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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Heiko Theißen.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
#function(pcgs,D,pnt,acts,act[,dict]) -- allow to reuse dictionary
# in case of many short orbits
InstallGlobalFunction(Pcs_OrbitStabilizer,function(arg)
local pcgs,D,pnt,acts,act,
orb, # orbit
len, # lengths of orbit before each extension
d, # dictionary
S, rel, # stabilizer and induced pcgs
img, pos, # image of <pnt> and its position in <orb>
stb, # stabilizing element, a word in <pcgs>
depths, # depths of stabilizer generators
i, ii, j, k; # loop variables
pcgs:=arg[1];D:=arg[2];pnt:=arg[3];acts:=arg[4];act:=arg[5];
pnt:=Immutable(pnt);
if Length(arg)>5 then
d:=arg[6];
else
d:=NewDictionary(pnt,true,D);
fi;
orb := [ pnt ];
AddDictionary(d,pnt,1);
len := ListWithIdenticalEntries( Length( pcgs ) + 1, 0 );
len[ Length( len ) ] := 1;
S := [ ];
depths:=[];
rel := [ ];
for i in Reversed( [ 1 .. Length( pcgs ) ] ) do
Info(InfoAction,3,"Step ",i,": ",Length(orb));
img := act( pnt, acts[ i ] );
MakeImmutable(img);
pos := LookupDictionary( d, img );
if pos = fail then
# The current generator moves the orbit as a block.
Add( orb, img );
AddDictionary(d,img,Length(orb));
for j in [ 2 .. len[ i + 1 ] ] do
img := act( orb[ j ], acts[ i ] );
MakeImmutable(img);
Add( orb, img );
AddDictionary(d,img,Length(orb));
od;
for k in [ 3 .. RelativeOrders( pcgs )[ i ] ] do
for j in Length( orb ) + [ 1 - len[ i + 1 ] .. 0 ] do
img := act( orb[ j ], acts[ i ] );
MakeImmutable(img);
Add( orb, img );
AddDictionary(d,img,Length(orb));
od;
od;
else
# The current generator leaves the orbit invariant.
stb := ListWithIdenticalEntries( Length( pcgs ), 0 );
stb[ i ] := 1;
ii := i + 2;
while pos <> 1 do
while len[ ii ] >= pos do
ii := ii + 1;
od;
stb[ ii - 1 ] := -QuoInt( pos - 1, len[ ii ] );
pos := ( pos - 1 ) mod len[ ii ] + 1;
od;
Add( S, LinearCombinationPcgs( pcgs, stb ) );
Add(depths,i);
Add( rel, RelativeOrders( pcgs )[ i ] );
fi;
len[ i ] := Length( orb );
od;
return rec( orbit := orb, length := len, stabpcs := Reversed( S ),
depths:=Reversed(depths),
relords := Reversed( rel ),dictionary:=d );
end);
InstallGlobalFunction(Pcgs_OrbitStabilizer,function(pcgs,D,pnt,acts,act)
local pcs, new,dep;
pcs := Pcs_OrbitStabilizer( pcgs, D,pnt, acts, act );
dep:=pcs.depths;
if not IsIdenticalObj(pcgs,ParentPcgs(pcgs)) then
if IsBound(pcgs!.depthsInParent) then
dep:=pcgs!.depthsInParent{dep};
new:=InducedPcgsByPcSequenceNC(ParentPcgs(pcgs),pcs.stabpcs,dep);
else
# cannot translate depths to parent without calling `depth'.
new:=InducedPcgsByPcSequenceNC(ParentPcgs(pcgs),pcs.stabpcs);
fi;
else
new:=InducedPcgsByPcSequenceNC(ParentPcgs(pcgs),pcs.stabpcs,dep);
fi;
SetRelativeOrders( new, pcs.relords );
return rec( orbit := pcs.orbit, stabpcgs := new, lengths:=pcs.length,
dictionary:=pcs.dictionary);
end);
#############################################################################
##
#F StabilizerPcgs( <pcgs>, <pnt> [,<acts>] [,<act>] ) . . . . . . . . . .
##
## computes the stabilizer in the group generated by <pcgs> of the point
## <pnt>. If given <acts> are elements by which <pcgs> acts, <act> is
## the acting function. This function returns a pcgs for the stabilizer
## which is induced by the `ParentPcgs' of <pcgs>, that is it is compatible
## with <pcgs>.
InstallGlobalFunction(StabilizerPcgs,function(arg)
local acts,act;
acts:=arg[1];
act:=OnPoints;
if Length(arg)=3 then
if IsFunction(arg[3]) then
act:=arg[3];
else
acts:=arg[3];
fi;
elif Length(arg)=4 then
acts:=arg[3];
act:=arg[4];
fi;
# catch the case of vectors and prepare for hashing.
if IsRowVector(arg[2]) and Length(acts)>0 and IsMatrix(acts[1]) then
return Pcgs_OrbitStabilizer(arg[1],NaturalActedSpace(acts,[arg[2]]),
arg[2],acts,act).stabpcgs;
else
return Pcgs_OrbitStabilizer(arg[1],false,arg[2],acts,act).stabpcgs;
fi;
end);
# This function does the same as the following method, however it does dot
# create an immutable copy of the orbit (which could be expensive)
BindGlobal("Pcgs_MutableOrbitStabilizerOp",function( G, D,pnt, pcgs, acts, act )
local S,stab;
S:=Pcgs_OrbitStabilizer(pcgs,D,pnt,acts,act);
stab := SubgroupByPcgs( G, S.stabpcgs );
# this setting is already done by `SubgroupByPcgs'.
#SetInducedPcgs(ParentPcgs(pcgs),stab,S.stabpcgs);
return rec( orbit := S.orbit, stabilizer := stab );
end);
#############################################################################
##
#M OrbitStabilizerOp( <G>, <D>, <pnt>, <pcgs>, <acts>, <act> ) . . . by pcgs
##
InstallMethod( OrbitStabilizerOp,
"G, D, pnt, pcgs, acts, act", true,
[ IsGroup, IsList, IsObject, IsPrimeOrdersPcgs, IsList, IsFunction ],
0,
function( G,D, pnt, pcgs, acts, act )
return Immutable(Pcgs_MutableOrbitStabilizerOp(G,D,pnt,pcgs,acts,act));
end );
InstallOtherMethod( OrbitStabilizerOp,
"G, pnt, pcgs, acts, act", true,
[ IsGroup, IsObject, IsPrimeOrdersPcgs, IsList, IsFunction ], 0,
function( G, pnt, pcgs, acts, act )
return Immutable(Pcgs_MutableOrbitStabilizerOp(G,false,pnt,pcgs,acts,act));
end );
InstallMethod( OrbitStabilizerAlgorithm,"for pcgs",true,
[IsGroup,IsObject,IsObject,IsPcgs,
IsList,IsRecord],0,
function(G,D,blist,pcgs,acts,pntact)
local S,stab,i,pnt,act;
pnt:=pntact.pnt;
if IsBound(pntact.act) then
act:=pntact.act;
else
act:=pntact.opr;
fi;
S:=Pcgs_OrbitStabilizer(pcgs,D,pnt,acts,act);
stab := SubgroupByPcgs( G, S.stabpcgs );
# tick off if necessary
if IsList(blist) then
if IsPositionDictionary(S.dictionary) then
UniteBlist(blist,S.dictionary!.blist);
else
for i in S.orbit do
blist[PositionCanonical(D,i)]:=true;
od;
fi;
fi;
return rec( orbit := S.orbit, stabilizer := stab );
end);
#############################################################################
##
#F SetCanonicalRepresentativeOfExternalOrbitByPcgs( <xset> ) . . . . . . . .
##
InstallGlobalFunction( SetCanonicalRepresentativeOfExternalOrbitByPcgs,
function( xset )
local G, D, pnt, pcgs, acts, act,
orb, bit, # orbit, as list and bit-list
len, # lengths of orbit before each extension
stab, S, # stabilizer and induced pcgs
img, pos, # image of <pnt> and its position in <D> (or <orb>)
min, mpos, # minimal value of <pos>, position in <orb>
stb, # stabilizing element, a word in <pcgs>
oper, # operating element, a word in <pcgs>
i, ii, j, k;# loop variables
G := ActingDomain( xset );
D := HomeEnumerator( xset );
pnt := Representative( xset );
if IsExternalSetDefaultRep( xset ) then
pcgs := Pcgs( G );
acts := pcgs;
act := FunctionAction( xset );
else
pcgs := xset!.generators;
acts := xset!.operators;
act := xset!.funcOperation;
fi;
orb := [ pnt ];
len := ListWithIdenticalEntries( Length( pcgs ) + 1, 0 );
len[ Length( len ) ] := 1;
min := Position( D, pnt ); mpos := 1;
bit := BlistList( [ 1 .. Length( D ) ], [ min ] );
S := [ ];
for i in Reversed( [ 1 .. Length( pcgs ) ] ) do
img := act( pnt, acts[ i ] );
pos := PositionCanonical( D, img );
if not bit[ pos ] then
# The current generator moves the orbit as a block.
Add( orb, img ); bit[ pos ] := true;
if pos < min then
min := pos; mpos := Length( orb );
fi;
for j in [ 2 .. len[ i + 1 ] ] do
img := act( orb[ j ], acts[ i ] );
pos := PositionCanonical( D, img );
Add( orb, img ); bit[ pos ] := true;
if pos < min then
min := pos; mpos := Length( orb );
fi;
od;
for k in [ 3 .. RelativeOrders( pcgs )[ i ] ] do
for j in Length( orb ) + [ 1 - len[ i + 1 ] .. 0 ] do
img := act( orb[ j ], acts[ i ] );
pos := PositionCanonical( D, img );
Add( orb, img ); bit[ pos ] := true;
if pos < min then
min := pos; mpos := Length( orb );
fi;
od;
od;
else
# The current generator leaves the orbit invariant.
pos := Position( orb, img );
stb := ListWithIdenticalEntries( Length( pcgs ), 0 );
stb[ i ] := 1;
ii := i + 2;
while pos <> 1 do
while len[ ii ] >= pos do
ii := ii + 1;
od;
stb[ ii - 1 ] := -QuoInt( pos - 1, len[ ii ] );
pos := ( pos - 1 ) mod len[ ii ] + 1;
od;
Add( S, LinearCombinationPcgs( pcgs, stb ) );
fi;
len[ i ] := Length( orb );
od;
SetEnumerator( xset, orb );
# Construct the operator for the minimal point at <mpos>.
oper := ListWithIdenticalEntries( Length( pcgs ), 0 );
ii := 2;
while mpos <> 1 do
while len[ ii ] >= mpos do
ii := ii + 1;
od;
mpos := mpos - 1;
oper[ ii - 1 ] := -QuoInt( mpos, len[ ii ] );
mpos := mpos mod len[ ii ] + 1;
od;
SetCanonicalRepresentativeOfExternalSet( xset, D[ min ] );
if not HasActorOfExternalSet( xset ) then
SetActorOfExternalSet( xset,
LinearCombinationPcgs( pcgs, oper ) ^ -1 );
fi;
# <S> is a reversed IGS.
if not HasStabilizerOfExternalSet( xset ) then
S := InducedPcgsByPcSequenceNC( ParentPcgs(pcgs), Reversed( S ) );
stab := SubgroupByPcgs( G, S );
# SetRelativeOrders( S, Reversed( rel ) );
# SetInducedPcgs(ParentPcgs(pcgs),stab,S);
# if ParentPcgs( pcgs ) = HomePcgs( stab ) then
# SetInducedPcgsWrtHomePcgs( stab, S );
# else
# SetPcgs( stab, S );
# fi;
SetStabilizerOfExternalSet( xset, stab );
fi;
end );
#############################################################################
##
#M Enumerator( <xorb> ) . . . . . . . . . . . . . . . . . . . . . . . . . .
##
InstallMethod( Enumerator, "<xorb by pcgs>", true,
[ IsExternalOrbit and IsExternalSetByPcgs ], 0,
function( xorb )
local orbstab;
orbstab := OrbitStabilizer( xorb, Representative( xorb ) );
SetStabilizerOfExternalSet( xorb, orbstab.stabilizer );
return orbstab.orbit;
end );
#############################################################################
##
#M CanonicalRepresentativeOfExternalSet( <xorb> ) . . . . . . . . . . . . .
##
InstallMethod( CanonicalRepresentativeOfExternalSet,
"via `ActorOfExternalSet'", true,
[ IsExternalOrbit and IsExternalSetByPcgs ], 0,
function( xorb )
local oper;
oper := ActorOfExternalSet( xorb );
if HasCanonicalRepresentativeOfExternalSet( xorb ) then
return CanonicalRepresentativeOfExternalSet( xorb );
else
return FunctionAction( xorb )( Representative( xorb ), oper );
fi;
end );
#############################################################################
##
#M ActorOfExternalSet( <xorb> ) . . . . . . . . . . . . . . . . . . . . .
##
InstallMethod( ActorOfExternalSet, true,
[ IsExternalOrbit and IsExternalSetByPcgs ], 0,
function( xorb )
SetCanonicalRepresentativeOfExternalOrbitByPcgs( xorb );
return ActorOfExternalSet( xorb );
end );
#############################################################################
##
#M StabilizerOfExternalSet( <xorb> ) . . . . . stabilizer of representative
##
InstallMethod( StabilizerOfExternalSet, true,
[ IsExternalOrbit and IsExternalSetByPcgs ], 0,
function( xorb )
local orbstab;
orbstab := OrbitStabilizer( xorb, Representative( xorb ) );
SetEnumerator( xorb, orbstab.orbit );
return orbstab.stabilizer;
end );
#############################################################################
##
#M OrbitOp( <G>, <D>, <pnt>, <pcgs>, <acts>, <act> ) . . . . . based on pcgs
##
BindGlobal("DoPcgsOrbitOp",function( G, D, pt, U, V, act )
local orb, v, img, len, i, j, k,d,nimg;
pt:=Immutable(pt);
d:=NewDictionary(pt,false,D);
orb := [ pt ];
AddDictionary(d,pt);
for i in Reversed( [ 1 .. Length( V ) ] ) do
v := V[ i ];
img := act( pt, v );
MakeImmutable(img);
if not KnowsDictionary(d,img) then
len := Length( orb );
Add( orb, img );
AddDictionary(d,img);
for j in [ 2 .. len ] do
nimg:=act( orb[ j ], v );
MakeImmutable(nimg);
Add( orb, nimg );
AddDictionary(d,nimg);
od;
for k in [ 3 .. RelativeOrders( U )[ i ] ] do
for j in [ Length( orb ) - len + 1 .. Length( orb ) ] do
nimg:=act( orb[ j ], v );
MakeImmutable(nimg);
Add( orb, nimg );
AddDictionary(d,nimg);
od;
od;
fi;
od;
return Immutable( orb );
end );
InstallMethod( OrbitOp,
"via prime order pcgs, with domain", true,
[ IsGroup, IsList, IsObject, IsPrimeOrdersPcgs, IsList, IsFunction ],
0,DoPcgsOrbitOp);
InstallOtherMethod( OrbitOp,
"action via prime order pcgs", true,
[ IsGroup, IsObject, IsPrimeOrdersPcgs, IsList, IsFunction ], 0,
function( G, pt, U, V, act )
return DoPcgsOrbitOp(G,false,pt,U,V,act);
end);
# #############################################################################
# ##
# #M RepresentativeActionOp( <G>, <D>, <d>, <e>, <pcgs>, <acts>, <act> ) .
# ##
# InstallOtherMethod( RepresentativeActionOp, true,
# [ IsGroup, IsList, IsObject, IsObject, IsPrimeOrdersPcgs,
# IsList, IsFunction ], 0,
# function( G, D, d, e, pcgs, acts, act )
# local dset, eset;
#
# dset := ExternalOrbit( G, D, d, pcgs, acts, act );
# eset := ExternalOrbit( G, D, e, pcgs, acts, act );
# return ActorOfExternalSet( dset ) /
# ActorOfExternalSet( eset );
# end );
#############################################################################
##
#M StabilizerOp( <G>, <D>, <pt>, <U>, <V>, <op> ) . . . . . . based on pcgs
##
InstallMethod( StabilizerOp,
"G, D, pnt, pcgs, acts, act, calling `Pcgs_MutableOrbitStabilizerOp'", true,
[ IsGroup, IsList, IsObject, IsPrimeOrdersPcgs,
IsList, IsFunction ], 0,
function( G, D, pt, U, V, op )
return Pcgs_MutableOrbitStabilizerOp( G, D,pt, U, V, op ).stabilizer;
end );
InstallOtherMethod( StabilizerOp,
"G, pnt, pcgs, acts, act, calling `Pcgs_MutableOrbitStabilizerOp'", true,
[ IsGroup, IsObject, IsPrimeOrdersPcgs,
IsList, IsFunction ], 0,
function( G, pt, U, V, op )
return Pcgs_MutableOrbitStabilizerOp( G, false,pt, U, V, op ).stabilizer;
end );
InstallOtherMethod( StabilizerOp,
"G (solv.), pnt, gens, gens, act", true,
[ IsGroup and CanEasilyComputePcgs, IsObject,
IsList,
IsList,
IsFunction ], 0,
function( G, pt, gens, acts, op )
if gens = acts then
return Pcgs_MutableOrbitStabilizerOp(G,false,pt,
Pcgs(G),Pcgs(G),op).stabilizer;
else
TryNextMethod();
fi;
end );
InstallMethod( StabilizerOp,
"G (solv.), D,pnt, gens, gens, act", true,
[ IsGroup and CanEasilyComputePcgs, IsListOrCollection,
IsObject,
IsList,
IsList,
IsFunction ], 0,
function( G,D, pt, gens, acts, op )
if gens = acts then
return Pcgs_MutableOrbitStabilizerOp(G,D,pt,Pcgs(G),Pcgs(G),op).stabilizer;
else
TryNextMethod();
fi;
end );
InstallOtherMethod( StabilizerOp,
"G (solv.), pnt, gens, gens, act", true,
[ IsGroup and CanEasilyComputePcgs, IsObject,
IsPrimeOrdersPcgs,
IsList,
IsFunction ], 0,
function( G, pt, gens, acts, op )
return Pcgs_MutableOrbitStabilizerOp(G,false,pt,gens,acts,op).stabilizer;
end );
InstallOtherMethod( StabilizerOp,
"G (solv.), D,pnt, gens, gens, act", true,
[ IsGroup and CanEasilyComputePcgs, IsObject,
IsObject,
IsPrimeOrdersPcgs,
IsList,
IsFunction ], 0,
function( G,D, pt, gens, acts, op )
return Pcgs_MutableOrbitStabilizerOp(G,D,pt,gens,acts,op).stabilizer;
end );
#############################################################################
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