/usr/share/gap/pkg/Polycyclic/doc/chap9.txt

  
  9 Matrix Representations
  
  This  chapter  describes functions which compute with matrix representations
  for pcp-groups. So far the routines in this package are only able to compute
  matrix representations for torsion-free nilpotent groups.
  
  
  9.1 Unitriangular matrix groups
  
  9.1-1 UnitriangularMatrixRepresentation
  
  UnitriangularMatrixRepresentation( G )  operation
  
  computes  a  faithful  representation of a torsion-free nilpotent group G as
  unipotent  lower  triangular matrices over the integers. The pc-presentation
  for  G  must  not contain any power relations. The algorithm is described in
  [dGN02].
  
  9.1-2 IsMatrixRepresentation
  
  IsMatrixRepresentation( G, matrices )  function
  
  checks  if  the map defined by mapping the i-th generator of the pcp-group G
  to the i-th matrix of matrices defines a homomorphism.
  
  
  9.2 Upper unitriangular matrix groups
  
  We  call  a  matrix  upper unitriangular if it is an upper triangular matrix
  with  ones on the main diagonal. The weight of an upper unitriangular matrix
  is the number of diagonals above the main diagonal that contain zeroes only.
  
  The  subgroup of all upper unitriangular matrices of GL(n,ℤ) is torsion-free
  nilpotent.  The  k-th  term  of  its  lower central series is the set of all
  matrices  of  weight  k-1.  The [122ℤX-rank of the k-th term of the lower central
  series modulo the (k+1)-th term is n-k.
  
  9.2-1 IsomorphismUpperUnitriMatGroupPcpGroup
  
  IsomorphismUpperUnitriMatGroupPcpGroup( G )  function
  
  takes  a group G generated by upper unitriangular matrices over the integers
  and computes a polycylic presentation for the group. The function returns an
  isomorphism  from  the  matrix  group  to  the  pcp group. Note that a group
  generated  by  upper  unitriangular  matrices  is  necessarily  torsion-free
  nilpotent.
  
  9.2-2 SiftUpperUnitriMatGroup
  
  SiftUpperUnitriMatGroup( G )  function
  
  takes  a group G generated by upper unitriangular matrices over the integers
  and  returns  a  recursive data structure L with the following properties: L
  contains a polycyclic generating sequence for G, using L one can decide if a
  given upper unitriangular matrix is contained in G, a given element of G can
  be  written  as  a  word  in  the  polycyclic  generating  sequence.  L is a
  representation  of  a  chain  of  subgroups of G refining the lower centrals
  series  of  G..  It  contains  for  each  subgroup  in  the  chain a minimal
  generating set.
  
  9.2-3 RanksLevels
  
  RanksLevels( L )  function
  
  takes  the  data  structure  returned  by  SiftUpperUnitriMat and prints the
  [122ℤX-rank of each the subgroup in L.
  
  9.2-4 MakeNewLevel
  
  MakeNewLevel( m )  function
  
  creates  one  level of the data structure returned by SiftUpperUnitriMat and
  initialises it with weight m.
  
  9.2-5 SiftUpperUnitriMat
  
  SiftUpperUnitriMat( gens, level, M )  function
  
  takes  the  generators  gens  of  an  upper  unitriangular  group,  the data
  structure   returned   level   by   SiftUpperUnitriMat   and  another  upper
  unitriangular  matrix  M.  It  sift  M  through  level  and  adds  M  at the
  appropriate  place  if  M  is  not  contained in the subgroup represented by
  level.
  
  The    function    SiftUpperUnitriMatGroup    illustrates    the    use   of
  SiftUpperUnitriMat.
  
    Example  
    InstallGlobalFunction( "SiftUpperUnitriMatGroup", function( G )
        local   firstlevel,  g;
    
        firstlevel := MakeNewLevel( 0 );
        for g in GeneratorsOfGroup(G) do
            SiftUpperUnitriMat( GeneratorsOfGroup(G), firstlevel, g );
        od;
        return firstlevel;
    end );
  
  
  9.2-6 DecomposeUpperUnitriMat
  
  DecomposeUpperUnitriMat( level, M )  function
  
  takes  the  data  structure  level returned by SiftUpperUnitriMatGroup and a
  upper  unitriangular matrix M and decomposes M into a word in the polycyclic
  generating sequence of level.
  

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