( irrtblC, x -> x[1] = const[1][1]
and x{ tblfustblC } in const );
Assert( 1, ( p = c and Length( ext ) = 1 ) or
( p <> c and Length( ext ) = c ),
"Extendible constituent is not unique.\n" );
# We can handle only a few cases where $e \neq 1$:
if e <> 1 and e = c - 1 then
# If $e = |C|-1$ then sum up all except one extension.
ext:= List( ext, x -> Sum( ext ) - x );
elif e <> 1 and e = c + 1 then
# If $e = |C|+1$ then sum up all plus one extension.
ext:= List( ext, x -> Sum( ext ) + x );
elif e <> 1 then
Error( "cannot handle a case where <e> > 1" );
fi;
for chitilde in ext do
irr:= [];
for j in [ 1 .. Length( tblKfustblKC ) ] do
irr[ tblKfustblKC[j] ]:= chi[j];
od;
irr{ outerKC }:= chitilde{ outerC };
Add( irreducibles, irr );
od;
else
# `chi' induces irreducibly.
irr:= [];
done[ Position( irrtblK, img ) ]:= true;
sum:= chi + img;
for j in [ 3 .. c ] do
img:= Permuted( img, aut );
done[ Position( irrtblK, img ) ]:= true;
sum:= sum + img;
od;
for j in [ 1 .. Length( tblKfustblKC ) ] do
irr[ tblKfustblKC[j] ]:= sum[j];
od;
irr{ outerKC }:= zero;
Add( irreducibles, irr );
fi;
fi;
od;
# Return the result.
Assert( 1, Length( irreducibles ) = Length( irreducibles[1] ),
Concatenation( "Not all irreducibles found (have ",
String( Length( irreducibles ) ), " of ",
String( Length( irreducibles[1] ) ), ")\n" ) );
return irreducibles;
end );
#############################################################################
##
#F CharacterTableOfTypeGS3( <tbl>, <tblC>, <tblK>, <aut>, <identifier> )
#F CharacterTableOfTypeGS3( <modtbl>, <modtblC>, <modtblK>, <ordtblKC>,
#F <identifier> )
##
InstallGlobalFunction( CharacterTableOfTypeGS3,
function( tbl, tblC, tblK, aut, identifier )
local p, # prime integer
tblfustblC, # class fusion from `tbl' into `tblC'
tblfustblK, # class fusion from `tbl' into `tblK'
tblKfustblKC, # class fusion from `tblK' into the desired table
tblCfustblKC, # class fusion from `tblC' into the desired table
outer, # positions of the classes of `tblC' \ `tbl'
i,
tblKC,
classes,
subclasses,
k,
orders,
suborders,
powermap,
pow,
oldfusC,
oldfusK;
# Fetch the underlying characteristic, and check the arguments.
p:= UnderlyingCharacteristic( tbl );
if UnderlyingCharacteristic( tblC ) <> p
or UnderlyingCharacteristic( tblK ) <> p then
Error( "UnderlyingCharacteristic values differ for <tbl>, <tblC>, ",
"<tblK>" );
elif 0 < p and not IsOrdinaryTable( aut ) then
Error( "enter the ordinary table of G.KC as the fourth argument" );
elif 0 = p and not IsPerm( aut ) then
Error( "enter a permutation as the fourth argument" );
fi;
# Fetch the stored fusions from `tbl'.
tblfustblC:= GetFusionMap( tbl, tblC );
tblfustblK:= GetFusionMap( tbl, tblK );
if tblfustblC = fail or tblfustblK = fail then
Error( "fusions <tbl> -> <tblC>, <tbl> -> <tblK> must be stored" );
fi;
outer:= Difference( [ 1 .. NrConjugacyClasses( tblC ) ], tblfustblC );
if 0 < p then
# We assume that the ordinary table of `tblKC' (given as the argument
# `aut') has the same ordering of classes as is guaranteed for the
# table to be constructed.
tblKC:= CharacterTableRegular( aut, p );
# Compute the restriction of the action to the `p'-regular classes.
tblKfustblKC:= GetFusionMap( tblK, tblKC );
if tblKfustblKC = fail then
Error( "fusion <tblK> -> <tblKC> must be stored" );
fi;
aut:= Product( List( Filtered( InverseMap( tblKfustblKC ), IsList ),
x -> MappingPermListList( x,
Concatenation( x{ [ 2 .. Length(x) ] },
[ x[1] ] ) ) ),
() );
# Fetch fusions for the result.
tblKfustblKC:= GetFusionMap( tblK, tblKC );
tblCfustblKC:= GetFusionMap( tblC, tblKC );
else
# Compute the needed fusions into `tblKC'.
tblKfustblKC:= InverseMap( Set( Orbits( Group( aut ),
[ 1 .. NrConjugacyClasses( tblK ) ] ) ) );
tblCfustblKC:= CompositionMaps( tblKfustblKC,
CompositionMaps( tblfustblK, InverseMap( tblfustblC ) ) );
tblCfustblKC{ outer }:= [ 1 .. Length( outer ) ]
+ Maximum( tblKfustblKC );
# Initialize the record for the character table `tblKC'.
tblKC:= rec( UnderlyingCharacteristic := 0,
Identifier := identifier,
Size := Size( tblK ) * Size( tblC ) / Size( tbl ) );
# Compute class lengths and centralizer orders.
classes:= ListWithIdenticalEntries( Maximum( tblCfustblKC ), 0 );
subclasses:= SizesConjugacyClasses( tblK );
for i in [ 1 .. Length( subclasses) ] do
classes[ tblKfustblKC[i] ]:= classes[ tblKfustblKC[i] ]
+ subclasses[i];
od;
subclasses:= SizesConjugacyClasses( tblC );
k:= Size( tblK ) / Size( tbl );
for i in outer do
classes[ tblCfustblKC[i] ]:= classes[ tblCfustblKC[i] ]
+ k * subclasses[i];
od;
tblKC.SizesConjugacyClasses:= classes;
tblKC.SizesCentralizers:= List( classes, x -> tblKC.Size / x );
# Compute element orders.
orders:= [];
suborders:= OrdersClassRepresentatives( tblK );
for i in [ 1 .. Length( tblKfustblKC ) ] do
orders[ tblKfustblKC[i] ]:= suborders[i];
od;
suborders:= OrdersClassRepresentatives( tblC );
for i in outer do
orders[ tblCfustblKC[i] ]:= suborders[i];
od;
tblKC.OrdersClassRepresentatives:= orders;
# Convert the record to a table object.
ConvertToLibraryCharacterTableNC( tblKC );
# Put the power maps together.
powermap:= ComputedPowerMaps( tblKC );
for p in PrimeDivisors( Size( tblKC ) ) do
pow:= InitPowerMap( tblKC, p );
TransferDiagram( PowerMap( tblC, p ), tblCfustblKC, pow );
TransferDiagram( PowerMap( tblK, p ), tblKfustblKC, pow );
powermap[p]:= pow;
Assert( 1, ForAll( pow, IsInt ),
Concatenation( Ordinal( p ),
" power map not uniquely determined" ) );
od;
fi;
# Compute the irreducibles.
SetIrr( tblKC,
List( IrreducibleCharactersOfTypeGS3( tbl, tblC, tblK, aut,
tblfustblC, tblfustblK, tblCfustblKC, tblKfustblKC,
outer ),
chi -> Character( tblKC, chi ) ) );
if IsOrdinaryTable( tblKC ) then
oldfusC:= ShallowCopy( ComputedClassFusions( tblC ) );
StoreFusion( tblC, tblCfustblKC, tblKC );
oldfusK:= ShallowCopy( ComputedClassFusions( tblK ) );
StoreFusion( tblK, tblKfustblKC, tblKC );
SetConstructionInfoCharacterTable( tblKC,
ConstructGS3Info( tblC, tblK, tblKC ).list );
if Length( oldfusC ) < Length( ComputedClassFusions( tblC ) ) then
Unbind( ComputedClassFusions( tblC )[
Length( ComputedClassFusions( tblC ) ) ] );
fi;
if Length( oldfusK ) < Length( ComputedClassFusions( tblK ) ) then
Unbind( ComputedClassFusions( tblK )[
Length( ComputedClassFusions( tblK ) ) ] );
fi;
fi;
SetInfoText( tblKC, "constructed using `CharacterTableOfTypeGS3'" );
# Return the result.
return rec( table := tblKC,
tblCfustblKC := tblCfustblKC,
tblKfustblKC := tblKfustblKC );
end );
#############################################################################
##
#F PossibleActionsForTypeGS3( <tbl>, <tblC>, <tbl3> )
##
#T Do we need a function that computes also compatible fusions if necessary?
#T (The condition is that the orbits on the classes of <tbl> describe an
#T action of S_3.)
##
InstallGlobalFunction( PossibleActionsForTypeGS3, function( tbl, tblC, tblK )
local tfustC, tfustK, c, elms, inner, linK, i, vals, c1, c2, newelms,
inv, orbs, orb;
# Check that the function is applicable.
tfustC:= GetFusionMap( tbl, tblC );
if tfustC = fail then
Error( "class fusion <tbl> -> <tblC> must be stored on <tbl>" );
fi;
tfustK:= GetFusionMap( tbl, tblK );
if tfustK = fail then
Error( "class fusion <tbl> -> <tblK> must be stored on <tbl>" );
fi;
# The automorphism must have order `c'.
c:= Size( tblC ) / Size( tbl );
elms:= Filtered( Elements( AutomorphismsOfTable( tblK ) ),
#T better avoid computing all elements
x -> Order( x ) = c );
Info( InfoCharacterTable, 1,
Length( elms ), " automorphism(s) of order ", c );
if Length( elms ) <= 1 then
return elms;
fi;
# The automorphism must permute the outer cosets of `tblK'.
inner:= Set( tfustK );
linK:= Filtered( Irr( tblK ),
chi -> IsSubset( ClassPositionsOfKernel( chi ), inner ) );
linK:= Difference( linK, [ TrivialCharacter( tblK ) ] );
elms:= Filtered( elms, x -> Permuted( linK[1], x ) = linK[2] );
Info( InfoCharacterTable, 1,
Length( elms ), " automorphism(s) permuting the cosets" );
if Length( elms ) <= 1 then
return elms;
fi;
# The automorphism respects the fusion of classes of `tbl' into `tblC'.
for i in InverseMap( tfustC ) do
if IsList( i ) then
vals:= SortedList( tfustK{ i } );
c1:= vals[1];
c2:= vals[2];
if c1 <> c2 then
RemoveSet( vals, c1 );
newelms:= Filtered( elms, x -> c1^x in vals );
if newelms <> elms then
elms:= newelms;
Info( InfoCharacterTable, 1,
Length( elms ), " automorphism(s) fusing ", c1, " and ",
c2 );
if Length( elms ) <= 1 then
return elms;
fi;
fi;
fi;
fi;
od;
# Two inner classes that are not fused in `tblC'
# cannot be conjugate in `tKC'.
# (Note that the centralizer order in `tblC' is `c' times larger than
# in `tbl', and this extra factor does not occur in the centralizer
# order in `tblK'.)
inv:= InverseMap( tfustK );
orbs:= Union( List( elms, i -> Filtered( Orbits( Group( i ), inner ),
orb -> Length( orb ) = c ) ) );
orbs:= Filtered( orbs,
x -> ( ForAll( inv{ x }, IsInt )
and Number( Set( tfustC{ inv{ x } } ) ) > 1 )
or ( ForAll( inv{ x }, IsList )
and ForAny( inv[ x[1] ],
y -> SizesCentralizers( tbl )[y]
< SizesCentralizers( tblC )[ tfustC[y] ] ) ) );
for orb in orbs do
c1:= orb[1];
c2:= orb[2];
newelms:= Filtered( elms, x -> c1^x <> c2 );
if newelms <> elms then
elms:= newelms;
Info( InfoCharacterTable, 1,
Length( elms ),
" automorphism(s) not fusing ", c1, " and ", c2 );
if Length( elms ) <= 1 then
return elms;
fi;
fi;
od;
# Return the result.
return elms;
end );
#############################################################################
##
## 3. Character Tables of Groups of Structure $G.2^2$
##
#############################################################################
##
#F PossibleActionsForTypeGV4( <tblG>, <tblsG2> )
##
InstallGlobalFunction( PossibleActionsForTypeGV4,
function( tblG, tblsG2 )
local tfust2, perms, actonG, fixedinG, i, j, k, inv, stabs, elms,
domains, triples, elm, comp, nccl, filt2, filt3;
# Check that the function is applicable.
tfust2:= List( tblsG2, t2 -> GetFusionMap( tblG, t2 ) );
if fail in tfust2 then
Error( "class fusions <tblG> -> <tblsG2> must be stored on <tblG>" );
elif ForAny( tblsG2, t2 -> Size( t2 ) <> 2 * Size( tblG ) ) then
Error( "<tblG> must have index 2 in all tables in <tblsG2>" );
fi;
# For computing compatible actions on the tables in `tblsG2',
# we rearrange these tables such that the fusions are sorted.
# In particular, the classes of <tblG> shall come first.
perms:= [];
actonG:= [];
fixedinG:= [];
for i in [ 1 .. 3 ] do
perms[i]:= [];
j:= 1;
for k in [ 1 .. Length( tfust2[i] ) ] do
if not IsBound( perms[i][ tfust2[i][k] ] ) then
perms[i][ tfust2[i][k] ]:= j;
j:= j+1;
fi;
od;
for k in [ 1 .. NrConjugacyClasses( tblsG2[i] ) ] do
if not IsBound( perms[i][k] ) then
perms[i][k]:= j;
j:= j+1;
fi;
od;
perms[i]:= PermList( perms[i] );
tfust2[i]:= OnTuples( tfust2[i], perms[i] );
inv:= InverseMap( tfust2[i] );
actonG[i]:= Product( List( Filtered( inv, IsList ),
x -> (x[1], x[2]) ),
() );
fixedinG[i]:= Filtered( inv, IsInt );
od;
# Check that the three fusions are compatible in the sense that
# the product of the three permutations induced on the classes of
# <tblG> is the identity.
if not IsOne( Product( actonG ) ) then
Info( InfoCharacterTable, 1,
"the three subgroup fusions are not compatible" );
return [];
fi;
# The automorphisms must have order at most 2.
fixedinG:= Intersection( fixedinG );
stabs:= [];
for i in [ 1 .. 3 ] do
stabs[i]:= Stabilizer( AutomorphismsOfTable( tblsG2[i] ),
OnTuples( tfust2[i]{ fixedinG }, perms[i]^-1 ), OnTuples );
od;
elms:= List( stabs, H -> Filtered( H, x -> Order( x ) <= 2 ) );
Info( InfoCharacterTable, 1,
Product( List( elms, Length ) ),
" triple(s) of automorphisms of order <= 2" );
# Two classes of $G$ that are not conjugate in any $G.2_i$
# are not conjugate in $G.2^2$.
# (By the compatibility, we need to test nonconjugacy only in $G.2_1$.)
elms[1]:= Filtered( elms[1],
x -> ForAll( tfust2[1]{ fixedinG },
p -> p^( x^perms[1] ) = p ) );
Info( InfoCharacterTable, 1,
Product( List( elms, Length ) ),
" triple(s) of automorphisms respecting inner classes" );
# The automorphisms must act compatibly on `tblG', and
# they must result in the same number of classes for $G.2^2$.
# (Note that the class number corresponds to the number of cycles.)
domains:= List( tblsG2, t -> [ 1 .. NrConjugacyClasses( t ) ] );
triples:= [];
for elm in elms[1] do
comp:= CompositionMaps( InverseMap(
OrbitsPerms( [ elm^perms[1] ], domains[1] ) ), tfust2[1] );
nccl:= 2 * NrConjugacyClasses( tblsG2[1] )
- 3 * NrMovedPointsPerm( elm ) / 2;
filt2:= Filtered( elms[2], x -> comp = CompositionMaps(
InverseMap( OrbitsPerms( [ x^perms[2] ], domains[2] ) ),
tfust2[2] )
and nccl = 2 * NrConjugacyClasses( tblsG2[2] )
- 3 * NrMovedPointsPerm( x ) / 2 );
filt3:= Filtered( elms[3], x -> comp = CompositionMaps(
InverseMap( OrbitsPerms( [ x^perms[3] ], domains[3] ) ),
tfust2[3] )
and nccl = 2 * NrConjugacyClasses( tblsG2[3] )
- 3 * NrMovedPointsPerm( x ) / 2 );
Append( triples, Cartesian( [ elm ], filt2, filt3 ) );
od;
Info( InfoCharacterTable, 1,
Length( triples ),
" triple(s) of automorphisms acting compatibly" );
# Return the result.
return triples;
end );
#############################################################################
##
#F PossibleCharacterTablesOfTypeGV4( <tblG>, <tblsG2>, <acts>, <identifier>
#F [, <tblGfustblsG2>] )
#F PossibleCharacterTablesOfTypeGV4( <modtblG>, <modtblsG2>, <ordtblGV4>
#F [, <ordtblsG2fusordtblG4>] )
##
InstallGlobalFunction( PossibleCharacterTablesOfTypeGV4,
function( arg )
local tblG, tblsG2, ordtblGV4, GfusG2, acts, identifier, char, tblGV4,
G2fusGV4, classes, cosets, G2fusGV4outer, i, k, tblfusordtbl, rest,
defectzero, intrest, G2fusGV4inner, ncclinner, tblrec,
subclasses, orders, suborders, map, powermap, p, pow, irr, ind,
indirr, triv, done, bad, num2, ext, numinv, poss1, poss2, chi,
intmat1, intmat2, todo, minus, nexttodo, poss, j, modrest;
# Get and check the arguments.
if Length( arg ) = 3 and
IsBrauerTable( arg[1] ) and IsList( arg[2] )
and IsOrdinaryTable( arg[3] ) then
tblG := arg[1];
tblsG2 := arg[2];
ordtblGV4 := arg[3];
GfusG2 := List( tblsG2, t -> GetFusionMap( tblG, t ) );
elif Length( arg ) = 4 and
IsBrauerTable( arg[1] ) and IsList( arg[2] )
and IsOrdinaryTable( arg[3] ) and IsList( arg[4] ) then
tblG := arg[1];
tblsG2 := arg[2];
ordtblGV4 := arg[3];
GfusG2 := List( tblsG2, t -> GetFusionMap( tblG, t ) );
G2fusGV4 := arg[4];
elif Length( arg ) = 4 and
IsOrdinaryTable( arg[1] ) and IsList( arg[2] ) and IsList( arg[3] )
and IsString( arg[4] ) then
tblG := arg[1];
tblsG2 := arg[2];
acts := arg[3];
identifier := arg[4];
GfusG2 := List( tblsG2, t -> GetFusionMap( tblG, t ) );
elif Length( arg ) = 5 and
IsCharacterTable( arg[1] ) and IsList( arg[2] ) and IsList( arg[3] )
and IsString( arg[4] ) and IsList( arg[5] ) then
tblG := arg[1];
tblsG2 := arg[2];
acts := arg[3];
identifier := arg[4];
GfusG2 := arg[5];
else
Error( "usage: PossibleCharacterTablesOfTypeGV4( <tlbG>, <tblsG2>, ",
"<acts>, <identifier>[, <fusions>] ) or\n",
"PossibleCharacterTablesOfTypeGV4( <modtblG>, <modtblsG2>, ",
"<ordtblGV4>[, <fusions>] )" );
fi;
if fail in GfusG2 then
Error( "the class fusions <tblG> -> <tblsG2> must be stored" );
fi;
# Fetch the underlying characteristic.
char:= UnderlyingCharacteristic( tblG );
if 0 < char then
# We assume that the ordinary table of `tblGV4' (given as an argument)
# has the same ordering of classes as is guaranteed for the
# table to be constructed.
tblGV4:= CharacterTableRegular( ordtblGV4, char );
if not IsBound( G2fusGV4 ) then
# Fetch the three fusions if they were not entered.
G2fusGV4:= List( tblsG2, t -> GetFusionMap( t, tblGV4 ) );
if fail in G2fusGV4 then
Error( "fusions <tblsG2> -> <tblGV4> must be stored" );
fi;
else
# Transfer the given fusions to the Brauer tables.
G2fusGV4:= List( [ 1 .. 3 ],
i -> CompositionMaps(
InverseMap( GetFusionMap( tblGV4, ordtblGV4 ) ),
CompositionMaps( G2fusGV4[i],
GetFusionMap( tblsG2[i],
OrdinaryCharacterTable( tblsG2[i] ) ) ) ) );
fi;
acts:= List( G2fusGV4,
map -> Product( List( Filtered( InverseMap( map ), IsList ),
pair -> ( pair[1], pair[2] ) ), () ) );
classes:= ShallowCopy( SizesConjugacyClasses( tblGV4 ) );
cosets:= Intersection( G2fusGV4 );
cosets:= List( G2fusGV4, map -> Difference( map, cosets ) );
G2fusGV4outer:= List( G2fusGV4, ShallowCopy );
for i in [ 1 .. 3 ] do
for k in GfusG2[i] do
Unbind( G2fusGV4outer[i][k] );
od;
od;
# We will use that defect zero characters must occur.
tblfusordtbl:= GetFusionMap( tblGV4, ordtblGV4 );
rest:= List( Irr( ordtblGV4 ), x -> x{ tblfusordtbl } );
defectzero:= Filtered( rest, x -> Size( tblGV4 ) / x[1] mod char <> 0 );
intrest:= IntegralizedMat( rest );
else
if not ( Length( acts ) = 3 and ForAll( acts, IsPerm ) ) then
Error( "<acts> must contain three permutations" );
fi;
# Construct the three fusions into $G.2^2$, via the three embeddings.
# The classes of $G$ come first in each map; note that we must choose
# the classes of $G$ compatibly in all three maps.
G2fusGV4inner:= [];
G2fusGV4inner[1]:= InverseMap( Set( Orbits( Group( acts[1] ),
Set( GfusG2[1] ) ) ) );
G2fusGV4inner[2]:= CompositionMaps( G2fusGV4inner[1],
CompositionMaps( GfusG2[1], InverseMap( GfusG2[2] ) ) );
G2fusGV4inner[3]:= CompositionMaps( G2fusGV4inner[1],
CompositionMaps( GfusG2[1], InverseMap( GfusG2[3] ) ) );
ncclinner:= Maximum( G2fusGV4inner[1] );
G2fusGV4outer:= [];
G2fusGV4outer[1]:= InverseMap( Set( Orbits( Group( acts[1] ),
Difference( [ 1 .. NrConjugacyClasses( tblsG2[1] ) ],
Set( GfusG2[1] ) ) ) ) ) + ncclinner;
G2fusGV4outer[2]:= InverseMap( Set( Orbits( Group( acts[2] ),
Difference( [ 1 .. NrConjugacyClasses( tblsG2[2] ) ],
Set( GfusG2[2] ) ) ) ) ) + Maximum( G2fusGV4outer[1] );
G2fusGV4outer[3]:= InverseMap( Set( Orbits( Group( acts[3] ),
Difference( [ 1 .. NrConjugacyClasses( tblsG2[3] ) ],
Set( GfusG2[3] ) ) ) ) ) + Maximum( G2fusGV4outer[2] );
cosets:= List( G2fusGV4outer, Set );
# Compute class lengths, centralizer orders, and element orders.
G2fusGV4:= G2fusGV4inner + G2fusGV4outer;
classes:= ListWithIdenticalEntries( Maximum( G2fusGV4[3] ), 0 );
subclasses:= SizesConjugacyClasses( tblsG2[1] );
orders:= [];
suborders:= OrdersClassRepresentatives( tblsG2[1] );
for i in [ 1 .. Length( G2fusGV4inner[1] ) ] do
if IsBound( G2fusGV4inner[1][i] ) then
classes[ G2fusGV4inner[1][i] ]:= classes[ G2fusGV4inner[1][i] ]
+ subclasses[i];
orders[ G2fusGV4inner[1][i] ]:= suborders[i];
fi;
od;
for k in [ 1 .. 3 ] do
subclasses:= SizesConjugacyClasses( tblsG2[k] );
suborders:= OrdersClassRepresentatives( tblsG2[k] );
map:= G2fusGV4outer[k];
for i in [ 1 .. Length( map ) ] do
if IsBound( map[i] ) then
classes[ map[i] ]:= classes[ map[i] ] + subclasses[i];
orders[ map[i] ]:= suborders[i];
fi;
od;
od;
# Initialize the record for the character table `tblGV4'.
tblrec:= rec( UnderlyingCharacteristic := 0,
Identifier := identifier,
Size := 4 * Size( tblG ),
SizesConjugacyClasses := Immutable( classes ),
OrdersClassRepresentatives := orders );
tblrec.SizesCentralizers:= List( classes, x -> tblrec.Size / x );
# Convert the record to a table object.
tblGV4:= ConvertToCharacterTableNC( ShallowCopy( tblrec ) );
# Put the power maps together.
powermap:= ComputedPowerMaps( tblGV4 );
for p in PrimeDivisors( Size( tblGV4 ) ) do
pow:= InitPowerMap( tblGV4, p );
for k in [ 1 .. 3 ] do
TransferDiagram( PowerMap( tblsG2[k], p ), G2fusGV4[k], pow );
od;
powermap[p]:= pow;
Assert( 1, ForAll( pow, IsInt ),
Concatenation( Ordinal( p ),
" power map not uniquely determined" ) );
od;
tblrec.ComputedPowerMaps:= ComputedPowerMaps( tblGV4 );
fi;
# Compute the irreducibles, starting from the irreducibles of $G$.
# First we compute the known extensions of the trivial character,
# then add the characters which are induced from some table in `tblsG2',
# then try to determine the extensions of the remaining characters.
irr:= List( Irr( tblG ), ValuesOfClassFunction );
ind:= List( [ 1 .. 3 ],
i -> InducedClassFunctionsByFusionMap( tblG, tblsG2[i], irr,
GfusG2[i] ) );
indirr:= List( [ 1 .. NrConjugacyClasses( tblG ) ],
k -> List( [ 1 .. 3 ],
i -> ind[i][k] in Irr( tblsG2[i] ) ) );
irr:= [];
triv:= Position( Irr( tblG ), TrivialCharacter( tblG ) );
if char = 2 then
irr[ triv ]:= [ 0 * classes + 1 ];
else
irr[ triv ]:= List( [ 1 .. 4 ], i -> 0 * classes + 1 );
irr[ triv ][3]{ cosets[1] }:= 0 * cosets[1] - 1;
irr[ triv ][4]{ cosets[1] }:= 0 * cosets[1] - 1;
irr[ triv ][2]{ cosets[2] }:= 0 * cosets[2] - 1;
irr[ triv ][4]{ cosets[2] }:= 0 * cosets[2] - 1;
irr[ triv ][2]{ cosets[3] }:= 0 * cosets[3] - 1;
irr[ triv ][3]{ cosets[3] }:= 0 * cosets[3] - 1;
fi;
done:= [ triv ];
bad:= [];
for i in [ 1 .. NrConjugacyClasses( tblG ) ] do
if not i in done then
num2:= Number( indirr[i], x -> x = false );
if num2 = 2 then
# This cannot happen, so the actions must be wrong.
Info( InfoCharacterTable, 1,
"PossibleCharacterTablesOfTypeGV4: contradiction, ",
"imposs. inertia subgroup" );
return [];
elif num2 = 0 then
# The character has inertia subgroup $G$.
irr[i]:= Induced( tblsG2[1], tblGV4, [ ind[1][i] ], G2fusGV4[1] );
AddSet( done, i );
AddSet( done, Position( ind[1], ind[1][i], i ) );
AddSet( done, Position( ind[2], ind[2][i], i ) );
AddSet( done, Position( ind[3], ind[3][i], i ) );
elif num2 = 1 then
# The character has inertia subgroup one of the $G.2_k$.
k:= Position( indirr[i], false );
ext:= Filtered( Irr( tblsG2[k] ),
x -> x{ GfusG2[k] } = Irr( tblG )[i] );
irr[i]:= Induced( tblsG2[k], tblGV4, ext, G2fusGV4[k] );
k:= ( ( k+1 ) mod 3 ) + 1;
AddSet( done, i );
AddSet( done, Position( ind[k], ind[k][i], i ) );
else
# The character has inertia subgroup $G.2^2$.
ext:= List( [ 1 .. 3 ], j -> Filtered( Irr( tblsG2[j] ),
x -> x{ GfusG2[j] } = Irr( tblG )[i] ) );
numinv:= Number( [ 1 .. 3 ],
x -> Permuted( ext[x][1], acts[x] ) = ext[x][1] );
ext:= ext[1];
if numinv in [ 1, 2 ] then
Info( InfoCharacterTable, 1,
"PossibleCharacterTablesOfTypeGV4: contradiction, ",
"impossible inertia subgroup" );
return [];
elif Permuted( ext[1], acts[1] ) <> ext[1] then
# The character induces from any of the $G.2_i$.
irr[i]:= Induced( tblsG2[1], tblGV4, ext{[1]}, G2fusGV4[1] );
AddSet( done, i );
else
# In characteristic $2$, we get a unique extension.
# Otherwise the character extends $4$-fold,
# and we have two possibilities for combining the different
# extensions to the tables in `tblsG2'.
ext:= List( ext, chi -> CompositionMaps( chi,
InverseMap( G2fusGV4[1] ) ) );
if char = 2 then
irr[i]:= ext;
AddSet( done, i );
else
poss1:= [ ShallowCopy( ext[1] ),
ShallowCopy( ext[1] ),
ShallowCopy( ext[2] ),
ShallowCopy( ext[2] ) ];
ext:= Filtered( Irr( tblsG2[2] ),
x -> x{ GfusG2[2] } = Irr( tblG )[i] );
for k in [ 1 .. Length( G2fusGV4outer[2] ) ] do
if IsBound( G2fusGV4outer[2][k] ) then
poss1{ [ 1 .. 4 ] }[ G2fusGV4outer[2][k] ]:=
ext[1][k] * [ 1, -1, 1, -1 ];
fi;
od;
ext:= Filtered( Irr( tblsG2[3] ),
x -> x{ GfusG2[3] } = Irr( tblG )[i] );
for k in [ 1 .. Length( G2fusGV4outer[3] ) ] do
if IsBound( G2fusGV4outer[3][k] ) then
poss1{ [ 1 .. 4 ] }[ G2fusGV4outer[3][k] ]:=
ext[1][k] * [ 1, -1, -1, 1 ];
fi;
od;
poss2:= List( poss1, ShallowCopy );
for chi in poss2 do
chi{ cosets[3] }:= - chi{ cosets[3] };
od;
if 0 < char and Size( tblGV4 ) / ext[1][1] mod char <> 0 then
if ForAll( poss1, x -> x in defectzero ) then
irr[i]:= poss1;
elif ForAll( poss2, x -> x in defectzero ) then
irr[i]:= poss2;
else
Error( "inconsistency involving defect zero characters" );
fi;
AddSet( done, i );
elif 0 < char then
# Check whether the possibilities are in the Z-span of the
# restricted ordinary characters.
intmat1:= IntegralizedMat( [ poss1[1] ],
intrest.inforec ).mat[1];
intmat2:= IntegralizedMat( [ poss2[1] ],
intrest.inforec ).mat[1];
if intmat1 = fail or
SolutionIntMat( intrest.mat, intmat1 ) = fail then
if intmat2 = fail or
SolutionIntMat( intrest.mat, intmat2 ) = fail then
# No combination of Brauer characters fits.
return [];
fi;
irr[i]:= poss2;
AddSet( done, i );
elif intmat2 = fail or
SolutionIntMat( intrest.mat, intmat2 ) = fail then
irr[i]:= poss1;
AddSet( done, i );
else
irr[i]:= [ poss1, poss2 ];
fi;
else
irr[i]:= [ poss1, poss2 ];
fi;
fi;
fi;
fi;
fi;
od;
# Deal with the extension case.
todo:= Difference( [ 1 .. NrConjugacyClasses( tblG ) ], done );
if char = 0 then
# For each set of four extensions of one character,
# check the scalar products with the characters $\chi^{2-}$,
# for all known irreducible (nonlinear) characters $\chi$.
pow:= ComputedPowerMaps( tblGV4 )[2];
minus:= Set( List( Union( Filtered( irr,
x -> x[1] <> 1 and
NestingDepthA( x ) = 2 ) ),
chi -> MinusCharacter( chi, pow, 2 ) ) );
nexttodo:= todo;
repeat
todo:= ShallowCopy( nexttodo );
for i in todo do
# Try to exclude one of the two possibilities via scalar products.
poss1:= NonnegIntScalarProducts( tblGV4, minus, irr[i][1][1] );
poss2:= NonnegIntScalarProducts( tblGV4, minus, irr[i][2][1] );
if not poss1 and not poss2 then
# Something must be wrong, for example the given actions.
Info( InfoCharacterTable, 1,
"PossibleCharacterTablesOfTypeGV4: contradiction, ",
"incompat. scalar products" );
return [];
elif poss1 and not poss2 then
irr[i]:= irr[i][1];
UniteSet( minus,
Set( List( irr[i], chi -> MinusCharacter( chi, pow, 2 ) ) ) );
RemoveSet( nexttodo, i );
elif poss2 and not poss1 then
irr[i]:= irr[i][2];
UniteSet( minus,
Set( List( irr[i], chi -> MinusCharacter( chi, pow, 2 ) ) ) );
RemoveSet( nexttodo, i );
fi;
od;
until todo = nexttodo;
# Form all combinations of extensions that are still possible.
poss:= [ irr ];
for i in todo do
poss:= Concatenation( [ List( poss, ShallowCopy ),
List( poss, ShallowCopy ) ] );
for j in [ 1 .. Length( poss ) / 2 ] do
poss[j][i]:= irr[i][1];
od;
for j in [ Length( poss ) / 2 + 1 .. Length( poss ) ] do
poss[j][i]:= irr[i][2];
od;
od;
for i in [ 1 .. Length( poss ) ] do
# Check that the irreducibles are closed under multiplication
# with linear characters,
# and that the power maps are admissible.
# Note that `PossiblePowerMaps' is not sufficient here,
# we check whether all symmetrizations decompose.
# An example where this excludes a candidate table is
# `2.U4(3).(2^2)_{133}'.
# Note that for large primes, constructing the symmetrizations
# is not feasible.
# An example where this happens is the table of `S4(9).2^2',
# the group order is divisible by 41.
tblGV4:= ConvertToCharacterTableNC( ShallowCopy( tblrec ) );
SetIrr( tblGV4, List( Concatenation( Compacted( poss[i] ) ),
chi -> Character( tblGV4, chi ) ) );
if ForAll( Irr( tblGV4 ), x -> ForAll( LinearCharacters( tblGV4 ),
y -> y * x in Irr( tblGV4 ) ) )
and ForAll( PrimeDivisors( Size( tblGV4 ) ),
p -> p > 20 or
ForAll( Symmetrizations( tblGV4, Irr( tblGV4 ), p ),
x -> NonnegIntScalarProducts( tblGV4,
Irr( tblGV4 ), x ) ) ) then
SetInfoText( tblGV4,
"constructed using `PossibleCharacterTablesOfTypeGV4'" );
AutomorphismsOfTable( tblGV4 );
poss[i]:= rec( table:= tblGV4, G2fusGV4:= G2fusGV4 );
else
Unbind( poss[i] );
fi;
od;
poss:= Compacted( poss );
else
# `char'-modular case:
# Form all combinations.
#T improve: consider blockwise, and perhaps for increasing degree
poss:= [ irr ];
for i in todo do
poss:= Concatenation( [ List( poss, ShallowCopy ),
List( poss, ShallowCopy ) ] );
for j in [ 1 .. Length( poss ) / 2 ] do
poss[j][i]:= poss[j][i][1];
od;
for j in [ Length( poss ) / 2 + 1 .. Length( poss ) ] do
poss[j][i]:= poss[j][i][2];
od;
od;
# Check each combination.
for i in [ 1 .. Length( poss ) ] do
# Test the decomposability of ordinary irreducibles.
poss[i]:= Concatenation( Compacted( poss[i] ) );
if fail in Decomposition( poss[i], rest, "nonnegative" ) then
Unbind( poss[i] );
else
# Test the decomposability of the restrictions
# to the subgroups of index two.
for j in [ 1 .. 3 ] do
modrest:= List( poss[i], x -> x{ G2fusGV4[j] } );
if fail in Decomposition( Irr( tblsG2[j] ), modrest,
"nonnegative" ) then
Unbind( poss[i] );
break;
fi;
od;
fi;
od;
poss:= Compacted( poss );
for i in [ 1 .. Length( poss ) ] do
tblGV4:= CharacterTableRegular( ordtblGV4, char );
SetIrr( tblGV4, List( poss[i],
chi -> Character( tblGV4, chi ) ) );
SetInfoText( tblGV4,
"constructed using `PossibleCharacterTablesOfTypeGV4'" );
AutomorphismsOfTable( tblGV4 );
poss[i]:= rec( table:= tblGV4, G2fusGV4:= G2fusGV4 );
od;
fi;
return poss;
end );
#############################################################################
##
#F PossibleActionsForTypeGA( <tblG>, <tblGA> )
##
InstallGlobalFunction( PossibleActionsForTypeGA,
function( tblG, tblGA )
local tfustA, A, elms, i, newelms;
# Check that the function is applicable.
tfustA:= GetFusionMap( tblG, tblGA );
if tfustA = fail then
Error( "class fusion <tblG> -> <tblGA> must be stored on <tblG>" );
fi;
# The automorphism must have order dividing `A'.
A:= Size( tblGA ) / Size( tblG );
elms:= Filtered( Elements( AutomorphismsOfTable( tblG ) ),
#T better avoid computing all elements
x -> A mod Order( x ) = 0 );
Info( InfoCharacterTable, 1,
Length( elms ), " automorphism(s) of order dividing ", A );
if Length( elms ) <= 1 then
return elms;
fi;
# The automorphism respects the fusion of classes of `tblG' into `tblGA'.
for i in InverseMap( tfustA ) do
if IsList( i ) then
newelms:= Filtered( elms, x -> OnSets( i, x ) = i and
OnPoints( i[1], x ) <> i[1] );
else
newelms:= Filtered( elms, x -> OnPoints( i, x ) = i );
fi;
if newelms <> elms then
elms:= newelms;
Info( InfoCharacterTable, 1,
Length( elms ), " automorphism(s) acting on ", i );
if Length( elms ) <= 1 then
return elms;
fi;
fi;
od;
# Return the result.
return elms;
end );
#############################################################################
##
#F ConstructMGAInfo( <tblmGa>, <tblmG>, <tblGa> )
##
InstallGlobalFunction( ConstructMGAInfo, function( tblmGa, tblmG, tblGa )
local factfus, subfus, kernel, nccl, irr, plan, chi, rest, nonfaith,
proj, zero, faithful, entry, sum, i, perm;
factfus:= GetFusionMap( tblmGa, tblGa );
subfus:= GetFusionMap( tblmG, tblmGa );
if factfus = fail or subfus = fail then
Error( "fusions <tblmG> -> <tblmGa> -> <tblGa> must be stored" );
fi;
kernel:= ClassPositionsOfKernel( factfus );
nccl:= NrConjugacyClasses( tblmG );
irr:= Irr( tblmG );
plan:= [];
for chi in Irr( tblmGa ) do
if not IsSubset( ClassPositionsOfKernel( chi ), kernel ) then
rest:= chi{ subfus };
Add( plan, Filtered( [ 1 .. nccl ],
i -> ScalarProduct( tblmG, rest, irr[i] ) <> 0 ) );
fi;
od;
nonfaith:= List( Irr( tblGa ), chi -> chi{ factfus } );
proj:= ProjectionMap( subfus );
zero:= [ 1 .. NrConjugacyClasses( tblmGa ) ] * 0;
faithful:= [];
for entry in plan do
# Note that `proj' need not be dense.
sum:= Sum( irr{ entry } );
chi:= ShallowCopy( zero );
for i in [ 1 .. Length( chi ) ] do
if IsBound( proj[i] ) then
chi[i]:= sum[ proj[i] ];
fi;
od;
Add( faithful, chi );
od;
perm:= Sortex( Concatenation( nonfaith, faithful ) ) /
Sortex( ShallowCopy( Irr( tblmGa ) ) );
return [ "ConstructMGA",
Identifier( tblmG ), Identifier( tblGa ), plan, perm ];
end );
#############################################################################
##
#F ConstructProj( <tbl>, <irrinfo> )
#F ConstructProjInfo( <tbl>, <kernel> )
##
InstallGlobalFunction( ConstructProj, function( tbl, irrinfo )
local i, j, factor, fus, mult, irreds, linear, omegasquare, I,
d, name, factfus, proj, adjust, Adjust,
ext, lin, chi, faith, nccl, partner, divs, prox, foll,
vals;
nccl:= Length( tbl.SizesCentralizers );
factor:= CharacterTableFromLibrary( irrinfo[1][1] );
fus:= First( tbl.ComputedClassFusions,
fus -> fus.name = irrinfo[1][1] ).map;
mult:= tbl.SizesCentralizers[1] / Size( factor );
irreds:= List( Irr( factor ), x -> ValuesOfClassFunction( x ){ fus } );
linear:= Filtered( irreds, x -> x[1] = 1 );
linear:= Filtered( linear, x -> ForAny( x, y -> y <> 1 ) );
# some roots of unity
omegasquare:= E(3)^2;
I:= E(4);
# Loop over the divisors of `mult' (a divisor of 12).
# Note the succession for `mult = 12'!
if mult <> 12 then
divs:= Difference( DivisorsInt( mult ), [ 1 ] );
else
divs:= [ 2, 4, 3, 6, 12 ];
fi;
for d in divs do
# Construct the faithful irreducibles for an extension by `d'.
# For that, we split and adjust the portion of characters (stored
# on the small table `factor') as if we would create this extension,
# and then we blow up these characters to the whole table.
name:= irrinfo[d][1];
partner:= irrinfo[d][2];
proj:= First( ProjectivesInfo( factor ), x -> x.name = name );
faith:= List( proj.chars, y -> y{ fus } );
if name = tbl.Identifier then
factfus:= [ 1 .. Length( tbl.SizesCentralizers ) ];
else
factfus:= First( tbl.ComputedClassFusions, x -> x.name = name ).map;
fi;
adjust:= First( tbl.ComputedClassFusions,
r -> r.name = Identifier( factor ) ).map;
proj:= ProjectionMap( CompositionMaps( adjust,
InverseMap( factfus ) ) );
Add( proj, Length( factfus ) + 1 ); # for termination of loop
adjust:= [];
for i in [ 1 .. Length( proj ) - 1 ] do
for j in [ proj[i] .. proj[i+1]-1 ] do
adjust[ j ]:= proj[i];
od;
od;
# Now we have to multiply the values on certain classes `j' with
# roots of unity, depending on the value of `d':
#T Note that we do not have the factor fusion from d.G to G available,
#T since the only tables we have are those of mult.G and G,
#T together with the projective characters for the various intermediate
#T tables!
#T -> wait: in the sit. m.G -->> d.G -->> G,
#T we know the fusions m.G -->> d.G and m.G -->> G and thus can
#T compute d.G -->> G as a composition!
Adjust:= [];
for i in [ 1 .. d-1 ] do
Adjust[i]:= Filtered( [ 1 .. Length( factfus ) ],
x -> adjust[ factfus[x] ] = factfus[x] - i );
od;
#T this means to adjust also in many zero columns;
#T if d = 6 and a class has only 2 or 3 preimages, the second preimage class
#T need not be adjusted for the faithful characters ...
# d = 2: classes in `Adjust[1]' multiply with `-1'
# d = 3: classes in `Adjust[x]' multiply
# with `E(3)^x' for the proxy cohort,
# with `E(3)^(2*x)' for the follower cohort
# d = 4: classes in `Adjust[x]' multiply
# with `E(4)^x' for the proxy cohort,
# with `(-E(4))^x' for the follower cohort,
# d = 6: classes in `Adjust[x]' multiply with `(-E(3))^x'
# d = 12: classes in `Adjust[x]' multiply with `(E(12)^7)^x'
#
# (*Note* that follower cohorts of classes never occur in projective
# ATLAS tables ... )
# Determine proxy classes and follower classes:
if Length( linear ) in [ 2, 5 ] then # out in [ 3, 6 ]
prox:= [];
foll:= [];
chi:= irreds[ Length( linear ) ];
for i in [ 1 .. nccl ] do
if chi[i] = omegasquare then
Add( foll, i );
else
Add( prox, i );
fi;
od;
elif Length( linear ) = 3 then # out = 4
prox:= [];
foll:= [];
chi:= irreds[2];
for i in [ 1 .. nccl ] do
if chi[i] = -I then Add( foll, i ); else Add( prox, i ); fi;
od;
else
prox:= [ 1 .. nccl ];
foll:= [];
fi;
if d = 2 then
# special case without Galois partners
for chi in faith do
for i in Adjust[1] do chi[i]:= - chi[i]; od;
Add( irreds, chi );
for lin in linear do
ext:= List( [ 1 .. nccl ], x -> lin[x] * chi[x] );
if not ext in irreds then Add( irreds, ext ); fi;
od;
od;
elif d = 12 then
# special case with three Galois partners and `lin = []'
vals:= [ E(12)^7, - omegasquare, - I, E(3), E(12)^11, -1,
-E(12)^7, omegasquare, I, -E(3), -E(12)^11 ];
for j in [ 1 .. Length( faith ) ] do
chi:= faith[j];
for i in [ 1 .. 11 ] do
chi{ Adjust[i] }:= vals[i] * chi{ Adjust[i] };
od;
Add( irreds, chi );
for i in partner[j] do
Add( irreds, List( chi, x -> GaloisCyc( x, i ) ) );
od;
od;
else
if d = 3 then
Adjust{ [ 1, 2 ] }:= [ Union( Intersection( Adjust[1], prox ),
Intersection( Adjust[2], foll ) ),
Union( Intersection( Adjust[2], prox ),
Intersection( Adjust[1], foll ) ) ];
vals:= [ E(3), E(3)^2 ];
elif d = 4 then
Adjust{ [ 1, 3 ] }:= [ Union( Intersection( Adjust[1], prox ),
Intersection( Adjust[3], foll ) ),
Union( Intersection( Adjust[3], prox ),
Intersection( Adjust[1], foll ) ) ];
vals:= [ I, -1, -I ];
elif d = 6 then
vals:= [ -E(3), omegasquare, -1, E(3), - omegasquare ];
fi;
for j in [ 1 .. Length( faith ) ] do
chi:= faith[j];
for i in [ 1 .. d-1 ] do
chi{ Adjust[i] }:= vals[i] * chi{ Adjust[i] };
od;
Add( irreds, chi );
for lin in linear do
ext:= List( [ 1 .. nccl ], x -> lin[x] * chi[x] );
if not ext in irreds then Add( irreds, ext ); fi;
od;
chi:= List( chi, x -> GaloisCyc( x, partner[j] ) );
Add( irreds, chi );
for lin in linear do
ext:= List( [ 1 .. nccl ], x -> lin[x] * chi[x] );
if not ext in irreds then Add( irreds, ext ); fi;
od;
od;
fi;
od;
tbl.Irr:= irreds;
end );
#############################################################################
##
#F CharacterTableSortedWRTCentralExtension( <tbl>, <facttbl>, <kernel> )
##
## This function is intended only as a subroutine of 'ConstructProjInfo'.
##
BindGlobal( "CharacterTableSortedWRTCentralExtension",
function( tbl, facttbl, kernel )
local classes, orders, mult, faithpos, sort, powers, fusion, inv,
mapping, i, preimorders, min, cand, count, first, j, divs,
portions;
# Check that the kernel is a central cyclic subgroup.
classes:= SizesConjugacyClasses( tbl );
orders:= OrdersClassRepresentatives( tbl );
mult:= Sum( classes{ kernel } );
faithpos:= First( kernel, i -> orders[i] = mult );
if 12 mod mult <> 0
or faithpos = fail
or Length( kernel ) <> mult then
Error( "only cyclic central ext. by a group of order dividing 12" );
fi;
# First sort the table w.r.t. the factor table.
sort:= SortedCharacterTable( tbl, facttbl, kernel );
if sort = fail then
Error( "<tbl> and <facttbl> do not fit w.r.t. <kernel>" );
fi;
kernel:= [ 1 .. mult ];
ResetFilterObj( sort, HasClassPermutation );
# Permute the classes of the kernel such that the $i$-th class
# is the $(i-1)$-th power of a generator class.
orders:= OrdersClassRepresentatives( sort );
faithpos:= First( kernel, i -> orders[i] = mult );
powers:= List( [ 1 .. mult-1 ], i -> PowerMap( sort, i, faithpos ) );
if powers <> [ 2 .. mult ] then
sort:= CharacterTableWithSortedClasses( sort,
PermList( Concatenation( [ 1 ], powers ) ) );
ResetFilterObj( sort, HasClassPermutation );
fi;
# Get the fusion to the factor group.
fusion:= GetFusionMap( sort, facttbl );
# Permute the classes such that the preimages of each class in the factor
# group are ordered in such a way that
# - the first preimage has minimal element order and among those classes
# with the minimal order has the fewest number of irrationalities,
# and for the classes where these two are minimal has more positive
# character values, and
# - the $i$-th preimage is obtained by
# multiplying the first preimage with the root of unity on the $i$-th
# class of the kernel.
# We assume that for the classes of the factor groups, the table is
# already sorted compatibly.
# So we have to consider only those cases where a full splitting occurs.
classes:= SizesConjugacyClasses( sort );
orders:= OrdersClassRepresentatives( sort );
inv:= InverseMap( fusion );
mapping:= [];
for i in [ 1 .. Length( inv ) ] do
if IsInt( inv[i] ) then
Add( mapping, inv[i] );
else
preimorders:= orders{ inv[i] };
min:= Minimum( preimorders );
cand:= Filtered( inv[i], j -> orders[j] = min );
if 1 < Length( cand ) then
count:= List( cand,
j -> Number( Irr( sort ), x -> not IsInt( x[j] ) ) );
min:= Minimum( count );
cand:= cand{ Filtered( [ 1 .. Length( cand ) ],
j -> count[j] = min ) };
if 1 < Length( cand ) then
count:= List( cand,
j -> Number( Irr( sort ), x -> IsNegInt( x[j] ) ) );
min:= Minimum( count );
cand:= cand{ Filtered( [ 1 .. Length( cand ) ],
j -> count[j] = min ) };
fi;
fi;
first:= cand[1];
for j in [ 1 .. mult ] do
Add( mapping, First( inv[i], k -> ForAll( Irr( sort ),
x -> x[k] * x[1] = x[j] * x[ first ] ) ) );
od;
fi;
od;
# Now distribute the irreducibles not having <kernel> in their kernels.
# Note the succession for `mult = 12'!
if mult <> 12 then
divs:= DivisorsInt( mult );
else
divs:= [ 1, 2, 4, 3, 6, 12 ];
fi;
portions:= List( Reversed( divs ),
d -> Filtered( Irr( sort ),
x -> Length( Intersection( kernel,
ClassPositionsOfKernel( x ) ) ) = d ) );
ResetFilterObj( sort, HasIrr );
SetIrr( sort, Concatenation( portions ) );
# Sort the classes.
return CharacterTableWithSortedClasses( sort, PermList( mapping ) );
end );
InstallGlobalFunction( ConstructProjInfo, function( tbl, kernel )
local fusions, fus, facttable,
sort,
mult, # order of the central subgroup `kernel'
faithpos, # position of a cyclic generator of the kernel
nsg, # class positions of subgroups of `kernel'
faith, # corresponding group orders
names, # names of factors by these subgroups
fusrec, # loop over fusions
faithchars, # faithful characters for each subgroup
chi, # loop over irreducibles of `tbl'
ker, # kernel of `chi'
proj,
nccl,
linear,
partners,
i,
new,
gal,
rest,
projectives,
info,
pi;
# Get the factor table.
fusions:= ComputedClassFusions( tbl );
fus:= First( fusions, x -> ClassPositionsOfKernel( x.map ) = kernel );
facttable:= CharacterTable( fus.name );
# Permute the classes and characters.
tbl:= CharacterTableSortedWRTCentralExtension( tbl, facttable, kernel );
kernel:= [ 1 .. Length( kernel ) ];
faithpos:= 2;
nsg:= Filtered( ClassPositionsOfNormalSubgroups( tbl ),
x -> IsSubset( kernel, x ) );
faith:= List( nsg, l -> Length( kernel ) / Length( l ) );
SortParallel( faith, nsg );
if Length( kernel ) = 12 then
nsg:= Permuted( nsg, (3,4) );
faith:= Permuted( faith, (3,4) );
fi;
names:= [];
fusions:= ComputedClassFusions( tbl );
for i in [ 1 .. Length( nsg )-1 ] do
fusrec:= First( fusions,
r -> ClassPositionsOfKernel( r.map ) = nsg[i] );
if fusrec = fail then
Error( "factor fusion with kernel ", nsg[i], " not stored" );
fi;
names[i]:= fusrec.name;
od;
names[ Length( nsg ) ]:= Identifier( tbl );
# Distribute the irreducibles according to their kernels.
# Take only those irreducibles
# of $3.G$ with value `E(3)' times the degree on the first nonid. class,
# of $4.G$ with value `E(4)' times the degree on the first nonid. class,
# of $6.G$ with value `E(6)^5' times the deg. on the first nonid. class,
# of $12.G$ with value `E(12)^7' times the deg. on the first nonid. class,
faithchars:= List( nsg, l -> [] );
for chi in Irr( tbl ) do
ker:= ClassPositionsOfKernel( chi );
for i in [ 1 .. Length( nsg ) ] do
if IsSubset( ker, nsg[i] ) then
if faith[i] <= 2
or ( faith[i] = 3 and chi[ faithpos ] = E(3) * chi[1] )
or ( faith[i] = 4 and chi[ faithpos ] = E(4) * chi[1] )
or ( faith[i] = 6 and chi[ faithpos ] = E(6)^5 * chi[1] )
or ( faith[i] = 12 and chi[ faithpos ] = E(12)^7 * chi[1] ) then
Add( faithchars[i], chi );
break;
fi;
fi;
od;
od;
# Remove characters obtained by multiplication with linear characters
# of the factor group,
# and create the result info.
fus:= First( fusions, x -> ClassPositionsOfKernel( x.map ) = kernel ).map;
proj:= ProjectionMap( fus );
nccl:= Length( proj );
linear:= List( Filtered( faithchars[1], chi -> chi[1] = 1 ),
lambda -> lambda{ proj } );
projectives:= [];
info:= [ [ names[1], [] ] ];
for i in [ 2 .. Length( nsg ) ] do
new:= [];
gal:= [];
for chi in faithchars[i] do
rest:= chi{ proj };
if ForAll( linear, lambda -> not List( [ 1 .. nccl ],
j -> lambda[j] * rest[j] ) in new ) then
Add( new, rest );
if 2 < faith[i] then
partners:= GaloisPartnersOfIrreducibles( tbl, [ chi ], faith[i] );
if faith[i] <> 12 then
partners:= partners[1];
fi;
Append( gal, partners );
#T works for 12 ??
fi;
fi;
od;
info[ faith[i] ]:= [ names[i], gal ];
Add( projectives, rec( name:= names[i], chars:= new ) );
od;
if HasClassPermutation( tbl ) then
pi:= ClassPermutation( tbl );
ResetFilterObj( tbl, HasClassPermutation );
else
pi:= ();
fi;
SetConstructionInfoCharacterTable( tbl, [ "ConstructProj", info ] );
# Return the result.
return rec( tbl := tbl,
projectives := projectives,
perm := pi,
info := info );
end );
#############################################################################
##
#F ConstructDirectProduct( <tbl>, <factors>[, <permclasses>, <permchars>] )
##
InstallGlobalFunction( ConstructDirectProduct, function( arg )
local tbl, factors, t, i;
tbl:= arg[1];
factors:= arg[2];
t:= CallFuncList( CharacterTableFromLibrary, factors[1] );
for i in [ 2 .. Length( factors ) ] do
t:= CharacterTableDirectProduct( t,
CallFuncList( CharacterTableFromLibrary, factors[i] ) );
od;
if 2 < Length( arg ) then
t:= CharacterTableWithSortedClasses( t, arg[3] );
t:= CharacterTableWithSortedCharacters( t, arg[4] );
# We must keep the class permutation obtained this way
# since it is contained in the `ConstructionInfo' data,
# and hence Brauer tables derived from the factors will respect it.
fi;
TransferComponentsToLibraryTableRecord( t, tbl );
if 1 < Length( factors ) then
Append( tbl.ComputedClassFusions, ComputedClassFusions( t ) );
fi;
end );
#############################################################################
##
#F ConstructCentralProduct( <tbl>, <factors>, <Dclasses>
#F [, <permclasses>, <permchars>] )
##
InstallGlobalFunction( ConstructCentralProduct, function( arg )
local tbl, factors, Dclasses, t, i, perms;
tbl:= arg[1];
factors:= arg[2];
Dclasses:= arg[3];
t:= CallFuncList( CharacterTableFromLibrary, factors[1] );
for i in [ 2 .. Length( factors ) ] do
t:= CharacterTableDirectProduct( t,
CallFuncList( CharacterTableFromLibrary, factors[i] ) );
od;
t:= CharacterTableFactorGroup( t, Dclasses );
perms:= Filtered( arg, IsPerm );
if Length( perms ) = 2 then
t:= CharacterTableWithSortedClasses( t, perms[1] );
t:= CharacterTableWithSortedCharacters( t, perms[2] );
fi;
TransferComponentsToLibraryTableRecord( t, tbl );
end );
#T improve this function! (do not unpack the direct product!)
#############################################################################
##
#F ConstructSubdirect( <tbl>, <factors>, <choice> )
##
InstallGlobalFunction( ConstructSubdirect, function( tbl, factors, choice )
local t, i;
t:= CallFuncList( CharacterTableFromLibrary, factors[1] );
for i in [ 2 .. Length( factors ) ] do
t:= CharacterTableDirectProduct( t,
CallFuncList( CharacterTableFromLibrary, factors[i] ) );
od;
t:= CharacterTableOfNormalSubgroup( t, choice );
TransferComponentsToLibraryTableRecord( t, tbl );
end );
#############################################################################
##
#F ConstructWreathSymmetric( <tbl>, <subname>, <n>
#F [, <permclasses>, <permchars>] )
##
InstallGlobalFunction( ConstructWreathSymmetric, function( arg )
local tbl, sub, t;
tbl:= arg[1];
sub:= CallFuncList( CharacterTableFromLibrary, arg[2] );
t:= CharacterTableWreathSymmetric( sub, arg[3] );
if 3 < Length( arg ) then
t:= CharacterTableWithSortedClasses( t, arg[4] );
t:= CharacterTableWithSortedCharacters( t, arg[5] );
if not IsBound( tbl.ClassPermutation ) then
# Do *not* inherit the permutation from the construction!
tbl.ClassPermutation:= ();
fi;
fi;
TransferComponentsToLibraryTableRecord( t, tbl );
# if 1 < Length( factors ) then
# Append( tbl.ComputedClassFusions, ComputedClassFusions( t ) );
# fi;
end );
#############################################################################
##
#F ConstructIsoclinic( <tbl>, <factors>[, <nsg>[, <centre>]]
#F [, <permclasses>, <permchars>] )
#F ConstructIsoclinic( <tbl>, <factors>[, <arec>]
#F [, <permclasses>, <permchars>] )
##
InstallGlobalFunction( ConstructIsoclinic, function( arg )
local tbl, factors, t, i, perms;
tbl:= arg[1];
factors:= arg[2];
t:= CallFuncList( CharacterTableFromLibrary, factors[1] );
for i in [ 2 .. Length( factors ) ] do
t:= CharacterTableDirectProduct( t,
CallFuncList( CharacterTableFromLibrary, factors[i] ) );
od;
if Length( arg ) = 2 or not IsRecord( arg[3] ) then
t:= CallFuncList( CharacterTableIsoclinic,
Concatenation( [ t ],
Filtered( arg{ [ 3 .. Length( arg ) ] }, IsList ) ) );
else
t:= CharacterTableIsoclinic( t, arg[3] );
fi;
perms:= Filtered( arg, IsPerm );
if Length( perms ) = 2 then
t:= CharacterTableWithSortedClasses( t, perms[1] );
t:= CharacterTableWithSortedCharacters( t, perms[2] );
fi;
TransferComponentsToLibraryTableRecord( t, tbl );
end );
#############################################################################
##
## 4. Character Tables of Groups of Structure $2^2.G$
##
#############################################################################
##
#F PossibleCharacterTablesOfTypeV4G( <tblG>, <tbls2G>, <id>[, <fusions>] )
#F PossibleCharacterTablesOfTypeV4G( <tblG>, <tbl2G>, <aut>, <id> )
##
InstallGlobalFunction( PossibleCharacterTablesOfTypeV4G, function( arg )
local tblG, tbls2G, aut, identifier, tbls2GfustblG, tbl2G, tbl2GfustblG,
invfus, fus, pow2, 2pow2, i, lst, int, j, powermaps, primes, inv,
p, firstfus, testfus, oldfus, pos, classes, tblV4G, tblGprojtblV4G,
irr, ker, irr1, irr2, faith, sortedfaith, parafus, indet, pointer,
max, entry, descendants, result, choice, map, lirr1, lirr2, error,
split;
# Get and check the arguments.
if Length( arg ) = 3 and IsOrdinaryTable( arg[1] ) and IsList( arg[2] )
and IsString( arg[3] ) then
tblG := arg[1];
tbls2G := arg[2];
aut := fail;
identifier := arg[3];
# Get the three factor fusions.
tbls2GfustblG:= List( tbls2G, t -> GetFusionMap( t, tblG ) );
if fail in tbls2GfustblG then
Error( "the factor fusions <tbls2G> -> <tblG> must be stored" );
fi;
elif Length( arg ) = 4 and IsOrdinaryTable( arg[1] )
and IsList( arg[2] )
and IsString( arg[3] )
and IsList( arg[4] ) then
tblG := arg[1];
tbls2G := arg[2];
identifier := arg[3];
tbls2GfustblG := arg[4];
elif Length( arg ) = 4 and IsOrdinaryTable( arg[1] )
and IsOrdinaryTable( arg[2] )
and IsPerm( arg[3] )
and IsString( arg[4] ) then
tblG := arg[1];
tbl2G := arg[2];
aut := arg[3];
identifier := arg[4];
# Get the three factor fusions.
tbl2GfustblG:= GetFusionMap( tbl2G, tblG );
if tbl2GfustblG = fail then
Error( "the factor fusion to <tblG> must be stored" );
fi;
tbls2GfustblG:= List( [ 0 .. 2 ],
i -> OnTuples( tbl2GfustblG, aut^i ) );
tbls2G:= ListWithIdenticalEntries( 3, tbl2G );
else
Error( "usage: PossibleCharacterTablesOfTypeV4G(<tbl>,<tbls>,<id>),\n",
"PossibleCharacterTablesOfTypeV4G(<tbl>,<tbls>,<id>,<fusions>),\n",
"PossibleCharacterTablesOfTypeV4G(<tblG>,<tbl2G>,<aut>,<id>)" );
fi;
# Construct the classes of $2^2.G$, via the three factor fusions.
invfus:= List( tbls2GfustblG, InverseMap );
fus:= [ [], [], [] ];
pow2:= PowerMap( tblG, 2 );
2pow2:= List( tbls2G, t -> PowerMap( t, 2 ) );
for i in [ 1 .. NrConjugacyClasses( tblG ) ] do
# Deal with the preimages of class `i' in `tblG'.
lst:= Filtered( [ 1 .. 3 ], j -> IsList( invfus[j][i] ) );
int:= Difference( [ 1 .. 3 ], lst );
if Length( lst ) = 0 then
# no splitting
for j in [ 1 .. 3 ] do
fus[j][i]:= [ [ invfus[j][i] ] ];
od;
elif Length( lst ) = 1 then
# exactly one splitting in step 1, so the other two split in step 2.
fus[ lst[1] ][i]:= [ invfus[ lst[1] ][i] ];
fus[ int[1] ][i]:= [ invfus[ int[1] ]{ [ i, i ] } ];
fus[ int[2] ][i]:= [ invfus[ int[2] ]{ [ i, i ] } ];
elif Length( lst ) = 3 then
# splitting in all three cases, we have the problem of identifying!
# (the first two fusions can be chosen,
# the third leads in general to two possibilities)
fus[1][i]:= [ invfus[1][i]{ [ 1, 1, 2, 2 ] } ];
fus[2][i]:= [ invfus[2][i]{ [ 1, 2, 1, 2 ] } ];
fus[3][i]:= [ invfus[3][i]{ [ 1, 2, 2, 1 ] },
invfus[3][i]{ [ 2, 1, 1, 2 ] } ];
else
# The tables do not fit together (`lst' must have length 0, 1, or 3)
Info( InfoCharacterTable, 1,
"PossibleCharacterTablesOfTypeV4G: inconsistent splitting\n",
"#I of classes at position ", i, " in ", tblG );
return [];
fi;
od;
# Initialize power maps using the first table,
# and check the consistency with the power maps in the second table.
powermaps:= [];
primes:= PrimeDivisors( Size( tbls2G[1] ) );
fus[1]:= Concatenation( Concatenation( fus[1] ) );
inv:= InverseMap( fus[1] );
for p in primes do
powermaps[p]:= CompositionMaps( inv, CompositionMaps(
PowerMap( tbls2G[1], p ), fus[1] ) );
PowerMap( tbls2G[2], p );
PowerMap( tbls2G[3], p );
od;
fus[2]:= Concatenation( Concatenation( fus[2] ) );
if not TestConsistencyMaps( powermaps, fus[2],
ComputedPowerMaps( tbls2G[2] ) ) then
Info( InfoCharacterTable, 1,
"PossibleCharacterTablesOfTypeV4G: inconsistent power maps\n",
"#I of the first two factors" );
return [];
fi;
# Try to resolve ambiguities using the power maps in the third factor.
# For example, this check determines the class among the four preimages
# that is fixed under the order three automorphisms (if there is one)
# if the image in the factor group is a 2nd power of a fixed class.
#T Is this true?
# And the case 2 image/preimage of a fixed class in case 2 under an odd
# power map must be fixed.
firstfus:= Concatenation( List( fus[3], x -> x[1] ) );
testfus:= Parametrized( [ firstfus,
Concatenation( List( fus[3], x -> x[ Length( x ) ] ) ) ] );
oldfus:= List( testfus, ShallowCopy );
if not TestConsistencyMaps( powermaps, testfus,
ComputedPowerMaps( tbls2G[3] ) ) then
Info( InfoCharacterTable, 1,
"PossibleCharacterTablesOfTypeV4G: inconsistent power maps\n",
"#I of the first and third factor" );
return [];
fi;
for i in [ 1 .. Length( testfus ) ] do
if IsInt( testfus[i] ) and IsList( oldfus[i] ) then
pos:= PositionProperty( invfus[3],
x -> IsList( x ) and testfus[i] in x );
if Length( fus[3][ pos ] ) = 2 then
if testfus[i] = firstfus[i] then
fus[3][ pos ]:= [ fus[3][ pos ][1] ];
else
fus[3][ pos ]:= [ fus[3][ pos ][2] ];
fi;
fi;
fi;
od;
# Create the table head data of $2^2.G$, using the first fusion.
classes:= [];
inv:= CompositionMaps( inv, InverseMap( tbls2GfustblG[1] ) );
for i in [ 1 .. NrConjugacyClasses( tblG ) ] do
if IsInt( inv[i] ) then
Add( classes, 4 * SizesConjugacyClasses( tblG )[i] );
elif Length( inv[i] ) = 2 then
Append( classes, 2 * SizesConjugacyClasses( tblG ){ [ i, i ] } );
else
Append( classes, SizesConjugacyClasses( tblG ){ [ i, i, i, i ] } );
fi;
od;
tblV4G:= rec( Identifier:= identifier,
UnderlyingCharacteristic:= 0,
Size:= 2 * Size( tbls2G[1] ),
SizesConjugacyClasses:= classes,
ComputedPowerMaps:= powermaps,
);
# Construct the first two portions of irreducible characters of $2^2.G$.
tblGprojtblV4G:= ProjectionMap(
CompositionMaps( tbls2GfustblG[1], fus[1] ) );
irr:= List( Irr( tbls2G[1] ), chi -> chi{ fus[1] } );
ker:= ClassPositionsOfKernel( tbls2GfustblG[1] );
irr1:= Filtered( Irr( tbls2G[1] ),
chi -> chi[ ker[1] ] <> chi[ ker[2] ] );
ker:= ClassPositionsOfKernel( tbls2GfustblG[2] );
irr2:= Filtered( Irr( tbls2G[2] ),
chi -> chi[ ker[1] ] <> chi[ ker[2] ] );
Append( irr, List( irr2, chi -> chi{ fus[2] } ) );
# Take the third portion of irreducibles.
ker:= ClassPositionsOfKernel( tbls2GfustblG[3] );
faith:= Filtered( Irr( tbls2G[3] ),
chi -> chi[ ker[1] ] <> chi[ ker[2] ] );
# Sort them such that those which distinguish most of the
# possibilities come first.
sortedfaith:= ShallowCopy( faith );
parafus:= Parametrized( [
Concatenation( List( fus[3], x -> x[ 1 ] ) ),
Concatenation( List( fus[3], x -> x[ Length( x ) ] ) ) ] );
testfus:= CompositionMaps( parafus, tblGprojtblV4G );
indet:= List( faith,
chi -> Indeterminateness( CompositionMaps( chi, testfus ) ) );
SortParallel( - indet, sortedfaith );
sortedfaith:= sortedfaith{ Filtered( [ 1 .. Length( sortedfaith ) ],
i -> 1 < indet[i] ) };
# Loop over the possible fusions onto the third factor table.
# First filter out those for which the power maps are compatible.
pointer:= [];
max:= 0;
for i in [ 1 .. Length( fus[3] ) ] do
entry:= fus[3][i];
if Length( entry ) = 1 then
pointer[i]:= 0;
else
pointer[i]:= max + [ 1 .. Length( entry[1] ) ];
fi;
max:= max + Length( entry[1] );
od;
descendants:= function( parafus, pointer )
local result, pos, entry, parafus1, pointer1, i, pos2;
result:= [];
pos:= PositionProperty( pointer, IsList );
if pos = fail then
if TestConsistencyMaps( powermaps, parafus,
ComputedPowerMaps( tbls2G[3] ) ) then
result[1]:= parafus;
fi;
return result;
fi;
for entry in fus[3][ pos ] do
parafus1:= List( parafus, ShallowCopy );
parafus1{ pointer[ pos ] }:= entry;
if TestConsistencyMaps( powermaps, parafus1,
ComputedPowerMaps( tbls2G[3] ) ) then
pointer1:= ShallowCopy( pointer );
for i in [ 1 .. Length( pointer ) ] do
if IsList( pointer1[i] ) then
pos2:= PositionProperty( parafus1{ pointer1[i] }, IsInt );
if pos2 <> fail then
if parafus1[ pointer1[i][ pos2 ] ] = fus[3][i][1][ pos2 ] then
parafus1{ pointer1[i] }:= fus[3][i][1];
else
parafus1{ pointer1[i] }:= fus[3][i][2];
fi;
pointer1[i]:= 0;
fi;
fi;
od;
Append( result, descendants( parafus1, pointer1 ) );
fi;
od;
return result;
end;
result:= [];
for choice in descendants( parafus, pointer ) do
map:= CompositionMaps( fus[1], ProjectionMap( choice ) );
lirr1:= List( irr1, x -> CompositionMaps( x, map ) );
map:= CompositionMaps( fus[2], ProjectionMap( choice ) );
lirr2:= List( irr2, x -> CompositionMaps( x, map ) );
# Compute tensor products of characters in `tbls2G[1]' and `tblsG2[2]',
# in order to check the table; this takes place in `tbls2G[3]'.
error:= false;
for i in lirr1 do
if not ForAll( lirr2,
j -> NonnegIntScalarProducts( tbls2G[3], sortedfaith,
Tensored( [ i ], [ j ] )[1] ) ) then
error:= true;
break;
fi;
od;
if not error then
# Create the table object.
split:= ShallowCopy( tblV4G );
split.ComputedClassFusions:= [
rec( name:= Identifier( tbls2G[1] ), map:= fus[1],
specification:= "1" ),
rec( name:= Identifier( tbls2G[2] ), map:= fus[2],
specification:= "2" ),
rec( name:= Identifier( tbls2G[3] ), map:= choice,
specification:= "3" ) ];
split:= ConvertToLibraryCharacterTableNC( split );
SetIrr( split,
List( Concatenation( irr, List( faith, chi -> chi{ choice } ) ),
chi -> Character( split, chi ) ) );
if IsBound( tbl2G ) then
# case of one factor plus an automorphism of order three
SetConstructionInfoCharacterTable( split,
ConstructV4GInfo( split, [ 1 .. 4 ] ) );
else
# case of three factors
SetConstructionInfoCharacterTable( split,
[ "ConstructV4G", List( tbls2G, Identifier ) ] );
fi;
# Add the table to the result list.
SetInfoText( split,
"constructed using `PossibleCharacterTablesOfTypeV4G'" );
Add( result, split );
fi;
od;
return result;
end );
#############################################################################
##
#F BrauerTableOfTypeV4G( <ordtblV4G>, <modtbls2G> )
#F BrauerTableOfTypeV4G( <ordtblV4G>, <modtbl2G>, <aut> )
##
InstallGlobalFunction( BrauerTableOfTypeV4G, function( arg )
local ordtblV4G, modtbls2G, aut, p, modtblV4G, fus, irr, i, modfus, ker,
chars;
# Get the arguments.
ordtblV4G:= arg[1];
if Length( arg ) = 2 then
# three nonsisomorphic factors
modtbls2G:= arg[2];
else
# one factor and an automorphism of `ordtblV4G'
modtbls2G:= [ arg[2] ];
aut:= arg[3];
fi;
# Construct the table head of the Brauer table.
p:= UnderlyingCharacteristic( modtbls2G[1] );
modtblV4G:= CharacterTableRegular( ordtblV4G, p );
# Fetch the factor fusions and inflate the irreducible characters.
fus:= List( modtbls2G, x -> GetFusionMap( modtblV4G, x ) );
irr:= List( Irr( modtbls2G[1] ), x -> x{ fus[1] } );
if p <> 2 then
# (For `p = 2', we would run into an error in the `else' case.)
if Length( arg ) = 2 then
for i in [ 2 .. Length( modtbls2G ) ] do
Append( irr, Filtered( List( Irr( modtbls2G[i] ),
x -> x{ fus[i] } ),
x -> not x in irr ) );
od;
else
# Rewrite the permutation of the ordinary table to the Brauer table.
modfus:= GetFusionMap( modtblV4G, ordtblV4G );
aut:= PermList( CompositionMaps( InverseMap( modfus ),
OnTuples( modfus, aut ) ) );
# Compute a class of the result table that is in the kernel of the
# natural epimorphism to G but not in the kernel of the map to 2.G.
# We can take an image of the kernel of `fus[1]' under `aut'.
ker:= ClassPositionsOfKernel( fus[1] );
ker:= Difference( OnTuples( ker, aut ), ker )[1];
chars:= List( Filtered( irr, x -> x[1] <> x[ ker ] ),
x -> Permuted( x, aut ) );
Append( irr, chars );
Append( irr, List( chars, x -> Permuted( x, aut ) ) );
fi;
fi;
SetIrr( modtblV4G, List( irr, x -> Character( modtblV4G, x ) ) );
SetInfoText( modtblV4G, "constructed using `BrauerTableOfTypeV4G'" );
return modtblV4G;
end );
#############################################################################
##
#F ConstructV4G( <tbl>, <facttbl>, <aut> )
#F ConstructV4G( <tbl>, <facttbls> )
##
InstallGlobalFunction( ConstructV4G, function( arg )
local tbl, facttbls, aut, fus, i, ker, chars;
tbl:= arg[1];
if Length( arg ) = 2 then
facttbls:= arg[2];
else
facttbls:= [ arg[2] ];
aut:= arg[3];
fi;
fus:= List( facttbls, x -> First( tbl.ComputedClassFusions,
fus -> fus.name = x ).map );
facttbls:= List( facttbls, CharacterTableFromLibrary );
tbl.Irr:= List( Irr( facttbls[1] ),
x -> ValuesOfClassFunction( x ){ fus[1] } );
if Length( arg ) = 2 then
for i in [ 2 .. Length( facttbls ) ] do
Append( tbl.Irr, Filtered( List( Irr( facttbls[i] ),
x -> ValuesOfClassFunction( x ){ fus[i] } ),
x -> not x in tbl.Irr ) );
od;
else
# Compute a class of the result table that is in the kernel of the
# natural epimorphism to G but not in the kernel of the map to 2.G.
# We can take an image of the kernel of `fus[1]' under `aut'.
ker:= ClassPositionsOfKernel( fus[1] );
ker:= Difference( OnTuples( ker, aut ), ker )[1];
chars:= List( Filtered( tbl.Irr, x -> x[1] <> x[ ker ] ),
x -> Permuted( x, aut ) );
Append( tbl.Irr, chars );
Append( tbl.Irr, List( chars, x -> Permuted( x, aut ) ) );
fi;
end );
#############################################################################
##
#F ConstructV4GInfo( <tblV4G>, <kernel> )
##
## We want the permutation to be a table automorphism,
## because otherwise it is not clear that the induced action on p-regular
## classes will work.
## The permutation need not be unique.
## If there are several possibilities then we prefer one that keeps the
## ordering of the characters.
##
InstallGlobalFunction( ConstructV4GInfo, function( tblV4G, kernel )
local faith, portions, permuted, pi, fusion, cand, cand2;
faith:= Filtered( Irr( tblV4G ),
chi -> not IsSubset( ClassPositionsOfKernel( chi ), kernel ) );
portions:= List( Difference( kernel, [ 1 ] ),
i -> Filtered( faith,
chi -> i in ClassPositionsOfKernel( chi ) ) );
permuted:= List( portions{ [ 2, 3, 1 ] }, Set );
fusion:= First( ComputedClassFusions( tblV4G ),
x -> ClassPositionsOfKernel( x.map ) = kernel{ [ 1, 2 ] } );
cand:= Filtered( Elements( AutomorphismsOfTable( tblV4G ) ),
#T better avoid computing all elements
x -> Order( x ) = 3 and kernel[2]^x = kernel[3]
and ForAll( [ 1 .. 3 ],
i -> Set( List( portions[i],
l -> Permuted( l, x ) ) )
= permuted[i] ) );
if Length( cand ) <> 1 then
cand2:= Filtered( cand, x -> ForAll( [ 1 .. 3 ],
i -> List( portions[i], l -> Permuted( l, x ) )
= portions[ ( i mod 3 ) + 1 ] ) );
if not IsEmpty( cand2 ) then
cand:= cand2;
fi;
fi;
pi:= cand[1];
return [ "ConstructV4G", fusion.name, pi ];
end );
#############################################################################
##
#F ConstructGS3( <tbls3>, <tbl2>, <tbl3>, <ind2>, <ind3>, <ext>, <perm> )
##
InstallGlobalFunction( ConstructGS3,
function( tbls3, tbl2, tbl3, ind2, ind3, ext, perm )
local fus2, # fusion map `tbl2' in `tbls3'
fus3, # fusion map `tbl3' in `tbls3'
proj2, # projection $G.S3$ to $G.2$
pos, # position in `proj2'
proj2i, # inner part of projection $G.S3$ to $G.2$
proj2o, # outer part of projection $G.S3$ to $G.2$
proj3, # projection $G.S3$ to $G.3$
zeroon2, # zeros for part of $G.2 \setminus G$ in $G.S_3$
irr, # irreducible characters of `tbls3'
irr3, # irreducible characters of `tbl3'
irr2, # irreducible characters of `tbl2'
i, # loop over `ind2'
pair, # loop over `ind3' and `ext'
chi, # character
chii, # inner part of character
chio; # outer part of character
tbl2:= CharacterTableFromLibrary( tbl2 );
tbl3:= CharacterTableFromLibrary( tbl3 );
fus2:= First( ComputedClassFusions( tbl2 ),
fus -> fus.name = tbls3.Identifier ).map;
fus3:= First( ComputedClassFusions( tbl3 ),
fus -> fus.name = tbls3.Identifier ).map;
proj2:= ProjectionMap( fus2 );
pos:= First( [ 1 .. Length( proj2 ) ], x -> not IsBound( proj2[x] ) );
proj2i:= proj2{ [ 1 .. pos-1 ] };
pos:= First( [ pos .. Length( proj2 ) ], x -> IsBound( proj2[x] ) );
proj2o:= proj2{ [ pos .. Length( proj2 ) ] };
proj3:= ProjectionMap( fus3 );
zeroon2:= Zero( Difference( [ 1 .. Length( tbls3.SizesCentralizers ) ],
fus3 ) );
# Induce the characters given by `ind2' from `tbl2'.
irr:= InducedLibraryCharacters( tbl2, tbls3, Irr( tbl2 ){ ind2 }, fus2 );
# Induce the characters given by `ind3' from `tbl3'.
irr3:= List( Irr( tbl3 ), ValuesOfClassFunction );
Append( irr, List( ind3,
pair -> Concatenation( Sum( irr3{ pair } ){ proj3 }, zeroon2 ) ) );
# Put the extensions from `tbl' together.
irr2:= List( Irr( tbl2 ), ValuesOfClassFunction );
for pair in ext do
chii:= irr3[ pair[1] ]{ proj3 };
chio:= irr2[ pair[2] ]{ proj2o };
Add( irr, Concatenation( chii, chio ) );
Add( irr, Concatenation( chii, -chio ) );
od;
# Permute the characters with `perm'.
irr:= Permuted( irr, perm );
# Store the irreducibles.
tbls3.Irr:= irr;
end );
#############################################################################
##
#F ConstructGS3Info( <tbl2>, <tbl3>, <tbls3> )
##
InstallGlobalFunction( ConstructGS3Info, function( tbl2, tbl3, tbls3 )
local irr2, # irreducible characters of `tbl2'
irr3, # irreducible characters of `tbl3'
irrs3, # irreducible characters of `tbls3'
ind, # list of induced characters
ind2, # positions of irreducible characters of `tbl2'
# inducing irreducibly to `tbls3'
oldind, # auxiliary list
i, # loop over positions in `ind'
pos, # position in `ind' or `irr3'
ind3, # positions of pairs of irreducible characters of
# `tbl3' inducing irreducibly to `tbls3'
ext, # list of pairs corresponding to irreducibles of
# `tbls3' that are extensions from `tbl2' and `tbl3'
chi, # loop over `irrs3'
pos2, # position in `irr2'
rest, # one restricted character
irr,
fus3,
proj3,
zeroon2,
proj2,
proj2o,
pair,
chii,
chio,
perm;
irr2 := Irr( tbl2 );
irr3 := Irr( tbl3 );
irrs3 := Irr( tbls3 );
ind:= Induced( tbl2, tbls3, Irr( tbl2 ) );
ind2:= Filtered( [ 1 .. Length( ind ) ],
i -> Position( ind, ind[i] ) = i and ind[i] in irrs3 );
oldind:= ind;
ind:= Induced( tbl3, tbls3, Irr( tbl3 ) );
ind3:= [];
for i in [ 1 .. Length( ind ) ] do
if ind[i] in irrs3 and not ind[i] in oldind then
pos:= Position( ind, ind[i] );
if pos <> i then
Add( ind3, [ pos, i ] );
fi;
fi;
od;
ext:= [];
for chi in irrs3 do
rest:= Restricted( tbls3, tbl3, [ chi ] )[1];
pos:= Position( irr3, rest );
if pos <> fail and ForAll( ext, x -> x[1] <> pos ) then
rest:= Restricted( tbls3, tbl2, [ chi ] )[1];
pos2:= Position( irr2, rest );
if pos2 <> fail then
Add( ext, [ pos, pos2 ] );
fi;
fi;
od;
# Put the characters together, for computing the necessary permutation.
# (Use the same code as in `ConstructGS3'.
irr:= Induced( tbl2, tbls3, Irr( tbl2 ){ind2} );
fus3:= GetFusionMap( tbl3, tbls3 );
proj3:= ProjectionMap( fus3 );
zeroon2:= Zero( Difference( [ 1 .. NrConjugacyClasses( tbls3 ) ],
fus3 ) );
proj2:= ProjectionMap( GetFusionMap( tbl2, tbls3 ) );
pos:= First( [ 1 .. Length( proj2 ) ], x -> not IsBound( proj2[x] ) );
pos:= First( [ pos .. Length( proj2 ) ], x -> IsBound( proj2[x] ) );
proj2o:= proj2{ [ pos .. Length( proj2 ) ] };
Append( irr, List( ind3,
pair -> Concatenation( Sum( irr3{ pair } ){ proj3 }, zeroon2 ) ) );
for pair in ext do
chii := irr3[pair[1]]{proj3};
chio := irr2[pair[2]]{proj2o};
Add( irr, Concatenation( chii, chio ) );
Add( irr, Concatenation( chii, - chio ) );
od;
perm := Sortex( irr ) / Sortex( ShallowCopy( Irr( tbls3 ) ) );
# Return the result.
return rec( ind2:= ind2, ind3:= ind3, ext:= ext, perm := perm,
list:= [ "ConstructGS3",
Identifier( tbl2 ), Identifier( tbl3 ),
ind2, ind3, ext, perm ] );
end );
#############################################################################
##
#F ConstructPermuted( <tbl>, <libnam>[, <permclasses>, <permchars>] )
##
InstallGlobalFunction( ConstructPermuted,
function( arg )
local tbl, t, hasFusionToTom, hasMaxes;
tbl:= arg[1];
# There may be fusions into `tbl', so we must *NOT* transfer the
# class permutation from `t' in `TransferComponentsToLibraryTableRecord'.
if not IsBound( tbl.ClassPermutation ) then
tbl.ClassPermutation:= ();
fi;
# Get the permuted table.
t:= CallFuncList( CharacterTableFromLibrary, arg[2] );
if 2 < Length( arg ) and not IsOne( arg[3] ) then
t:= CharacterTableWithSortedClasses( t, arg[3] );
fi;
if 3 < Length( arg ) and not IsOne( arg[4] ) then
t:= CharacterTableWithSortedCharacters( t, arg[4] );
fi;
hasFusionToTom:= IsBound( tbl.FusionToTom );
hasMaxes:= IsBound( tbl.Maxes );
# Store the components in `tbl'.
TransferComponentsToLibraryTableRecord( t, tbl );
# Remove attribute values that may contradict the compatibility
# between several tables.
if not hasFusionToTom then
Unbind( tbl.FusionToTom ); # a permuted fusion would be needed
fi;
if not hasMaxes then
Unbind( tbl.Maxes ); # no fusions stored into the permuted table
fi;
end );
#############################################################################
##
#F ConstructAdjusted( <tbl>, <libnam>, <pairs>
#F [, <permclasses>, <permchars>] )
##
InstallGlobalFunction( ConstructAdjusted,
function( tbl, libnam, pairs, args... )
local t, pair;
# Get the permuted library table.
t:= CallFuncList( CharacterTableFromLibrary, libnam );
if 0 < Length( args ) and not IsOne( args[1] ) then
t:= CharacterTableWithSortedClasses( t, args[1] );
fi;
if 1 < Length( args ) and not IsOne( args[2] ) then
t:= CharacterTableWithSortedCharacters( t, args[2] );
fi;
# Set the components that shall be adjusted.
# (Make sure that the values of mutable attributes are mutable.)
for pair in pairs do
tbl.( pair[1] ):= ShallowCopy( pair[2] );
od;
# Transfer not adjusted defining components.
# We are *NOT* allowed to call `TransferComponentsToLibraryTableRecord',
# because `tbl' is expected to be *NOT* equivalent to `t'.
if not IsBound( tbl.SizesCentralizers ) then
tbl.SizesCentralizers:= SizesCentralizers( t );
fi;
if not IsBound( tbl.ComputedPowerMaps ) then
# The lists of power maps for the two tables should be independent.
tbl.ComputedPowerMaps:= ShallowCopy( ComputedPowerMaps( t ) );
fi;
if not IsBound( tbl.Irr ) then
tbl.Irr:= List( Irr( t ), ValuesOfClassFunction );
fi;
end );
#############################################################################
##
#F ConstructFactor( <tbl>, <libnam>, <kernel> )
##
InstallGlobalFunction( ConstructFactor, function( tbl, libnam, kernel )
local t;
# Construct the required table of the factor group.
t:= CharacterTableFactorGroup( CallFuncList( CharacterTableFromLibrary,
libnam ),
kernel );
# Store the components in `tbl'.
TransferComponentsToLibraryTableRecord( t, tbl );
end );
#############################################################################
##
#F ConstructClifford( <tbl>, <cliffordtable> )
##
InstallGlobalFunction( ConstructClifford, function( tbl, cliffordtable )
local i, j, n,
AnzTi,
tables,
ct, # list of lists of relevant characters,
# one for each inertia factor group
clmexp,
clmat,
matsize,
grps,
newct, # the list of irreducibles of `tbl'
rowct, # actual row
colct, # actual column
eintr,
chars,
linear,
chi, # loop over a character list
lin,
new;
# Get the character tables of the inertia groups,
# and store the relevant list of characters.
tables:= cliffordtable[2];
AnzTi:= Length( tables );
ct:= [];
for i in [ 1 .. AnzTi ] do
if tables[i][1] = "projectives" then
eintr:= CharacterTableFromLibrary( tables[i][2] );
else
eintr:= CallFuncList( CharacterTableFromLibrary, tables[i] );
fi;
if eintr = fail then
Error( "table of inertia factor group `", tables[i],
"' not in the library" );
fi;
if tables[i][1] = "projectives" then
# We must multiply the stored projectives with all linear characters
# of the factor group in order to get the full list.
chars:= First( ProjectivesInfo( eintr ),
x -> x.name = tables[i][3] ).chars;
ct[i]:= [];
linear:= List( Filtered( Irr( eintr ), x -> x[1] = 1 ),
ValuesOfClassFunction );
n:= NrConjugacyClasses( eintr );
for chi in chars do
for lin in linear do
new:= List( [ 1 .. n ], x -> chi[x] * lin[x] );
if not new in ct[i] then
Add( ct[i], new );
fi;
od;
od;
else
ct[i]:= List( Irr( eintr ), ValuesOfClassFunction );
fi;
# tables[i]:= eintr;
od;
# Construct the matrix of irreducible characters.
newct := List( tbl.SizesCentralizers, x -> [] );
colct := 0;
for i in cliffordtable[3] do
# Get the necessary components of the `i'-th Clifford matrix,
# and multiply it with the character tables of inertia factor groups.
clmexp := UnpackedCll( i );
clmat := clmexp.mat;
matsize := Length( clmat );
grps := clmexp.inertiagrps;
# Loop over the columns of the matrix.
for n in [ 1 .. matsize ] do
rowct := 0;
colct := colct + 1;
# Loop over the inertia factor groups.
for j in [ 1 .. AnzTi ] do
for chi in ct[j] do
rowct:= rowct + 1;
newct[rowct][colct]:= Sum( Filtered( [ 1 .. matsize ],
r -> grps[r] = j ),
#T this value is indep. of chi!
x -> clmat[x][n] * chi[ clmexp.fusionclasses[x] ]);
#T Eventually it should be possible to handle tables where not all
#T classes belonging to a Clifford matrix are expected to be
#T subsequent ...
#T (add an indirection by the fusion)
od;
od;
od;
od;
tbl.Irr := newct;
end );
#############################################################################
##
#F IBrOfExtensionBySingularAutomorphism( <modtbl>, <perm> )
#F IBrOfExtensionBySingularAutomorphism( <modtbl>, <orbits> )
#F IBrOfExtensionBySingularAutomorphism( <modtbl>, <ordexttbl> )
##
InstallGlobalFunction( IBrOfExtensionBySingularAutomorphism,
function( modtbl, ordexttbl )
local ordtbl, p, fus, orbits, extirr, nccl, chi, comp, sum, i;
ordtbl:= OrdinaryCharacterTable( modtbl );
p:= UnderlyingCharacteristic( modtbl );
# Get the fusion into `modexttbl' (without constructing `modexttbl').
if IsOrdinaryTable( ordexttbl ) then
# Check the consistency of the arguments.
if Size( ordexttbl ) <> p * Size( ordtbl ) then
Error( "<ordexttbl> is not an index <p> extension of <ordtbl>" );
fi;
# Compute the action.
fus:= GetFusionMap( ordtbl, ordexttbl );
if fus = fail then
Error( "fusion from <ordtbl> to <ordexttbl> is not stored" );
elif not Set( fus ) in ClassPositionsOfNormalSubgroups( ordexttbl ) then
Error( "<ordtbl> is not normal in <ordexttbl>" );
fi;
fus:= CompositionMaps( fus, GetFusionMap( modtbl, ordtbl ) );
# Compute the orbits of the automorphism on the classes of `modtbl'.
orbits:= Compacted( InverseMap( fus ) );
elif IsPerm( ordexttbl ) then
orbits:= Set( List( Orbits( Group( ordexttbl ),
[ 1 .. NrConjugacyClasses( modtbl ) ] ), Set ) );
elif IsList( ordexttbl ) then
orbits:= ordexttbl;
else
Error( "<ordexttbl> must be a char. table, a permutation, or a list" );
fi;
# Compute the irreducibles.
extirr:= [];
nccl:= Length( orbits );
for chi in Irr( modtbl ) do
comp:= CompositionMaps( chi, orbits );
if ForAny( comp, IsList ) then
# The character is not invariant, so it does not extend.
sum:= [];
for i in [ 1 .. nccl ] do
if IsList( comp[i] ) then
sum[i]:= Sum( chi{ orbits[i] }, 0 );
else
sum[i]:= p * comp[i];
fi;
od;
if not sum in extirr then
Add( extirr, sum );
fi;
else
# The character is invariant, so it extends uniquely.
Add( extirr, comp );
fi;
od;
# Return the result;
return extirr;
end );
#############################################################################
##
## 5. Character Tables of Subdirect Products of Index Two
##
## Besides the documented function
## `CharacterTableOfIndexTwoSubdirectProduct',
## we need the utilities `IrreducibleCharactersOfIndexTwoSubdirectProduct'
## and `ClassFusionsForIndexTwoSubdirectProduct'.
##
#############################################################################
##
#F IrreducibleCharactersOfIndexTwoSubdirectProduct( <irrH1xH2>, <irrG1xG2>,
#F <H1xH2fusG>, <GfusG1xG2> )
##
## We do not want to use the table head of the subdirect product because
## this function is also called by `ConstructIndexTwoSubdirectProduct',
## and there just a record is available from which the table is computed
## later.
##
BindGlobal( "IrreducibleCharactersOfIndexTwoSubdirectProduct",
function( irrH1xH2, irrG1xG2, H1xH2fusG, GfusG1xG2 )
local H1xH2fusG1xG2, restpos, i, rest, pos, irr, zero, proj1, perm,
proj2, chi, ind, j;
H1xH2fusG1xG2:= CompositionMaps( GfusG1xG2, H1xH2fusG );
# Compute which irreducibles of H1xH2 extend to G1xG2.
restpos:= List( irrH1xH2, x -> [] );
for i in [ 1 .. Length( irrG1xG2 ) ] do
rest:= ValuesOfClassFunction( irrG1xG2[i] ){ H1xH2fusG1xG2 };
pos:= Position( irrH1xH2, rest );
if pos <> fail then
Add( restpos[ pos ], i );
fi;
od;
irr:= [];
zero:= 0 * GfusG1xG2;
proj1:= ProjectionMap( H1xH2fusG );
perm:= Product( List( Filtered( InverseMap( H1xH2fusG ), IsList ),
l -> ( l[1], l[2] ) ), () );
proj2:= [];
for i in [ 1 .. Length( proj1 ) ] do
if IsBound( proj1[i] ) then
proj2[i]:= proj1[i]^perm;
fi;
od;
for i in [ 1 .. Length( irrH1xH2 ) ] do
if not IsEmpty( restpos[i] ) then
# The i-th irreducible of H1xH2 extends to G1xG2.
# Restrict these extensions to G.
Append( irr, DuplicateFreeList( List( irrG1xG2{ restpos[i] },
chi -> chi{ GfusG1xG2 } ) ) );
else
# The i-th irreducible character of H1xH2 has inertia subgroup one of
# H1xG2 or G1xH2, so it induces irreducibly to G.
# Compute the induced character (without using the table head).
chi:= irrH1xH2[i];
# The curly bracket operator works only for dense sublists.
# ind:= ShallowCopy( zero ) + chi{ proj1 } + chi{ proj2 };
ind:= ShallowCopy( zero );
for j in [ 1 .. Length( proj1 ) ] do
if IsBound( proj1[j] ) then
ind[j]:= ind[j] + chi[ proj1[j] ];
fi;
od;
for j in [ 1 .. Length( proj2 ) ] do
if IsBound( proj2[j] ) then
ind[j]:= ind[j] + chi[ proj2[j] ];
fi;
od;
if not ind in irr then
Add( irr, ind );
fi;
fi;
od;
return irr;
end );
#############################################################################
##
#F ClassFusionsForIndexTwoSubdirectProduct( <tblH1>, <tblG1>, <tblH2>,
#F <tblG2> )
##
## It is assumed that all tables are either ordinary tables or Brauer tables
## for the same characteristic.
##
## Note that the components `GfusG1xG2', `Gclasses', `Gorders' refer only to
## the classes inside the normal subgroup `<tblH1> * <tblH2>'.
##
DeclareGlobalFunction( "ClassFusionsForIndexTwoSubdirectProduct" );
InstallGlobalFunction( ClassFusionsForIndexTwoSubdirectProduct,
function( tblH1, tblG1, tblH2, tblG2 )
local p, H1classes, H2classes, H1orders, H2orders, H1fusG1, H2fusG2,
inv1, inv2, ncclH2, ncclG2, H1xH2fusG, GfusG1xG2,
Gclasses, Gorders, i1, i2, posG1xG2, len, pos,
ordH1, ordG1, ordH2, ordG2, info, modGfusordG, modfus2,
modG1xG2fusordG1xG2, modH1xH2fusordH1xH2;
p:= UnderlyingCharacteristic( tblH1 );
if p = 0 then
H1classes:= SizesConjugacyClasses( tblH1 );
H2classes:= SizesConjugacyClasses( tblH2 );
H1orders:= OrdersClassRepresentatives( tblH1 );
H2orders:= OrdersClassRepresentatives( tblH2 );
H1fusG1:= GetFusionMap( tblH1, tblG1 );
if H1fusG1 = fail then
H1fusG1:= RepresentativesFusions( tblH1,
PossibleClassFusions( tblH1, tblG1 ), tblG1 );
if Length( H1fusG1 ) <> 1 then
Error( "fusion <tblH1> to <tblG1> is not determined" );
fi;
fi;
H2fusG2:= GetFusionMap( tblH2, tblG2 );
if H2fusG2 = fail then
H2fusG2:= RepresentativesFusions( tblH2,
PossibleClassFusions( tblH2, tblG2 ), tblG2 );
if Length( H2fusG2 ) <> 1 then
Error( "fusion <tblH2> to <tblG2> is not determined" );
fi;
fi;
inv1:= InverseMap( H1fusG1 );
inv2:= InverseMap( H2fusG2 );
ncclH2:= Length( H2classes );
ncclG2:= NrConjugacyClasses( tblG2 );
H1xH2fusG:= [];
GfusG1xG2:= [];
Gclasses:= [];
Gorders:= [];
for i1 in [ 1 .. Length( inv1 ) ] do
if IsBound( inv1[ i1 ] ) then
for i2 in [ 1 .. Length( inv2 ) ] do
if IsBound( inv2[ i2 ] ) then
posG1xG2:= ( i1 - 1 ) * ncclG2 + i2;
if IsInt( inv1[ i1 ] ) then
if IsInt( inv2[ i2 ] ) then
# no fusion
len:= Length( GfusG1xG2 ) + 1;
H1xH2fusG[ ( inv1[ i1 ] - 1 ) * ncclH2 + inv2[ i2 ] ]:= len;
GfusG1xG2[ len ]:= posG1xG2;
Gclasses[ len ]:= H1classes[ inv1[ i1 ] ]
* H2classes[ inv2[ i2 ] ];
Gorders[ len ]:= LcmInt( H1orders[ inv1[ i1 ] ],
H2orders[ inv2[ i2 ] ] );
else
# fusion from H2 to G2
len:= Length( GfusG1xG2 ) + 1;
for pos in inv2[ i2 ] do
H1xH2fusG[ ( inv1[ i1 ] - 1 ) * ncclH2 + pos ]:= len;
od;
GfusG1xG2[ len ]:= posG1xG2;
Gclasses[ len ]:= 2 * H1classes[ inv1[ i1 ] ]
* H2classes[ inv2[ i2 ][1] ];
Gorders[ len ]:= LcmInt( H1orders[ inv1[ i1 ] ],
H2orders[ inv2[ i2 ][1] ] );
fi;
elif IsInt( inv2[ i2 ] ) then
# fusion from H1 to G1
len:= Length( GfusG1xG2 ) + 1;
for pos in inv1[ i1 ] do
H1xH2fusG[ ( pos - 1 ) * ncclH2 + inv2[ i2 ] ]:= len;
od;
GfusG1xG2[ len ]:= posG1xG2;
Gclasses[ len ]:= 2 * H1classes[ inv1[ i1 ][1] ]
* H2classes[ inv2[ i2 ] ];
Gorders[ len ]:= LcmInt( H1orders[ inv1[ i1 ][1] ],
H2orders[ inv2[ i2 ] ] );
else
# fusion in both factors (get two classes)
len:= Length( GfusG1xG2 ) + 1;
H1xH2fusG[ ( inv1[ i1 ][1]-1 ) * ncclH2 + inv2[i2][1] ]:= len;
H1xH2fusG[ ( inv1[ i1 ][2]-1 ) * ncclH2 + inv2[i2][2] ]:= len;
GfusG1xG2[ len ]:= posG1xG2;
Gclasses[ len ]:= 2 * H1classes[ inv1[ i1 ][1] ]
* H2classes[ inv2[ i2 ][1] ];
Gorders[ len ]:= LcmInt( H1orders[ inv1[ i1 ][1] ],
H2orders[ inv2[ i2 ][1] ] );
H1xH2fusG[ ( inv1[i1][1]-1 ) * ncclH2 + inv2[i2][2] ]:= len + 1;
H1xH2fusG[ ( inv1[i1][2]-1 ) * ncclH2 + inv2[i2][1] ]:= len + 1;
GfusG1xG2[ len + 1 ]:= posG1xG2;
Gclasses[ len + 1 ]:= Gclasses[ len ];
Gorders[ len + 1 ]:= Gorders[ len ];
fi;
fi;
od;
fi;
od;
else
ordH1:= OrdinaryCharacterTable( tblH1 );
ordG1:= OrdinaryCharacterTable( tblG1 );
ordH2:= OrdinaryCharacterTable( tblH2 );
ordG2:= OrdinaryCharacterTable( tblG2 );
# Compute the maps for the underlying ordinary tables.
info:= ClassFusionsForIndexTwoSubdirectProduct( ordH1, ordG1,
ordH2, ordG2 );
# Compute the embeddings of `p'-regular classes of G, H1xH2, G1xG2,
# without actually constructing these tables.
modGfusordG:= Filtered( [ 1 .. Length( info.Gorders ) ],
i -> info.Gorders[i] mod p <> 0 );
modfus2:= GetFusionMap( tblG2, ordG2 );
modG1xG2fusordG1xG2:= Concatenation(
List( GetFusionMap( tblG1, ordG1 ),
i -> modfus2 + ( i - 1 ) * NrConjugacyClasses( ordG2 ) ) );
modfus2:= GetFusionMap( tblH2, ordH2 );
modH1xH2fusordH1xH2:= Concatenation(
List( GetFusionMap( tblH1, ordH1 ),
i -> modfus2 + ( i - 1 ) * NrConjugacyClasses( ordH2 ) ) );
# Compute the maps for the Brauer tables.
H1xH2fusG:= CompositionMaps( InverseMap( modGfusordG ),
CompositionMaps( info.H1xH2fusG, modH1xH2fusordH1xH2 ) );
GfusG1xG2:= CompositionMaps( InverseMap( modG1xG2fusordG1xG2 ),
CompositionMaps( info.GfusG1xG2, modGfusordG ) );
Gclasses:= info.Gclasses{ modGfusordG };
Gorders:= info.Gorders{ modGfusordG };
fi;
return rec( H1xH2fusG:= H1xH2fusG,
GfusG1xG2:= GfusG1xG2,
Gclasses:= Gclasses,
Gorders:= Gorders,
);
end );
#############################################################################
##
#F CharacterTableOfIndexTwoSubdirectProduct( <tblH1>, <tblG1>,
#F <tblH2>, <tblG2>, <identifier> )
##
InstallGlobalFunction( CharacterTableOfIndexTwoSubdirectProduct,
function( tblH1, tblG1, tblH2, tblG2, identifier )
local char, ordtblG, permcols, info, H1fusG1, H2fusG2, H1xH2fusG,
GfusG1xG2, Gclasses, H1xH2, G1xG2, G1fusG1xG2, G2fusG1xG2,
H1fusG1xG2, H2fusG1xG2, nsg, outer, tblG, powermap, p, pow, i, j,
irrH1xH2, irrG1xG2, fus, result, H1fusH1xH2, H2fusH1xH2;
# Fetch the underlying characteristic, and check the arguments.
char:= UnderlyingCharacteristic( tblH1 );
if ForAny( [ tblG1, tblH2, tblG2 ],
t -> UnderlyingCharacteristic( t ) <> char ) then
Info( InfoCharacterTable, 1,
"CharacterTableOfIndexTwoSubdirectProduct:\n",
"#I UnderlyingCharacteristic values of input tables differ" );
return fail;
fi;
if char = 0 then
if not IsString( identifier ) then
Info( InfoCharacterTable, 1,
"CharacterTableOfIndexTwoSubdirectProduct:\n",
"#I <identifier> must be a string" );
return fail;
fi;
elif IsOrdinaryTable( identifier ) then
ordtblG:= identifier;
permcols:= ();
elif IsList( identifier ) and Length( identifier ) = 2
and IsOrdinaryTable( identifier[1] ) then
ordtblG:= identifier[1];
permcols:= identifier[2];
else
Info( InfoCharacterTable, 1,
"CharacterTableOfIndexTwoSubdirectProduct:\n",
"#I <identifier> must be the ordinary table of the result" );
return fail;
fi;
# Initialize auxiliary tables and fusions.
info:= ClassFusionsForIndexTwoSubdirectProduct( tblH1, tblG1, tblH2,
tblG2 );
H1fusG1:= GetFusionMap( tblH1, tblG1 );
H2fusG2:= GetFusionMap( tblH2, tblG2 );
H1xH2fusG:= info.H1xH2fusG;
GfusG1xG2:= info.GfusG1xG2;
if char = 0 then
Gclasses:= info.Gclasses;
# Compute the outer classes of G.
# For that, determine the unique index two subgroup
# that contains H1 and H2 but none of G1, G2.
H1xH2:= CharacterTableDirectProduct( tblH1, tblH2 );
G1xG2:= CharacterTableDirectProduct( tblG1, tblG2 );
if Identifier( tblG1 ) = Identifier( tblG2 ) then
G1fusG1xG2:= GetFusionMap( tblG1, G1xG2, "1" );
G2fusG1xG2:= GetFusionMap( tblG2, G1xG2, "2" );
else
G1fusG1xG2:= GetFusionMap( tblG1, G1xG2 );
G2fusG1xG2:= GetFusionMap( tblG2, G1xG2 );
fi;
H1fusG1xG2:= CompositionMaps( G1fusG1xG2, H1fusG1 );
H2fusG1xG2:= CompositionMaps( G2fusG1xG2, H2fusG2 );
nsg:= Filtered( ClassPositionsOfNormalSubgroups( G1xG2 ),
x -> Sum( SizesConjugacyClasses( G1xG2 ){ x } )
= Size( G1xG2 ) / 2
and not IsSubset( x, GetFusionMap( tblG1, G1xG2 ) )
and not IsSubset( x, GetFusionMap( tblG2, G1xG2 ) )
and IsSubset( x, H1fusG1xG2 )
and IsSubset( x, H2fusG1xG2 ) );
outer:= Difference( nsg[1],
ClassPositionsOfNormalClosure( G1xG2,
Union( H1fusG1xG2, H2fusG1xG2 ) ) );
Append( GfusG1xG2, outer );
Append( Gclasses, SizesConjugacyClasses( G1xG2 ){ outer } );
# Initialize the record for the character table `tblG'.
tblG:= rec(
UnderlyingCharacteristic := 0,
Identifier := identifier,
Size := 2 * Size( H1xH2 ),
SizesConjugacyClasses := Gclasses,
OrdersClassRepresentatives :=
OrdersClassRepresentatives( G1xG2 ){ GfusG1xG2 },
);
tblG.SizesCentralizers:= List( Gclasses, x -> tblG.Size / x );
# Convert the record to a table object.
ConvertToLibraryCharacterTableNC( tblG );
# Put the power maps together.
powermap:= ComputedPowerMaps( tblG );
for p in PrimeDivisors( Size( tblG ) ) do
pow:= InitPowerMap( tblG, p );
TransferDiagram( pow, GfusG1xG2, PowerMap( G1xG2, p ) );
TransferDiagram( PowerMap( H1xH2, p ), H1xH2fusG, pow );
powermap[p]:= pow;
Assert( 1, ForAll( pow, IsInt ),
Concatenation( Ordinal( p ),
" power map not uniquely determined" ) );
od;
# Store the factor fusions.
# (Note that the containment of classes modulo H1 and H2 can be decided
# in the bigger group G1xG2.
StoreFusion( tblG, CompositionMaps( GetFusionMap( G1xG2, tblG1 ),
GfusG1xG2 ),
tblG1 );
StoreFusion( tblG, CompositionMaps( GetFusionMap( G1xG2, tblG2 ),
GfusG1xG2 ),
tblG2 );
irrH1xH2:= Irr( H1xH2 );
irrG1xG2:= Irr( G1xG2 );
# Set the construction info,
# in order to enable the computation of Brauer tables.
SetConstructionInfoCharacterTable( tblG,
[ "ConstructIndexTwoSubdirectProduct",
Identifier( tblH1 ), Identifier( tblG1 ),
Identifier( tblH2 ), Identifier( tblG2 ),
outer, (), () ] );
else
# The table head is derived from the known ordinary table.
# All we need to construct are the irreducibles.
tblG:= CharacterTableRegular( ordtblG, char );
# Rewrite the class permutation to the p-regular classes.
fus:= GetFusionMap( tblG, ordtblG );
permcols:= SortingPerm( OnTuples( fus, permcols^-1 ) )^-1;
# Adjust the fusions to the sorted table head.
H1xH2fusG:= OnTuples( H1xH2fusG, permcols );
outer:= [];
for i in [ 1 .. NrConjugacyClasses( tblG1 ) ] do
for j in [ 1 .. NrConjugacyClasses( tblG2 ) ] do
if not ( i in H1fusG1 or j in H2fusG2 ) then
Add( outer, j + ( i - 1 ) * NrConjugacyClasses( tblG2 ) );
fi;
od;
od;
GfusG1xG2:= Permuted( Concatenation( GfusG1xG2, outer ), permcols );
# Form the irreducibles of the subgroup and the supergroup.
irrH1xH2:= KroneckerProduct( Irr( tblH1 ), Irr( tblH2 ) );
irrG1xG2:= KroneckerProduct( Irr( tblG1 ), Irr( tblG2 ) );
fi;
SetInfoText( tblG,
"constructed using `CharacterTableOfIndexTwoSubdirectProduct'" );
# Compute the irreducibles.
SetIrr( tblG, List( IrreducibleCharactersOfIndexTwoSubdirectProduct(
irrH1xH2, irrG1xG2, H1xH2fusG, GfusG1xG2 ),
chi -> Character( tblG, chi ) ) );
# Return the result.
result:= rec( table:= tblG );
if char = 0 then
if Identifier( tblH1 ) = Identifier( tblH2 ) then
H1fusH1xH2:= GetFusionMap( tblH1, H1xH2, "1" );
H2fusH1xH2:= GetFusionMap( tblH2, H1xH2, "2" );
else
H1fusH1xH2:= GetFusionMap( tblH1, H1xH2 );
H2fusH1xH2:= GetFusionMap( tblH2, H1xH2 );
fi;
result.H1fusG:= CompositionMaps( H1xH2fusG, H1fusH1xH2 );
result.H2fusG:= CompositionMaps( H1xH2fusG, H2fusH1xH2 );
result.outerfus:= outer;
fi;
return result;
end );
#############################################################################
##
#F ConstructIndexTwoSubdirectProduct( <tbl>, <tblH1>, <tblG1>, <tblH2>,
#F <tblG2>, <outerfus>, <permclasses>, <permchars> )
##
InstallGlobalFunction( ConstructIndexTwoSubdirectProduct,
function( tbl, tblH1, tblG1, tblH2, tblG2, outerfus,
permclasses, permchars )
local info, irreds;
tblH1:= CharacterTable( tblH1 );
tblG1:= CharacterTable( tblG1 );
tblH2:= CharacterTable( tblH2 );
tblG2:= CharacterTable( tblG2 );
info:= ClassFusionsForIndexTwoSubdirectProduct(
tblH1, tblG1, tblH2, tblG2 );
irreds:= IrreducibleCharactersOfIndexTwoSubdirectProduct(
KroneckerProduct( Irr( tblH1 ), Irr( tblH2 ) ),
KroneckerProduct( Irr( tblG1 ), Irr( tblG2 ) ),
info.H1xH2fusG, Concatenation( info.GfusG1xG2, outerfus ) );
tbl.Irr:= Permuted( List( irreds, chi -> Permuted( chi, permclasses ) ),
permchars );
end );
#############################################################################
##
#F ConstructIndexTwoSubdirectProductInfo( <tbl>[, <tblH1>, <tblG1>,
#F <tblH2>, <tblG2>] )
##
InstallGlobalFunction( ConstructIndexTwoSubdirectProductInfo,
function( arg )
local tbl, nsg, sizes, Gsize, result, i, j, k, r, fact1, fact2,
name1, sub1, name2, sub2, cand, tblH1, tblG1, tblH2, tblG2,
trans, rr;
# Get and check the arguments.
if Length( arg ) = 1 then
tbl:= arg[1];
# Check whether the table has the required structure.
if not IsEmpty( ClassPositionsOfDirectProductDecompositions( tbl ) ) then
# The structure is in fact better.
Info( InfoCharacterTable, 2,
"the table of `", Identifier( tbl ),
"' is a nontrivial direct product" );
return [];
fi;
nsg:= ClassPositionsOfNormalSubgroups( tbl );
sizes:= List( nsg, x -> Sum( SizesConjugacyClasses( tbl ){ x } ) );
Gsize:= Size( tbl ) / 2;
result:= [];
for i in Filtered( [ 1 .. Length( nsg ) ], x -> sizes[x] = Gsize ) do
for j in [ 1 .. Length( nsg ) ] do
if nsg[j] <> [ 1 ] then
# and IsSubset( nsg[i], nsg[j] ) !! (outside the k loop!)
for k in [ 1 .. j-1 ] do
if nsg[k] <> [ 1 ]
and Intersection( nsg[j], nsg[k] ) = [ 1 ]
and IsSubset( nsg[i], nsg[j] ) # move outside the k loop!
and IsSubset( nsg[i], nsg[k] )
and sizes[j] * sizes[k] = Gsize then
# One decomposition has been found.
r:= rec( kernels:= [ nsg[j], nsg[k] ],
kernelsizes:= [ sizes[j], sizes[k] ],
factors:= [ fail, fail ],
subgroups:= [ fail, fail ],
);
# Try to derive the character tables of the ingredients.
fact1:= First( ComputedClassFusions( tbl ),
r -> ClassPositionsOfKernel( r.map ) = nsg[j] );
if fact1 <> fail then
r.factors[1]:= fact1.name;
fact1:= CharacterTable( fact1.name );
fi;
fact2:= First( ComputedClassFusions( tbl ),
r -> ClassPositionsOfKernel( r.map ) = nsg[k] );
if fact2 <> fail then
r.factors[2]:= fact2.name;
fact2:= CharacterTable( fact2.name );
fi;
if fact1 <> fail and fact2 <> fail then
Unbind( rr );
for name1 in NamesOfFusionSources( fact1 ) do
sub1:= CharacterTable( name1 );
if sub1 <> fail and Size( sub1 ) = Size( fact1 ) / 2 then
for name2 in NamesOfFusionSources( fact2 ) do
sub2:= CharacterTable( name2 );
if sub2 <> fail and
Size( sub2 ) = Size( fact2 ) / 2 then
cand:= CharacterTableOfIndexTwoSubdirectProduct(
sub1, fact1, sub2, fact2, "test" );
trans:= TransformingPermutationsCharacterTables(
tbl, cand.table );
if trans <> fail then
rr:= ShallowCopy( r );
rr.subgroups:= [ name1, name2 ];
Add( result, rr );
fi;
fi;
od;
fi;
od;
if not IsBound( rr ) then
Add( result, r );
fi;
else
Add( result, r );
fi;
fi;
od;
fi;
od;
od;
return result;
elif Length( arg ) = 5 then
tblH1:= arg[2];
tblG1:= arg[3];
tblH2:= arg[4];
tblG2:= arg[5];
# Check the construction from the given tables.
cand:= CharacterTableOfIndexTwoSubdirectProduct(
tblH1, tblG1, tblH2, tblG2, "test" );
trans:= TransformingPermutationsCharacterTables( cand.table, arg[1] );
if trans <> fail then
return [ "ConstructIndexTwoSubdirectProduct",
Identifier( tblH1 ), Identifier( tblG1 ),
Identifier( tblH2 ), Identifier( tblG2 ),
cand.outerfus,
trans.columns, trans.rows ];
fi;
return fail;
else
Error( "usage: ConstructIndexTwoSubdirectProductInfo( <tbl>\n",
"[, <tblH1>, <tblG1>, <tblH2>, <tblG2>] )" );
fi;
end );
#############################################################################
##
## 8. Character Tables of Coprime Central Extensions
##
#############################################################################
##
#F CharacterTableOfCommonCentralExtension( <tblG>, <tblmG>, <tblnG>, <id> )
##
InstallGlobalFunction( CharacterTableOfCommonCentralExtension,
function( tblG, tblmG, tblnG, id )
local mGfusG, nGfusG, m, n, M, invm, invn, i, ordersG, ordersmG,
ordersnG, try, facttbl, factfusion, newinvm, newinvn, mnGfusmG,
mnGfusnG, lenm, lenn, j, imod, jmod, cents, invmnGfusmG, pow, p,
comp, tblmnG, ker, faithm, faithn, irr, centre, ordersmnG, zpos,
needed, faithmn, partners, chi;
# Check the arguments.
mGfusG:= GetFusionMap( tblmG, tblG );
nGfusG:= GetFusionMap( tblnG, tblG );
if mGfusG = fail or nGfusG = fail then
Error( "the fusions <tblmG>, <tblnG> ->> <tblG> must be stored" );
fi;
m:= Size( tblmG ) / Size( tblG );
n:= Size( tblnG ) / Size( tblG );
if not IsPrimeInt( m ) then
Error( "<tblmG> ->> <tblG> must be a prime order extension" );
elif not IsPrimeInt( n ) then
Error( "<tblnG> ->> <tblG> must be a prime order extension" );
elif m = n then
Error( "<tblmG>, <tblnG> ->> <tblG> must have coprime kernel" );
elif not IsSubset( ClassPositionsOfCentre( tblmG ),
ClassPositionsOfKernel( mGfusG ) ) then
Error( "<tblmG> must be a central extension of <tblG>" );
elif not IsSubset( ClassPositionsOfCentre( tblnG ),
ClassPositionsOfKernel( nGfusG ) ) then
Error( "<tblnG> must be a central extension of <tblG>" );
fi;
M:= m * n;
# Compute compatible fusions from $mn.G$ to $m.G$ and $n.G$.
invm:= InverseMap( mGfusG );
invn:= InverseMap( nGfusG );
for i in [ 1 .. Length( invm ) ] do
if IsInt( invm[i] ) then
invm[i]:= [ invm[i] ];
fi;
if IsInt( invn[i] ) then
invn[i]:= [ invn[i] ];
fi;
od;
# Note that $mn.G$ may have a cyclic central subgroup of order $M$
# larger than $m n$.
# We consider a largest possible cyclic central extension
# because then more cohorts of faithful characters exist;
# note that we have to compute only the characters in one family of
# Galois conjugate cohorts, and derive the others in the end.
# The second implication is that we can achieve the class ordering
# relative to the smaller factor group, as in the {\ATLAS} tables in
# the {\GAP} Character Table Library.
ordersG:= OrdersClassRepresentatives( tblG );
ordersmG:= OrdersClassRepresentatives( tblmG );
ordersnG:= OrdersClassRepresentatives( tblnG );
try:= Filtered( ClassPositionsOfCentre( tblG ),
x -> ordersG[x] * m in ordersmG{ invm[x] }
and ordersG[x] * n in ordersnG{ invn[x] }
and Length( invm[x] ) = m
and Length( invn[x] ) = n );
if 1 < Length( try ) then
# Compute the fusions onto the smaller factor group.
i:= Maximum( ordersG{ try } );
facttbl:= tblG / [ try[ Position( ordersG{ try }, i ) ] ];
factfusion:= GetFusionMap( tblG, facttbl );
M:= M * Size( tblG ) / Size( facttbl );
# Choose the class ordering w.r.t. the smaller factor group.
# For that, replace the inverse maps by compositions with the
# additional factor fusion, but be careful about compatible congurnces.
newinvm:= [];
newinvn:= [];
for i in InverseMap( factfusion ) do
if IsInt( i ) then
Add( newinvm, invm[i] );
Add( newinvn, invn[i] );
else
Add( newinvm, Concatenation( TransposedMat( invm{ i } ) ) );
Add( newinvn, Concatenation( TransposedMat( invn{ i } ) ) );
fi;
od;
invm:= newinvm;
invn:= newinvn;
fi;
mnGfusmG:= [];
mnGfusnG:= [];
for i in [ 1 .. Length( invm ) ] do
lenm:= Length( invm[i] );
lenn:= Length( invn[i] );
for j in [ 1 .. LcmInt( lenm, lenn ) ] do
# Take only those parameter pairs that are compatible
# with the fusions onto `tblmG' and `tblnG'.
imod:= j mod lenm; if imod = 0 then imod:= lenm; fi;
Add( mnGfusmG, invm[i][ imod ] );
jmod:= j mod lenn; if jmod = 0 then jmod:= lenn; fi;
Add( mnGfusnG, invn[i][ jmod ] );
od;
od;
# Create the table head.
cents:= [];
invmnGfusmG:= InverseMap( mnGfusmG );
for i in [ 1 .. Length( invmnGfusmG ) ] do
if IsInt( invmnGfusmG[i] ) then
cents[ invmnGfusmG[i] ]:= SizesCentralizers( tblmG )[i];
else
cents{ invmnGfusmG[i] }:= Length( invmnGfusmG[i] )
* SizesCentralizers( tblmG ){ List( invmnGfusmG[i], x -> i ) };
fi;
od;
pow:= [];
for p in PrimeDivisors( cents[1] ) do
pow[p]:= CompositionMaps( InverseMap( mnGfusmG ),
CompositionMaps( PowerMap( tblmG, p ), mnGfusmG ) );
comp:= CompositionMaps( InverseMap( mnGfusnG ),
CompositionMaps( PowerMap( tblnG, p ), mnGfusnG ) );
MeetMaps( pow[p], comp );
Assert( 1, ForAll( pow[p], IsInt ) );
od;
tblmnG:= ConvertToLibraryCharacterTableNC( rec(
Identifier := id,
InfoText :=
"constructed using `CharacterTableOfCommonCentralExtension'",
Size := Size( tblmG ) * n,
UnderlyingCharacteristic := 0,
SizesCentralizers := cents,
ComputedPowerMaps := pow,
OrdersClassRepresentatives := ElementOrdersPowerMap( pow ) ) );
StoreFusion( tblmnG, mGfusG{ mnGfusmG }, tblG );
StoreFusion( tblmnG, mnGfusmG, tblmG );
StoreFusion( tblmnG, mnGfusnG, tblnG );
# Transfer the known irreducibles.
ker:= ClassPositionsOfKernel( mGfusG );
faithm:= Filtered( Irr( tblmG ),
chi -> not IsSubset( ClassPositionsOfKernel( chi ), ker ) );
faithm:= List( faithm, chi -> Character( tblmnG, chi{ mnGfusmG } ) );
ker:= ClassPositionsOfKernel( nGfusG );
faithn:= Filtered( Irr( tblnG ),
chi -> not IsSubset( ClassPositionsOfKernel( chi ), ker ) );
faithn:= List( faithn, chi -> Character( tblmnG, chi{ mnGfusnG } ) );
if m < n then
irr:= List( Irr( tblmG ), chi -> Character( tblmnG, chi{ mnGfusmG } ) );
Append( irr, faithn );
else
irr:= List( Irr( tblnG ), chi -> Character( tblmnG, chi{ mnGfusnG } ) );
Append( irr, faithm );
fi;
# Fix a central class on which the values of missing characters
# can be prescribed as the degree times a fixed root of unity.
centre:= ClassPositionsOfCentre( tblmnG );
ordersmnG:= OrdersClassRepresentatives( tblmnG );
i:= Maximum( ordersmnG{ centre } );
zpos:= First( centre, x -> ordersmnG[x] = i
and ordersmG[ mnGfusmG[x] ] < ordersmnG[x]
and ordersnG[ mnGfusnG[x] ] < ordersmnG[x] );
# We use a heuristic for finding the irreducibles in one cohort.
needed:= ( NrConjugacyClasses( tblmnG ) - Length( irr ) ) / Phi( M );
faithmn:= IrreduciblesForCharacterTableOfCommonCentralExtension( tblmnG,
irr, zpos, needed );
# Create also the other cohorts.
partners:= GaloisPartnersOfIrreducibles( tblmnG, faithmn, M );
for i in [ 1 .. Length( faithmn ) ] do
chi:= faithmn[i];
Add( irr, chi );
for j in partners[i] do
Add( irr, Character( tblmnG, List( chi, x -> GaloisCyc( x, j ) ) ) );
od;
od;
if Length( irr ) = NrConjugacyClasses( tblmnG ) then
SetIrr( tblmnG, irr );
fi;
return rec( tblmnG := tblmnG,
IsComplete := Length( irr ) = NrConjugacyClasses( tblmnG ),
irreducibles := irr );
end );
#############################################################################
##
#F IrreduciblesForCharacterTableOfCommonCentralExtension(
#F <tblmnG>, <factirreducibles>, <zpos>, <needed> )
##
InstallGlobalFunction( IrreduciblesForCharacterTableOfCommonCentralExtension,
function( tblmnG, factirreducibles, zpos, needed )
local id, z, root, cohorts, faithmn, reducibles, i, ten, red, galois,
lll;
id:= Identifier( tblmnG );
Info( InfoCharacterTable, 1,
id, ": need ", needed, " faithful irreducibles" );
# Try to find the faithful irreducibles.
# We restrict our interest to one faithful cohort for each factor,
# and form tensor products.
# The faithful cohort is determined by the values of the central class
# `zpos' of maximal order whose image in both factor groups has smaller
# element order.
z:= OrdersClassRepresentatives( tblmnG )[ zpos ];
root:= E( z );
cohorts:= List( [ 1 .. z ],
i -> Filtered( factirreducibles,
x -> x[ zpos ] = x[1] * root^i ) );
# Take those combinations of two cohorts such that the tensor products
# lie in the target cohort.
faithmn:= [];
reducibles:= [];
for i in [ 1 .. Int( z / 2 ) ] do
if not IsEmpty( cohorts[i] ) and not IsEmpty( cohorts[ z+1-i ] ) then
ten:= TensorAndReduce( tblmnG, cohorts[i], cohorts[ z+1-i ],
faithmn, needed );
red:= ReducedX( tblmnG, ten, reducibles );
reducibles:= red.remainders;
if not IsEmpty( red.irreducibles ) then
Info( InfoCharacterTable, 1,
id, ": ", Length( red.irreducibles ),
" found by tensoring" );
Append( faithmn, red.irreducibles );
if needed <= Length( faithmn ) then
return faithmn;
fi;
fi;
fi;
od;
# Use Galois conjugates of the found faithful irreducibles
# to form tensor products that lie in the target cohort.
for i in [ 1 .. Int( z / 2 ) ] do
if GcdInt( i, z ) = 1 and not IsEmpty( faithmn )
and not IsEmpty( cohorts[ z+1-i ] ) then
galois:= List( faithmn, x -> List( x, y -> GaloisCyc( y, i ) ) );
ten:= TensorAndReduce( tblmnG, cohorts[ z+1-i ], galois,
faithmn, needed );
red:= ReducedX( tblmnG, ten, reducibles );
reducibles:= red.remainders;
if not IsEmpty( red.irreducibles ) then
Info( InfoCharacterTable, 1,
id, ": ", Length( red.irreducibles ),
" found by further tensoring" );
Append( faithmn, red.irreducibles );
if needed <= Length( faithmn ) then
return faithmn;
fi;
fi;
fi;
od;
# Use LLL.
lll:= LLL( tblmnG, reducibles );
if not IsEmpty( lll.irreducibles ) then
Info( InfoCharacterTable, 1,
id, ": ", Length( lll.irreducibles ), " found by LLL" );
Append( faithmn, lll.irreducibles );
fi;
if needed <= Length( faithmn ) then
return faithmn;
fi;
#T The following code was not needed up to now.
# #T make sure that lll.remainders are orthogonal to lll.irreducibles!
# if ForAny( lll.remainders, x -> ForAny( lll.irreducibles, y ->
# ScalarProduct( tblmnG, x, y ) <> 0 ) ) then
# Error( "nonorthogonal LLL run!" );
# fi;
#
# # Use a combination of tensor products and LLL.
# irreducibles:= faithmn;
# while not IsEmpty( irreducibles ) do
#
# newirreducibles:=[];
#
# # Use tensor products with the newly found faithful irreducibles.
# ten:= TensorAndReduce( tblmnG, factirreducibles, irreducibles,
# faithmn, needed );
# Info( InfoCharacterTable, 1,
# id, ": ", Length( ten.irreducibles ), " found by tensoring" );
# Append( newirreducibles, ten.irreducibles );
# Append( faithmn, ten.irreducibles );
# if needed <= Length( faithmn ) then
# return faithmn;
# fi;
#
# # Use LLL.
# lll:= LLL( tblmnG, ten.remainders );
# Info( InfoCharacterTable, 1,
# id, ": ", Length( lll.irreducibles ), " found by LLL" );
# Append( newirreducibles, lll.irreducibles );
# Append( faithmn, lll.irreducibles );
# if needed <= Length( faithmn ) then
# return faithmn;
# fi;
#
# irreducibles:= newirreducibles;
#
# od;
#
# # Use orthogonal embeddings.
# needed:= needed - Length( faithmn );
# if 0 < needed then
# mat:= MatScalarProducts( tblmnG, lll.remainders, lll.remainders );
# emb:= OrthogonalEmbeddingsSpecialDimension( tblmnG, lll.remainders,
# mat, needed );
# Info( InfoCharacterTable, 1,
# id, ": ", Length( emb.irreducibles ),
# " found by orth. embeddings" );
# UniteSet( faithmn, emb.irreducibles );
# fi;
# Sort the irreducibles.
faithmn:= SortedCharacters( tblmnG, faithmn, "degree" );
# Return the irreducibles.
return faithmn;
end );
#############################################################################
##
## 9. Miscellaneous
##
#############################################################################
##
#F ReducedX( <tbl>, <redresult>, <chars> )
##
## In each step, we start with a record irr1/red1 and a list red.
## After the step, we have a result record irr2/red2 and the list red1.
##
## red1,irr1 red
## | | |
## | --------
## | |
## red1 irr2,red2
## | | |
## -------- |
## | |
## red3,irr3 red2
## | | |
## | --------
## | |
## red3 irr4,red4
##
##
InstallGlobalFunction( ReducedX, function( tbl, redresult, chars )
local irreducibles, help;
irreducibles:= ShallowCopy( redresult.irreducibles );
while not IsEmpty( redresult.irreducibles ) do
help:= Reduced( tbl, redresult.irreducibles, chars );
chars:= redresult.remainders;
redresult:= help;
Append( irreducibles, redresult.irreducibles );
od;
# Return the result.
return rec( irreducibles := irreducibles,
remainders := Concatenation( redresult.remainders, chars ) );
end );
#############################################################################
##
#F TensorAndReduce( <tbl>, <chars1>, <chars2>, <irreducibles>, <needed> )
##
InstallGlobalFunction( TensorAndReduce,
function( tbl, chars1, chars2, irreducibles, needed )
local newirreducibles,
reducibles,
chi, # loop over `chars1'
psi, # loop over `chars2'
ten, # one tensor product
red;
irreducibles:= ShallowCopy( irreducibles );
newirreducibles:= [];
reducibles:= [];
for chi in chars1 do
for psi in chars2 do
ten:= Tensored( [ chi ], [ psi ] );
ten:= ReducedOrdinary( tbl, irreducibles, ten );
red:= ReducedX( tbl, ten, reducibles );
Append( irreducibles, red.irreducibles );
Append( newirreducibles, red.irreducibles );
reducibles:= red.remainders;
if needed <= Length( newirreducibles ) then
return rec( irreducibles := newirreducibles,
remainders := reducibles );
fi;
od;
od;
# Return the result.
return rec( irreducibles := newirreducibles,
remainders := reducibles );
end );
#T analogously: SymmetrizeAndReduce
# #############################################################################
# ##
# #M BrauerTableOp( <ordtbl>, <p> ) . . . . . . . . . . for the liftable case
# ##
# ## This method gets called before the ``last resort'' method from the
# ## GAP library.
# ## It checks the criteria from the library method, and calls 'TryNextMethod'
# ## if the library method would be successful;
# ## this way, constructions that have worked do not change due to the
# ## installation of the method below.
# ##
# ## If all <p>-modular irreducible Brauer characters lift to characteristic
# ## zero then we can write them down.
# ##
# InstallMethod( BrauerTableOp,
# [ "IsOrdinaryTable", "IsPosInt" ], 1, # above the library method
# function( tbl, p )
# local result, modtbls, source, bl, inv, choice, fusion, rest, b, brest,
# brestset, l;
#
# # Check the criteria from the library method first,
# # in order not to change constructions that have worked.
# result:= fail;
#
# if IsPSolvableCharacterTable( tbl, p ) then
# TryNextMethod();
# elif HasFactorsOfDirectProduct( tbl ) then
# modtbls:= List( FactorsOfDirectProduct( tbl ),
# t -> BrauerTable( t, p ) );
# if not fail in modtbls then
# TryNextMethod();
# fi;
# elif HasSourceOfIsoclinicTable( tbl ) then
# # Compute the isoclinic table of the Brauer table of the source table,
# # i.e., use the alternative path in the commutative diagram that is
# # given by forming the Brauer table and the isoclinic table.
# source:= SourceOfIsoclinicTable( tbl );
# modtbls:= BrauerTable( source[1], p );
# if modtbls <> fail then
# TryNextMethod();
# fi;
# fi;
#
# # Check whether the Sylow 'p'-subgroup is cyclic.
# result:= CharacterTableRegular( tbl, p );
# bl:= PrimeBlocks( tbl, p );
# inv:= InverseMap( bl.block );
# choice:= [];
# fusion:= GetFusionMap( result, tbl );
# rest:= List( Irr( tbl ), x -> ValuesOfClassFunction( x ){ fusion } );
# if not p^bl.defect[1] in OrdersClassRepresentatives( tbl ) then
# TryNextMethod();
# fi;
#
# # Now we know that any block has cyclic defect.
# for b in [ 1 .. Length( bl.defect ) ] do
# if bl.defect[b] = 0 then
# Add( choice, inv[b] );
# else
# brest:= rest{ inv[b] };
# brestset:= Set( brest );
# l:= Length( brestset );
# if l = 1 then
# # There is only one irreducible Brauer character in the block.
# Add( choice, inv[b][1] );
# elif Sum( brestset{ [ 1 .. l-1 ] } ) = brestset[l] then
# Append( choice,
# inv[b]{ List( [ 1 .. l-1 ],
# i -> Position( brest, brestset[i] ) ) } );
# else
# # Not all Brauer characters lift to characteristic zero.
# TryNextMethod();
# fi;
# fi;
# od;
#
# # The irreducibles shall be sorted as in the ordinary table.
# Sort( choice );
#
# # Set the attributes.
# SetIrr( result, List( choice, i -> Character( result, rest[i] ) ) );
# SetInfoText( result,
# "computed using that all Brauer characters lift to char. zero" );
# if HasClassParameters( tbl ) and result <> fail then
# SetClassParameters( result,
# ClassParameters( tbl ){ GetFusionMap( result, tbl ) } );
# fi;
#
# return result;
# end );
#############################################################################
##
#E
