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##
#W initqs.gi   LPRES    René Hartung
##

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##
#M  InitQuotientSystem ( <LpGroup> )
##
## computes a weighted nilpotent quotient system for the abelian quotient
## of <LpGroup>.
##
InstallMethod( InitQuotientSystem,
  "for an L-presented group",
  true,
  [ IsLpGroup ], 0,
  function(G)
  local ftl,  # FromTheLeftCollector for G/G'
          A,  # power relations from Hermite normal form
          Q,  # new quotient system
          n,  # number of generators of L
          ev,evn, # exponent vectors for spinning algorithm
          rel,  # loop variable for (iterated) relators
          map,  # loop variable for endomorphisms
          i,j,k, # loop variables
          HNF,  # Hermite normal form of the relations
          stack, # stack for the spinning algorithm
          endos, # endomorphisms as matrices
          obj, mat, # loop variables to determine the matrices
   Gens,  # position of new gens in the HNF
   Imgs;  # loop variable to build the endomorphism

  # number of new generators
  n:=Length(GeneratorsOfGroup(G));

  # determine the iterated relators
  stack:=[];
  HNF:=rec(mat:=[],Heads:=[]);
  for rel in IteratedRelatorsOfLpGroup(G) do
    ev:=ListWithIdenticalEntries(n,0);
    obj:=ExtRepOfObj(rel);
    for i in [1,3..Length(obj)-1] do
      ev[obj[i]]:=ev[obj[i]]+obj[i+1];
    od;
    Add(stack,ShallowCopy(ev));
    LPRES_AddRow(HNF,ev);
  od;

  # determine the endomorphisms
  endos:=[];
  for map in EndomorphismsOfLpGroup(G) do
    mat:=NullMat(n,n);

    # UnderlyingElement: map is an endomorphism of the free group;
    # x an object of the L-presented group
    obj:=List(GeneratorsOfGroup(G),x->ExtRepOfObj(UnderlyingElement(x)^map));
    for j in [1..n] do
      for k in [1,3..Length(obj[j])-1] do
        mat[j][obj[j][k]]:=mat[j][obj[j][k]]+obj[j][k+1];
      od;
    od;
    Add(endos,mat);
  od;

  # spinning algorithm
  while not Length(stack)=0 do
    ev:=stack[1];
    Remove( stack, 1);
    if not IsZero(ev) then
      for i in [1..Length(endos)] do
        evn:=ev*endos[i];
        if LPRES_AddRow(HNF,evn) then
          Add(stack,evn);
        fi;
      od;
    fi;
  od;

  # add the (fixed) relators
  for rel in FixedRelatorsOfLpGroup(G) do
    ev:=ListWithIdenticalEntries(n,0);
    obj:=ExtRepOfObj(rel);
    for i in [1,3..Length(obj)-1] do
      ev[obj[i]]:=ev[obj[i]]+obj[i+1];
    od;
    LPRES_AddRow(HNF,ev);
  od;

  # translate the Hermite normal form to power relations
  A:=LPRES_PowerRelationsOfHNF(HNF);

  # generators with power relation
  Gens:=HNF.Heads{Filtered([1..Length(HNF.Heads)],
                           x->HNF.mat[x][HNF.Heads[x]]<>1)};

  # infinite generators;
  Append(Gens,Filtered([1..n],x->not x in HNF.Heads));
  Sort(Gens);

  # if the group is perfect
  if Length(Gens)=0 then
    Q := rec( Lpres       := G,
              Pccol       := FromTheLeftCollector( 0 ),
              Imgs        := ListWithIdenticalEntries( n, [] ),
              Weights     := [],
              Definitions := [] );
    UpdatePolycyclicCollector( Q.Pccol );
    Imgs := List( Q.Imgs, x -> PcpElementByGenExpList( Q.Pccol, x ) );
    Q.Epimorphism := GroupHomomorphismByImagesNC( Q.Lpres,
                                        PcpGroupByCollectorNC( Q.Pccol),
                                        GeneratorsOfGroup( Q.Lpres), Imgs );

    return( Q );
  fi;

  # build quotient system
  Q:=rec();
  Q.Lpres:=G;

  # the new collector
  Q.Pccol:=FromTheLeftCollector(Length(Gens));
  for i in [1..Length(Gens)] do
    k:=Position(HNF.Heads,Gens[i]);
    if k<>fail then
      SetRelativeOrder(Q.Pccol,i, A[k][Gens[i]]);
      ev:=ShallowCopy(A[k]{Gens});
      ev[i]:=0;
      if not IsZero(ev) then
        SetPower(Q.Pccol,i,ObjByExponents(Q.Pccol,-ev));
      fi;
    fi;
  od;
  UpdatePolycyclicCollector(Q.Pccol);

  # the natural homomorphism onto the new presentation
  Q.Imgs:=[];
  for i in [1..n] do
    if i in Gens then
      k:=i-Length(Filtered([1..Length(HNF.Heads)],x->HNF.mat[x][HNF.Heads[x]]=1
                                                     and HNF.Heads[x]<i));
      Add(Q.Imgs,k);
    else
      Add(Q.Imgs,ObjByExponents(Q.Pccol,-A[Position(HNF.Heads,i)]{Gens}));
    fi;
  od;

  # the epimorphism into the new presentation
  Imgs:=[];
  for i in [1..Length(Q.Imgs)] do
    if IsInt(Q.Imgs[i]) then
      Add(Imgs,PcpElementByGenExpList(Q.Pccol,[Q.Imgs[i],1]));
    else
      Add(Imgs,PcpElementByGenExpList(Q.Pccol,Q.Imgs[i]));
    fi;
  od;
  Q.Epimorphism:=GroupHomomorphismByImagesNC(Q.Lpres,
                      PcpGroupByCollectorNC(Q.Pccol),
   GeneratorsOfGroup(Q.Lpres),Imgs);

  Q.Weights:=List([1..Length(Gens)],x->1);
  Q.Definitions:=Filtered([1..Length(Q.Imgs)],x->not IsList(Q.Imgs[x]));

  return(Q);
  end);

############################################################################
##
#M  AbelianInvariants ( <LpGroup> )
##
## computes the abelian invariants of <LpGroup>.
##
InstallMethod( AbelianInvariants,
  "for an L-presented group",
  [ IsLpGroup ],0,
  G -> AbelianInvariants( NilpotentQuotient(G,1) ) );

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