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############################################################################
##
#W buildnew.gi LPRES René Hartung
##
############################################################################
##
#F LPRES_BuildNewCollector ( )
##
## Builds a new collector with respect to the relations and the consistency
## relations for the covering group.
##
InstallGlobalFunction( LPRES_BuildNewCollector,
function(Q,ftl,HNF,weights,Defs,Imgs)
local A, # power relations with attention to the HNF
orders, # relative order of the 'old' collector
Gens, # new generators
QS, # new quotient system
rhs,rhsTails, # right hand side of the new relation
i,j,k, # loop variables
c, # nilpotency class
b; # position of the first (pseudo) generator/tail
# nilpotency class
c:=Maximum(weights);
# first new generator/pseudo-generator
b:=Position(weights,c);
# relative orders of the old quotient system
orders:=RelativeOrders(Q.Pccol);
# power relations from Hermite normal form
A:=LPRES_PowerRelationsOfHNF(HNF);
# find new generators in HNF
Gens:=HNF.Heads{Filtered([1..Length(HNF.Heads)],
x->HNF.mat[x][HNF.Heads[x]]<>1)};
Append(Gens, Filtered([1..Length(weights)-(b-1)],x->not x in HNF.Heads));
Sort(Gens);
if Length(Gens)=0 then
return(Q);
fi;
# new quotient system
QS:=rec();
QS.Lpres:=Q.Lpres;
QS.Pccol:=FromTheLeftCollector(b-1+Length(Gens));
# restore the new images
QS.Imgs:=[];
for i in [1..Length(Imgs)] do
if IsPosInt(Imgs[i]) then
QS.Imgs[i]:=Imgs[i];
elif IsList(Imgs[i]) then
rhs:=ExponentsByObj(ftl,Imgs[i]);
rhsTails:=rhs{[b..Length(rhs)]};
rhs:=Concatenation(rhs{[1..b-1]}, LPRES_RowReduce(rhsTails,HNF){Gens});
QS.Imgs[i]:=ObjByExponents(QS.Pccol,rhs);
else
Error("Strange entry in Imgs");
fi;
od;
# new definitions and new weights
QS.Definitions:=Concatenation(Defs{[1..b-1]}, Defs{Gens+(b-1)});
QS.Weights:=Concatenation(weights{[1..b-1]},weights{Gens+(b-1)});
# set the old power relations
for i in Filtered([1..Length(orders)],x-> orders[x]<>0) do
rhs:=ExponentsByObj(ftl,GetPower(ftl,i));
rhsTails:=rhs{[b..Length(rhs)]};
rhs:=Concatenation(rhs{[1..b-1]}, LPRES_RowReduce(rhsTails,HNF){Gens});
SetRelativeOrder(QS.Pccol,i,orders[i]);
SetPower(QS.Pccol,i,ObjByExponents(QS.Pccol,rhs));
od;
# set conjugacy relations
for i in [1..(b-1)-1] do
for j in [i+1..b-1] do
# a_j a_i = a_i a_j u_ij^++
rhs:=ExponentsByObj(ftl,GetConjugate(ftl,j,i));
rhsTails:=rhs{[b..Length(rhs)]};
if not IsZero(rhsTails) then
rhs:=Concatenation(rhs{[1..b-1]}, LPRES_RowReduce(rhsTails,HNF){Gens});
else
rhs:=Concatenation(rhs{[1..b-1]},
ListWithIdenticalEntries(Length(Gens),0));
fi;
SetConjugate(QS.Pccol,j,i,ObjByExponents(QS.Pccol,rhs));
if orders[i]=0 then
# a_j a_i^-1 = a_i^-1 a_j u_ij^-+
rhs:=ExponentsByObj(ftl,GetConjugate(ftl,j,-i));
rhsTails:=rhs{[b..Length(rhs)]};
if not IsZero(rhsTails) then
rhs:=Concatenation(rhs{[1..b-1]}, LPRES_RowReduce(rhsTails,HNF){Gens});
else
rhs:=Concatenation(rhs{[1..b-1]},
ListWithIdenticalEntries(Length(Gens),0));
fi;
SetConjugate(QS.Pccol,j,-i,ObjByExponents(QS.Pccol,rhs));
if orders[j]=0 then
# a_j^-1 a_i^-1 = a_i^-1 a_j^-1 u_ij^--
rhs:=ExponentsByObj(ftl,GetConjugate(ftl,-j,-i));
rhsTails:=rhs{[b..Length(rhs)]};
if not IsZero(rhsTails) then
rhs:=Concatenation(rhs{[1..b-1]}, LPRES_RowReduce(rhsTails,HNF){Gens});
else
rhs:=Concatenation(rhs{[1..b-1]},
ListWithIdenticalEntries(Length(Gens),0));
fi;
SetConjugate(QS.Pccol,-j,-i,ObjByExponents(QS.Pccol,rhs));
fi;
elif orders[j]=0 then
# a_j^-1 a_i = a_i a_j^-1 u_ij^+-
rhs:=ExponentsByObj(ftl,GetConjugate(ftl,-j,i));
rhsTails:=rhs{[b..Length(rhs)]};
if not IsZero(rhsTails) then
rhs:=Concatenation(rhs{[1..b-1]}, LPRES_RowReduce(rhsTails,HNF){Gens});
else
rhs:=Concatenation(rhs{[1..b-1]},
ListWithIdenticalEntries(Length(Gens),0));
fi;
SetConjugate(QS.Pccol,-j,i,ObjByExponents(QS.Pccol,rhs));
fi;
od;
od;
# set the new power relations for the tails
for i in Filtered([1..Length(HNF.Heads)],x->HNF.mat[x][HNF.Heads[x]]<>1) do
k:=Position(Gens,HNF.Heads[i]);
SetRelativeOrder(QS.Pccol,k+(b-1),HNF.mat[i][HNF.Heads[i]]);
rhs:=ListWithIdenticalEntries(Length(weights)-(b-1),0);
rhs[HNF.Heads[i]]:=HNF.mat[i][HNF.Heads[i]];
rhs:=Concatenation(ListWithIdenticalEntries(b-1,0),
LPRES_RowReduce(rhs,HNF){Gens});;
SetPower(QS.Pccol,k+(b-1),ObjByExponents(QS.Pccol,rhs));
od;
FromTheLeftCollector_SetCommute(QS.Pccol);
SetFilterObj(QS.Pccol,IsUpToDatePolycyclicCollector);
FromTheLeftCollector_CompleteConjugate(QS.Pccol);
FromTheLeftCollector_CompletePowers(QS.Pccol);
SetFilterObj(QS.Pccol,IsUpToDatePolycyclicCollector);
# SetFilterObj(QS.Pccol,UseLibraryCollector);
if LPRES_TEST_ALL then
if not IsConfluent(QS.Pccol) then
Error("presentation is not confluent");
fi;
fi;
# build the mapping QS.Epimorphism
Imgs:=[];
for i in [1..Length(QS.Imgs)] do
if IsInt(QS.Imgs[i]) then
Add(Imgs,PcpElementByGenExpList(QS.Pccol,[QS.Imgs[i],1]));
else
Add(Imgs,PcpElementByGenExpList(QS.Pccol,QS.Imgs[i]));
fi;
od;
QS.Epimorphism:=GroupHomomorphismByImagesNC(Q.Lpres,
PcpGroupByCollectorNC(QS.Pccol),
GeneratorsOfGroup(Q.Lpres),Imgs);
return(QS);
end);
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