[1X3. Algorithms of Orbit-Stabilizer Type[0X We introduce a way to calculate a sufficient part of an orbit and the stabilizer of a point. [1X3.1 Orbit Stabilizer for Crystallographic Groups[0X [1X3.1-1 OrbitStabilizerInUnitCubeOnRight[0X [2X> OrbitStabilizerInUnitCubeOnRight( [0X[3Xgroup, x[0X[2X ) _______________________[0Xmethod [6XReturns:[0X A record containing -- [9X.stabilizer[0X: the stabilizer of [3Xx[0X. -- [9X.orbit[0X set of vectors from [0,1)^n which represents the orbit. Let [3Xx[0X be a rational vector from [0,1)^n and [3Xgroup[0X 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 [3Xx[0X. 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 [10XOrbitStabilizerInUnitCubeOnRight[0X should be an element of the returned orbit (part). [4X--------------------------- Example ----------------------------[0X [4X [0X [4Xgap> S:=SpaceGroup(3,5);;[0X [4Xgap> OrbitStabilizerInUnitCubeOnRight(S,[1/2,0,9/11]); [0X [4Xrec( orbit := [ [ 0, 1/2, 2/11 ], [ 1/2, 0, 9/11 ] ], [0X [4X stabilizer := Group([ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [0X [4X [ 0, 0, 0, 1 ] ] ]) )[0X [4Xgap> OrbitStabilizerInUnitCubeOnRight(S,[0,0,0]); [0X [4Xrec( orbit := [ [ 0, 0, 0 ] ], stabilizer := <matrix group with 2 generators> )[0X [4X[0X [4X------------------------------------------------------------------[0X If you are interested in other parts of the orbit, you can use [2XVectorModOne[0X ([14X2.1-2[0X) for the base point and the functions [2XShiftedOrbitPart[0X ([14X3.1-9[0X), [2XTranslationsToOneCubeAroundCenter[0X ([14X3.1-10[0X) and [2XTranslationsToBox[0X ([14X3.1-11[0X) for the resulting orbit Suppose we want to calculate the part of the orbit of [10X[4/3,5/3,7/3][0X in the cube of sidelength [10X1[0X around this point: [4X--------------------------- Example ----------------------------[0X [4Xgap> S:=SpaceGroup(3,5);;[0X [4Xgap> p:=[4/3,5/3,7/3];;[0X [4Xgap> o:=OrbitStabilizerInUnitCubeOnRight(S,VectorModOne(p)).orbit;[0X [4X[ [ 1/3, 2/3, 1/3 ], [ 1/3, 2/3, 2/3 ] ][0X [4Xgap> box:=p+[[-1,1],[-1,1],[-1,1]];[0X [4X[ [ 1/3, 8/3, 7/3 ], [ 1/3, 8/3, 7/3 ], [ 1/3, 8/3, 7/3 ] ][0X [4Xgap> o2:=Concatenation(List(o,i->i+TranslationsToBox(i,box)));;[0X [4Xgap> # This is what we looked for. But it is somewhat large:[0X [4Xgap> Size(o2);[0X [4X54[0X [4X------------------------------------------------------------------[0X [1X3.1-2 OrbitStabilizerInUnitCubeOnRightOnSets[0X [2X> OrbitStabilizerInUnitCubeOnRightOnSets( [0X[3Xgroup, set[0X[2X ) _______________[0Xmethod [6XReturns:[0X A record containing -- [9X.stabilizer[0X: the stabilizer of [3Xset[0X. -- [9X.orbit[0X set of sets of vectors from [0,1)^n which represents the orbit. Calculates orbit and stabilizer of a set of vectors. Just as [2XOrbitStabilizerInUnitCubeOnRight[0X ([14X3.1-1[0X), it needs input from [0,1)^n. The returned orbit part [9X.orbit[0X is a set of sets such that every element of [9X.orbit[0X has a non-trivial intersection with the cube [0,1)^n. In general, these sets will not lie inside [0,1)^n completely. [4X--------------------------- Example ----------------------------[0X [4Xgap> S:=SpaceGroup(3,5);;[0X [4Xgap> OrbitStabilizerInUnitCubeOnRightOnSets(S,[[0,0,0],[0,1/2,0]]);[0X [4Xrec( orbit := [ [ [ -1/2, 0, 0 ], [ 0, 0, 0 ] ], [0X [4X [ [ 0, 0, 0 ], [ 0, 1/2, 0 ] ],[0X [4X [ [ 1/2, 0, 0 ], [ 1, 0, 0 ] ] ],[0X [4X stabilizer := Group([ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [0X [4X [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ] ]) )[0X [4X------------------------------------------------------------------[0X [1X3.1-3 OrbitPartInVertexSetsStandardSpaceGroup[0X [2X> OrbitPartInVertexSetsStandardSpaceGroup( [0X[3Xgroup, vertexset, allvertices[0X[2X ) [0Xmethod [6XReturns:[0X Set of subsets of [3Xallvertices[0X. If [3Xallvertices[0X is a set of vectors and [3Xvertexset[0X is a subset thereof, then [2XOrbitPartInVertexSetsStandardSpaceGroup[0X returns that part of the orbit of [3Xvertexset[0X which consists entirely of subsets of [3Xallvertices[0X. Note that,unlike the other [10XOrbitStabilizer[0X algorithms, this does not require the input to lie in some particular part of the space. [4X--------------------------- Example ----------------------------[0X [4Xgap> S:=SpaceGroup(3,5);;[0X [4Xgap> OrbitPartInVertexSetsStandardSpaceGroup(S,[[0,1,5],[1,2,0]],[0X [4X> 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]]));[0X [4X[ [ [ 0, 1, 5 ], [ 1, 2, 0 ] ], [ [ 1, 2, 6 ], [ 2, 3, 1 ] ] ][0X [4Xgap> OrbitPartInVertexSetsStandardSpaceGroup(S, [[1,2,0]],[0X [4X> 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]]));[0X [4X[ [ [ 0, 1, 5 ] ], [ [ 1, 1, 0 ] ], [ [ 1, 2, 0 ] ], [ [ 1, 2, 6 ] ], [ [ 2, 3, 1 ] ] ][0X [4X------------------------------------------------------------------[0X [1X3.1-4 OrbitPartInFacesStandardSpaceGroup[0X [2X> OrbitPartInFacesStandardSpaceGroup( [0X[3Xgroup, vertexset, faceset[0X[2X ) ____[0Xmethod [6XReturns:[0X Set of subsets of [3Xfaceset[0X. This calculates the orbit of a space group on sets restricted to a set of faces. If [3Xfaceset[0X is a set of sets of vectors and [3Xvertexset[0X is an element of [3Xfaceset[0X, then [2XOrbitPartInFacesStandardSpaceGroup[0X returns that part of the orbit of [3Xvertexset[0X which consists entirely of elements of [3Xfaceset[0X. Note that,unlike the other [10XOrbitStabilizer[0X algorithms, this does not require the input to lie in some particular part of the space. [1X3.1-5 OrbitPartAndRepresentativesInFacesStandardSpaceGroup[0X [2X> OrbitPartAndRepresentativesInFacesStandardSpaceGroup( [0X[3Xgroup, vertexset, faceset[0X[2X ) [0Xmethod [6XReturns:[0X A set of face-matrix pairs . This is a slight variation of [2XOrbitPartInFacesStandardSpaceGroup[0X ([14X3.1-4[0X) that also returns a representative for every orbit element. [4X--------------------------- Example ----------------------------[0X [4Xgap> S:=SpaceGroup(3,5);;[0X [4Xgap> OrbitPartInVertexSetsStandardSpaceGroup(S,[[0,1,5],[1,2,0]],[0X [4X> 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]]));[0X [4X[ [ [ 0, 1, 5 ], [ 1, 2, 0 ] ], [ [ 1, 2, 6 ], [ 2, 3, 1 ] ] ][0X [4Xgap> OrbitPartInFacesStandardSpaceGroup(S,[[0,1,5],[1,2,0]],[0X [4X> Set( [ [ [ 0, 1, 5 ], [ 1, 2, 0 ] ], [[1/17,6,1/2],[1,2,7]]]));[0X [4X[ [ [ 0, 1, 5 ], [ 1, 2, 0 ] ] ][0X [4Xgap> OrbitPartAndRepresentativesInFacesStandardSpaceGroup(S,[[0,1,5],[1,2,0]],[0X [4X> Set( [ [ [ 0, 1, 5 ], [ 1, 2, 0 ] ], [[1/17,6,1/2],[1,2,7]]]));[0X [4X[ [ [ [ 0, 1, 5 ], [ 1, 2, 0 ] ],[0X [4X [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ] ] ][0X [4X------------------------------------------------------------------[0X [1X3.1-6 StabilizerOnSetsStandardSpaceGroup[0X [2X> StabilizerOnSetsStandardSpaceGroup( [0X[3Xgroup, set[0X[2X ) ___________________[0Xmethod [6XReturns:[0X finite group of affine matrices (OnRight) Given a set [3Xset[0X of vectors and a space group [3Xgroup[0X in standard form, this method calculates the stabilizer of that set in the full crystallographic group. [4X--------------------------- Example ----------------------------[0X [4X[0X [4Xgap> G:=SpaceGroup(3,12);;[0X [4Xgap> v:=[ 0, 0,0 ];;[0X [4Xgap> s:=StabilizerOnSetsStandardSpaceGroup(G,[v]);[0X [4X<matrix group with 2 generators>[0X [4Xgap> s2:=OrbitStabilizerInUnitCubeOnRight(G,v).stabilizer;[0X [4X<matrix group with 2 generators>[0X [4Xgap> s2=s;[0X [4Xtrue[0X [4X[0X [4X------------------------------------------------------------------[0X [1X3.1-7 RepresentativeActionOnRightOnSets[0X [2X> RepresentativeActionOnRightOnSets( [0X[3Xgroup, set, imageset[0X[2X ) __________[0Xmethod [6XReturns:[0X Affine matrix. Returns an element of the space group S which takes the set [3Xset[0X to the set [3Ximageset[0X. The group must be in standard form and act on the right. [4X--------------------------- Example ----------------------------[0X [4Xgap> S:=SpaceGroup(3,5);;[0X [4Xgap> RepresentativeActionOnRightOnSets(G, [[0,0,0],[0,1/2,0]],[0X [4X> [ [ 0, 1/2, 0 ], [ 0, 1, 0 ] ]);[0X [4X[ [ 0, -1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 1, 0, 1 ] ][0X [4X------------------------------------------------------------------[0X [1X3.1-8 Getting other orbit parts[0X [5XHAPcryst[0X 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. [1X3.1-9 ShiftedOrbitPart[0X [2X> ShiftedOrbitPart( [0X[3Xpoint, orbitpart[0X[2X ) _______________________________[0Xmethod [6XReturns:[0X Set of vectors Takes each vector in [3Xorbitpart[0X to the cube unit cube centered in [3Xpoint[0X. [4X--------------------------- Example ----------------------------[0X [4Xgap> ShiftedOrbitPart([0,0,0],[[1/2,1/2,1/3],-[1/2,1/2,1/2],[19,3,1]]);[0X [4X[ [ 1/2, 1/2, 1/3 ], [ 1/2, 1/2, 1/2 ], [ 0, 0, 0 ] ][0X [4Xgap> ShiftedOrbitPart([1,1,1],[[1/2,1/2,1/2],-[1/2,1/2,1/2]]);[0X [4X[ [ 3/2, 3/2, 3/2 ] ][0X [4X------------------------------------------------------------------[0X [1X3.1-10 TranslationsToOneCubeAroundCenter[0X [2X> TranslationsToOneCubeAroundCenter( [0X[3Xpoint, center[0X[2X ) _________________[0Xmethod [6XReturns:[0X List of integer vectors This method returns the list of all integer vectors which translate [3Xpoint[0X into the box [3Xcenter[0X+[-1/2,1/2]^n [4X--------------------------- Example ----------------------------[0X [4Xgap> TranslationsToOneCubeAroundCenter([1/2,1/2,1/3],[0,0,0]);[0X [4X[ [ 0, 0, 0 ], [ 0, -1, 0 ], [ -1, 0, 0 ], [ -1, -1, 0 ] ][0X [4Xgap> TranslationsToOneCubeAroundCenter([1,0,1],[0,0,0]);[0X [4X[ [ -1, 0, -1 ] ][0X [4X------------------------------------------------------------------[0X [1X3.1-11 TranslationsToBox[0X [2X> TranslationsToBox( [0X[3Xpoint, box[0X[2X ) ____________________________________[0Xmethod [6XReturns:[0X 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 [3Xbox[0X. The box [3Xbox[0X has to be given as a list of pairs. [4X--------------------------- Example ----------------------------[0X [4Xgap> TranslationsToBox([0,0],[[1/2,2/3],[1/2,2/3]]);[0X [4X[ ][0X [4Xgap> TranslationsToBox([0,0],[[-3/2,1/2],[1,4/3]]);[0X [4X[ [ -1, 1 ], [ 0, 1 ] ][0X [4Xgap> TranslationsToBox([0,0],[[-3/2,1/2],[2,1]]);[0X [4XError, Box must not be empty called from[0X [4X...[0X [4X------------------------------------------------------------------[0X