chap3.txt

  
  3 Algorithms of Orbit-Stabilizer Type
  
  We  introduce  a  way  to  calculate  a  sufficient part of an orbit and the
  stabilizer of a point.
  
  
  3.1 Orbit Stabilizer for Crystallographic Groups
  
  3.1-1 OrbitStabilizerInUnitCubeOnRight
  
  > OrbitStabilizerInUnitCubeOnRight( group, x ) _______________________method
  Returns:  A record containing
  
  --              .stabilizer: the stabilizer of x.
  
  --              .orbit  set  of  vectors  from  [0,1)^n which represents the
                  orbit.
  
  Let  x be a rational vector from [0,1)^n and group a space group in standard
  form.  The  function then calculates the part of the orbit which lies inside
  the  cube [0,1)^n and the stabilizer of x. Observe that every element of the
  full  orbit  differs  from  a  point  in  the  returned orbit only by a pure
  translation.
  
  Note  that  the  restriction  to  points  from [0,1)^n makes sense if orbits
  should be compared and the vector passed to OrbitStabilizerInUnitCubeOnRight
  should be an element of the returned orbit (part).
  
  ---------------------------  Example  ----------------------------
       
    gap> S:=SpaceGroup(3,5);;
    gap> OrbitStabilizerInUnitCubeOnRight(S,[1/2,0,9/11]);   
    rec( orbit := [ [ 0, 1/2, 2/11 ], [ 1/2, 0, 9/11 ] ], 
      stabilizer := Group([ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], 
              [ 0, 0, 0, 1 ] ] ]) )
    gap> OrbitStabilizerInUnitCubeOnRight(S,[0,0,0]);     
    rec( orbit := [ [ 0, 0, 0 ] ], stabilizer := <matrix group with 2 generators> )
    
  ------------------------------------------------------------------
  
  If  you are interested in other parts of the orbit, you can use VectorModOne
  (2.1-2)  for  the  base  point  and  the functions ShiftedOrbitPart (3.1-9),
  TranslationsToOneCubeAroundCenter  (3.1-10)  and  TranslationsToBox (3.1-11)
  for the resulting orbit
  Suppose  we  want to calculate the part of the orbit of [4/3,5/3,7/3] in the
  cube of sidelength 1 around this point:
  
  ---------------------------  Example  ----------------------------
    gap> S:=SpaceGroup(3,5);;
    gap> p:=[4/3,5/3,7/3];;
    gap> o:=OrbitStabilizerInUnitCubeOnRight(S,VectorModOne(p)).orbit;
    [ [ 1/3, 2/3, 1/3 ], [ 1/3, 2/3, 2/3 ] ]
    gap> box:=p+[[-1,1],[-1,1],[-1,1]];
    [ [ 1/3, 8/3, 7/3 ], [ 1/3, 8/3, 7/3 ], [ 1/3, 8/3, 7/3 ] ]
    gap> o2:=Concatenation(List(o,i->i+TranslationsToBox(i,box)));;
    gap> # This is what we looked for. But it is somewhat large:
    gap> Size(o2);
    54
  ------------------------------------------------------------------
  
  3.1-2 OrbitStabilizerInUnitCubeOnRightOnSets
  
  > OrbitStabilizerInUnitCubeOnRightOnSets( group, set ) _______________method
  Returns:  A record containing
  
  --              .stabilizer: the stabilizer of set.
  
  --              .orbit  set of sets of vectors from [0,1)^n which represents
                  the orbit.
  
  Calculates   orbit   and   stabilizer   of   a   set  of  vectors.  Just  as
  OrbitStabilizerInUnitCubeOnRight  (3.1-1),  it needs input from [0,1)^n. The
  returned  orbit  part  .orbit  is  a  set of sets such that every element of
  .orbit  has  a  non-trivial  intersection with the cube [0,1)^n. In general,
  these sets will not lie inside [0,1)^n completely.
  
  ---------------------------  Example  ----------------------------
    gap> S:=SpaceGroup(3,5);;
    gap> OrbitStabilizerInUnitCubeOnRightOnSets(S,[[0,0,0],[0,1/2,0]]);
    rec( orbit := [ [ [ -1/2, 0, 0 ], [ 0, 0, 0 ] ], 
                    [ [ 0, 0, 0 ], [ 0, 1/2, 0 ] ],
                    [ [ 1/2, 0, 0 ], [ 1, 0, 0 ] ] ],
      stabilizer := Group([ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], 
                            [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ] ]) )
  ------------------------------------------------------------------
  
  3.1-3 OrbitPartInVertexSetsStandardSpaceGroup
  
  > OrbitPartInVertexSetsStandardSpaceGroup( group, vertexset, allvertices ) method
  Returns:  Set of subsets of allvertices.
  
  If  allvertices  is a set of vectors and vertexset is a subset thereof, then
  OrbitPartInVertexSetsStandardSpaceGroup  returns  that  part of the orbit of
  vertexset   which   consists   entirely  of  subsets  of  allvertices.  Note
  that,unlike  the other OrbitStabilizer algorithms, this does not require the
  input to lie in some particular part of the space.
  
  ---------------------------  Example  ----------------------------
    gap> S:=SpaceGroup(3,5);;
    gap> OrbitPartInVertexSetsStandardSpaceGroup(S,[[0,1,5],[1,2,0]],
    > Set([[1,2,0],[2,3,1],[1,2,6],[1,1,0],[0,1,5],[3/5,7,12],[1/17,6,1/2]]));
    [ [ [ 0, 1, 5 ], [ 1, 2, 0 ] ], [ [ 1, 2, 6 ], [ 2, 3, 1 ] ] ]
    gap> OrbitPartInVertexSetsStandardSpaceGroup(S, [[1,2,0]],
    > Set([[1,2,0],[2,3,1],[1,2,6],[1,1,0],[0,1,5],[3/5,7,12],[1/17,6,1/2]]));
    [ [ [ 0, 1, 5 ] ], [ [ 1, 1, 0 ] ], [ [ 1, 2, 0 ] ], [ [ 1, 2, 6 ] ], [ [ 2, 3, 1 ] ] ]
  ------------------------------------------------------------------
  
  3.1-4 OrbitPartInFacesStandardSpaceGroup
  
  > OrbitPartInFacesStandardSpaceGroup( group, vertexset, faceset ) ____method
  Returns:  Set of subsets of faceset.
  
  This  calculates  the  orbit of a space group on sets restricted to a set of
  faces.
  If  faceset  is  a  set  of  sets  of vectors and vertexset is an element of
  faceset,  then  OrbitPartInFacesStandardSpaceGroup  returns that part of the
  orbit of vertexset which consists entirely of elements of faceset.
  Note that,unlike the other OrbitStabilizer algorithms, this does not require
  the input to lie in some particular part of the space.
  
  3.1-5 OrbitPartAndRepresentativesInFacesStandardSpaceGroup
  
  > OrbitPartAndRepresentativesInFacesStandardSpaceGroup( group, vertexset, faceset ) method
  Returns:  A set of face-matrix pairs .
  
  This  is  a  slight  variation of OrbitPartInFacesStandardSpaceGroup (3.1-4)
  that also returns a representative for every orbit element.
  
  ---------------------------  Example  ----------------------------
    gap> S:=SpaceGroup(3,5);;
    gap> OrbitPartInVertexSetsStandardSpaceGroup(S,[[0,1,5],[1,2,0]],
    > Set([[1,2,0],[2,3,1],[1,2,6],[1,1,0],[0,1,5],[3/5,7,12],[1/17,6,1/2]]));
    [ [ [ 0, 1, 5 ], [ 1, 2, 0 ] ], [ [ 1, 2, 6 ], [ 2, 3, 1 ] ] ]
    gap> OrbitPartInFacesStandardSpaceGroup(S,[[0,1,5],[1,2,0]],
    > Set( [ [ [ 0, 1, 5 ], [ 1, 2, 0 ] ], [[1/17,6,1/2],[1,2,7]]]));
    [ [ [ 0, 1, 5 ], [ 1, 2, 0 ] ] ]
    gap> OrbitPartAndRepresentativesInFacesStandardSpaceGroup(S,[[0,1,5],[1,2,0]],
    > Set( [ [ [ 0, 1, 5 ], [ 1, 2, 0 ] ], [[1/17,6,1/2],[1,2,7]]]));
    [ [ [ [ 0, 1, 5 ], [ 1, 2, 0 ] ],
          [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ] ] ]
  ------------------------------------------------------------------
  
  3.1-6 StabilizerOnSetsStandardSpaceGroup
  
  > StabilizerOnSetsStandardSpaceGroup( group, set ) ___________________method
  Returns:  finite group of affine matrices (OnRight)
  
  Given  a  set  set of vectors and a space group group in standard form, this
  method  calculates  the  stabilizer of that set in the full crystallographic
  group.
  
  ---------------------------  Example  ----------------------------
    
    gap> G:=SpaceGroup(3,12);;
    gap> v:=[ 0, 0,0 ];;
    gap> s:=StabilizerOnSetsStandardSpaceGroup(G,[v]);
    <matrix group with 2 generators>
    gap> s2:=OrbitStabilizerInUnitCubeOnRight(G,v).stabilizer;
    <matrix group with 2 generators>
    gap> s2=s;
    true
    
  ------------------------------------------------------------------
  
  3.1-7 RepresentativeActionOnRightOnSets
  
  > RepresentativeActionOnRightOnSets( group, set, imageset ) __________method
  Returns:  Affine matrix.
  
  Returns  an  element of the space group S which takes the set set to the set
  imageset. The group must be in standard form and act on the right.
  
  ---------------------------  Example  ----------------------------
    gap> S:=SpaceGroup(3,5);;
    gap> RepresentativeActionOnRightOnSets(G, [[0,0,0],[0,1/2,0]],
    >        [ [ 0, 1/2, 0 ], [ 0, 1, 0 ] ]);
    [ [ 0, -1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 1, 0, 1 ] ]
  ------------------------------------------------------------------
  
  
  3.1-8 Getting other orbit parts
  
  HAPcryst  does  not  calculate the full orbit but only the part of it having
  coefficients  between  -1/2  and  1/2.  The  other parts of the orbit can be
  calculated using the following functions.
  
  3.1-9 ShiftedOrbitPart
  
  > ShiftedOrbitPart( point, orbitpart ) _______________________________method
  Returns:  Set of vectors
  
  Takes each vector in orbitpart to the cube unit cube centered in point.
  
  ---------------------------  Example  ----------------------------
    gap> ShiftedOrbitPart([0,0,0],[[1/2,1/2,1/3],-[1/2,1/2,1/2],[19,3,1]]);
    [ [ 1/2, 1/2, 1/3 ], [ 1/2, 1/2, 1/2 ], [ 0, 0, 0 ] ]
    gap> ShiftedOrbitPart([1,1,1],[[1/2,1/2,1/2],-[1/2,1/2,1/2]]);
    [ [ 3/2, 3/2, 3/2 ] ]
  ------------------------------------------------------------------
  
  3.1-10 TranslationsToOneCubeAroundCenter
  
  > TranslationsToOneCubeAroundCenter( point, center ) _________________method
  Returns:  List of integer vectors
  
  This  method  returns  the list of all integer vectors which translate point
  into the box center+[-1/2,1/2]^n
  
  ---------------------------  Example  ----------------------------
    gap> TranslationsToOneCubeAroundCenter([1/2,1/2,1/3],[0,0,0]);
    [ [ 0, 0, 0 ], [ 0, -1, 0 ], [ -1, 0, 0 ], [ -1, -1, 0 ] ]
    gap> TranslationsToOneCubeAroundCenter([1,0,1],[0,0,0]);
    [ [ -1, 0, -1 ] ]
  ------------------------------------------------------------------
  
  3.1-11 TranslationsToBox
  
  > TranslationsToBox( point, box ) ____________________________________method
  Returns:  An iterator of integer vectors or the empty iterator
  
  Given  a vector v and a list of pairs, this function returns the translation
  vectors  (integer vectors) which take v into the box box. The box box has to
  be given as a list of pairs.
  
  ---------------------------  Example  ----------------------------
    gap> TranslationsToBox([0,0],[[1/2,2/3],[1/2,2/3]]);
    [  ]
    gap> TranslationsToBox([0,0],[[-3/2,1/2],[1,4/3]]);
    [ [ -1, 1 ], [ 0, 1 ] ]
    gap> TranslationsToBox([0,0],[[-3/2,1/2],[2,1]]);
    Error, Box must not be empty called from
    ...
  ------------------------------------------------------------------