We introduce a way to calculate a sufficient part of an orbit and the
stabilizer of a point.
<Section><Heading>Orbit Stabilizer for Crystallographic Groups</Heading>
<ManSection>
<Meth Name="OrbitStabilizerInUnitCubeOnRight"
Arg="group, x"/>
<Returns>
A record containing
<List>
<Item>
<K>.stabilizer</K>: the stabilizer of <Arg>x</Arg>.
</Item>
<Item>
<K>.orbit</K> set of vectors from <M>[0,1)^n</M> which
represents the orbit.
</Item>
</List>
</Returns>
<Description>
Let <Arg>x</Arg> be a rational vector from <M>[0,1)^n</M> and
<Arg>group</Arg> a space group in standard form.
The function then calculates the part of the orbit which lies inside the
cube <M>[0,1)^n</M> and the stabilizer of <Arg>x</Arg>. Observe that every element of the full orbit differs from a point in the returned orbit only
by a pure translation.
</Description>
</ManSection>
Note that the restriction to points from <M>[0,1)^n</M> makes sense if orbits
should be compared and the vector passed to
<C>OrbitStabilizerInUnitCubeOnRight</C> should be an element of the returned
orbit (part).
If you are interested in other parts of the orbit, you can use <Ref
Meth="VectorModOne"/> for the base point and the functions <Ref
Meth="ShiftedOrbitPart"/>, <Ref Meth="TranslationsToOneCubeAroundCenter"/> and
<Ref Meth="TranslationsToBox"/> for the resulting orbit<Br/>
Suppose we want to calculate the part of the orbit of <C>[4/3,5/3,7/3]</C> in
the cube of sidelength <C>1</C> 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
</Example>
<ManSection>
<Meth Name="OrbitStabilizerInUnitCubeOnRightOnSets"
Arg="group, set"/>
<Returns>
A record containing
<List>
<Item>
<K>.stabilizer</K>: the stabilizer of <Arg>set</Arg>.
</Item>
<Item>
<K>.orbit</K> set of sets of vectors from <M>[0,1)^n</M> which
represents the orbit.
</Item>
</List>
</Returns>
<Description>
Calculates orbit and stabilizer of a set of vectors. Just as <Ref
Meth="OrbitStabilizerInUnitCubeOnRight"></Ref>, it needs input from
<M>[0,1)^n</M>.
The returned orbit part <K>.orbit</K> is a set of sets such that every element of <K>.orbit</K> has a non-trivial intersection with the
cube <M>[0,1)^n</M>. In general, these sets will not lie inside
<M>[0,1)^n</M> completely.
</Description>
</ManSection>
<ManSection>
<Meth Name="OrbitPartInVertexSetsStandardSpaceGroup"
Arg="group vertexset allvertices"/>
<Returns>
Set of subsets of <A>allvertices</A>.
</Returns>
<Description>
If <A>allvertices</A> is a set of vectors and <A>vertexset</A> is
a subset thereof, then <Ref
Meth="OrbitPartInVertexSetsStandardSpaceGroup"></Ref> returns
that part of the orbit of <A>vertexset</A> which consists entirely of
subsets of <A>allvertices</A>.
Note that,unlike the other <C>OrbitStabilizer</C> algorithms, this does not
require the input to lie in some particular part of the space.
</Description>
</ManSection>
<ManSection>
<Meth Name="OrbitPartInFacesStandardSpaceGroup"
Arg="group vertexset faceset"/>
<Returns>
Set of subsets of <A>faceset</A>.
</Returns>
<Description>
This calculates the orbit of a space group on sets restricted to a set of
faces.<Br/>
If <A>faceset</A> is a set of sets of vectors and <A>vertexset</A> is
an element of <A>faceset</A>, then <Ref
Meth="OrbitPartInFacesStandardSpaceGroup"></Ref> returns
that part of the orbit of <A>vertexset</A> which consists entirely of
elements of <A>faceset</A>.<Br/>
Note that,unlike the other <C>OrbitStabilizer</C> algorithms, this does not
require the input to lie in some particular part of the space.
</Description>
</ManSection>
<ManSection>
<Meth Name="OrbitPartAndRepresentativesInFacesStandardSpaceGroup"
Arg="group vertexset faceset"/>
<Returns>
A set of face-matrix pairs .
</Returns>
<Description>
This is a slight variation of
<Ref Meth="OrbitPartInFacesStandardSpaceGroup"></Ref>
that also returns a representative for every orbit element.
</Description>
</ManSection>
<ManSection>
<Meth Name="StabilizerOnSetsStandardSpaceGroup"
Arg="group set"/>
<Returns>finite group of affine matrices (OnRight)</Returns>
<Description>
Given a set <A>set</A> of vectors and a space group <A>group</A> in
standard form, this method calculates the stabilizer of that set in
the full crystallographic group.<Br/>
</Description>
</ManSection>
<Example>
<![CDATA[
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
]]>
</Example>
<ManSection>
<Meth Name="RepresentativeActionOnRightOnSets"
Arg="group set imageset"/>
<Returns>
Affine matrix.
</Returns>
<Description>
Returns an element of the space group
<M>S</M> which takes the set <A>set</A> to the set
<A>imageset</A>. The group must be in standard form and act on the right.
</Description>
</ManSection>
<Subsection><Heading>Getting other orbit parts</Heading>
<Package>HAPcryst</Package> does not calculate the full orbit but only the part
of it having coefficients between <M>-1/2</M> and <M>1/2</M>. The other parts
of the orbit can be calculated using the following functions.
</Subsection>
<ManSection>
<Meth Name="ShiftedOrbitPart" Arg="point, orbitpart"/>
<Returns>Set of vectors </Returns>
<Description>
Takes each vector in <A>orbitpart</A> to the cube unit cube centered in
<A>point</A>.
</Description>
</ManSection>
<ManSection>
<Meth Name="TranslationsToOneCubeAroundCenter" Arg="point, center"/>
<Returns>List of integer vectors</Returns>
<Description>
This method returns the list of all integer vectors which translate
<A>point</A> into the box <A>center</A><M>+[-1/2,1/2]^n</M>
</Description>
</ManSection>
<ManSection>
<Meth Name="TranslationsToBox" Arg="point, box"/> <!-- <Returns>List of integer vectors or the empty list</Returns>-->
<Returns>An iterator of integer vectors or the empty iterator</Returns>
<Description>
Given a vector <M>v</M> and a list of pairs, this function returns the
translation vectors (integer vectors) which take <M>v</M> into the box
<A>box</A>. The box <A>box</A> has to be given as a list of pairs.
</Description>
</ManSection>
<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
...
</Example>
</Section>
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