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#############################################################################
##
#W spacegroups.gi HAPcryst package Marc Roeder
##
##
##
##
#Y Copyright (C) 2006 Marc Roeder
#Y
#Y This program is free software; you can redistribute it and/or
#Y modify it under the terms of the GNU General Public License
#Y as published by the Free Software Foundation; either version 2
#Y of the License, or (at your option) any later version.
#Y
#Y This program is distributed in the hope that it will be useful,
#Y but WITHOUT ANY WARRANTY; without even the implied warranty of
#Y MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
#Y GNU General Public License for more details.
#Y
#Y You should have received a copy of the GNU General Public License
#Y along with this program; if not, write to the Free Software
#Y Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA
##
InstallMethod(PointGroupRepresentatives,"for crystallographic groups on right",
[IsAffineCrystGroupOnLeftOrRight],
function(group)
local linearPart, walkthetreeright, walkthetreeleft,
generators, dim, affinepointgroupels, pointgroupels;
##########
# local methods for fast access to translational and linear part:
#
linearPart:=function(mat)
return mat{[1..dim-1]}{[1..dim-1]};
end;
##########
# A recursive method to generate representatives for the point group.
# This also finds a translation basis.
# Note that the results are stored in global (not belonging to
# "walkthetree") variables.
#
walkthetreeright:=function(element,gens)
local g, endofbranch, newel, newellin, differences,
newtrans;
for g in gens
do
newel:=element*g;
newellin:=linearPart(newel);
if not (newellin in pointgroupels)
then
AddSet(affinepointgroupels,newel);
AddSet(pointgroupels,newellin);
walkthetreeright(newel,gens);
fi;
od;
end;
walkthetreeleft:=function(element,gens)
local g, endofbranch, newel, newellin, differences,
newtrans;
for g in gens
do
newel:=g*element;
newellin:=linearPart(newel);
if not (newellin in pointgroupels)
then
AddSet(affinepointgroupels,newel);
AddSet(pointgroupels,newellin);
walkthetreeleft(newel,gens);
fi;
od;
end;
##########
#And here is the main program:
#
generators:=GeneratorsOfGroup(group);
dim:=DimensionOfMatrixGroup(group);
##########
# Now generate the pointgroup and representatives for it.
#
affinepointgroupels:=[IdentityMat(dim)];
pointgroupels:=[IdentityMat(dim-1)];
if IsAffineCrystGroupOnRight(group)
then
walkthetreeright(generators[1]^0,generators);
if not Size(PointGroup(AffineCrystGroupOnRight(generators)))=Size(affinepointgroupels)
then
Error("generation of point group failed!");
fi;
else
walkthetreeleft(generators[1]^0,generators);
if not Size(PointGroup(AffineCrystGroupOnLeft(generators)))=Size(affinepointgroupels)
then
Error("generation of point group failed!");
fi;
fi;
return Set(affinepointgroupels);
end);
#############################################################################
#
# Let $S$ be an $n$-dimensional crystallographic group in standard form.
# For a given point $x$ in $[0,1]^n$, we determine the stabilizer in $S$
# and the part of the orbit which lies inside $[0,1]^n$.
#
# If the linear part of the group elements are orthogonal with respect to
# the euclidian scalar product, all elements of the orbit lying in $[-1,1]^n$
# (and a few more) are returned. In all other cases, this is not guaranteed.
#
InstallMethod(OrbitStabilizerInUnitCubeOnRight,"for space groups on right",
[IsGroup,IsVector],
function(group,x)
local orbitpart, stabilizer, xaff, pointgroupreps, g,
imageaff, image, translation;
if not IsAffineCrystGroupOnRight(group) and IsStandardSpaceGroup(group)
then
Error("Group must be an AffineCrystGroupOnRight in standard form");
fi;
orbitpart:=[];
stabilizer:=[];
if not VectorModOne(x)=x
then
Error("please choose a vector from [0,1)^n");
fi;
xaff:=Concatenation(x,[1]);
Add(orbitpart,x);
pointgroupreps:=PointGroupRepresentatives(group);
for g in pointgroupreps
do
imageaff:=xaff*g;
image:=VectorModOne(imageaff){[1..Size(imageaff)-1]};
if not image in orbitpart
then
AddSet(orbitpart,image);
elif imageaff=xaff
then
Add(stabilizer,g);
elif image=x
then
translation:=IdentityMat(Size(xaff));
translation[Size(translation)]:=-imageaff+xaff;
translation[Size(translation)][Size(translation)]:=1;
Add(stabilizer,g*translation);
fi;
od;
return rec(orbit:=orbitpart,stabilizer:=Subgroup(group,stabilizer));
end);
#############################################################################
##
#O OrbitStabilzerInUniCubeOnRightOnSets(group,set)
##
InstallMethod(OrbitStabilizerInUnitCubeOnRightOnSets,
"for space groups on right",
[IsGroup,IsDenseList],
function(group,set)
local dim, stepsFromZeroOne, orbitpart, stabilizer, setaff,
pointgroupreps, g, imageaff, point, transvector,
image;
if not IsStandardSpaceGroup(group) and IsAffineCrystGroupOnRight(group)
then
Error("group must be a StandardSpaceGroup acting on right");
fi;
dim:=DimensionOfMatrixGroup(group);
if not Set(set,VectorModOne)=set or not Set(set,Size)=[dim-1]
then
Error("please choose a set of vectors from [0,1)^n");
fi;
if not IsSet(set) and ForAll(set,IsVector)
then
Error("set must be a Set of Vectors");
fi;
stepsFromZeroOne:=function(q)
local r;
r:=Int(q);
if q<0
then
return r-1;
else
return r;
fi;
end;
orbitpart:=[];
stabilizer:=[];
setaff:=Set(set,x->Concatenation(x,[1]));
Add(orbitpart,set);
pointgroupreps:=PointGroupRepresentatives(group);
for g in pointgroupreps
do
imageaff:=Set(setaff,s->s*g);
if imageaff=setaff
then
Add(stabilizer,g);
fi;
for point in imageaff
do
transvector:=List(point{[1..Size(point)-1]},
i->stepsFromZeroOne(i));
image:=Set(imageaff,i->i{[1..Size(point)-1]}-transvector);
if not image in orbitpart
then
AddSet(orbitpart,image);
elif image=set
then
Add(stabilizer,g*TranslationOnRightFromVector(-transvector));
fi;
od;
od;
return rec(orbit:=orbitpart,stabilizer:=Group(stabilizer));
end);
#############################################################################
##
#O OrbitPartInVertexSetsStandardSpaceGroup
##
# And a special function not only for polyhedrae:
#this takes a list of vertices (vectors) and returns the part of the
# orbit which consists of vertex sets.
#
# This doesn't even use that the vertices are all in [-1,1].
# It just works for arbitray polyhedra.
#
# works better for small <vertexset> than for large.
#
InstallMethod(OrbitPartInVertexSetsStandardSpaceGroup,
[IsGroup,IsDenseList,IsDenseList],
function(group,vertexset,allvertices)
local orbit, dim, pointGroupReps, setaff, newimagesize,
newimage, g, image, point, trans, index,
stillinvertices;
if not IsStandardSpaceGroup(group) or not IsAffineCrystGroupOnRight(group)
then
TryNextMethod();
fi;
if not (IsSet(vertexset)
and IsMatrix(vertexset)
and IsSet(allvertices)
and IsSubset(allvertices,vertexset))
then
Error("<vertexset> must be a subset of <allvertices> which must be a set of vectors");
fi;
dim:=Size(allvertices[1]);
orbit:=[vertexset];
pointGroupReps:=PointGroupRepresentatives(group);
setaff:=Set(vertexset,x->Concatenation(x,[1]));
newimagesize:=Size(setaff);
newimage:=[1..newimagesize];
for g in pointGroupReps
do
# imageaff:=Set(setaff,s->s*g);
image:=Set(setaff*g,i->i{[1..dim]});
point:=image[1];#{[1..dim]};
for trans in allvertices-point
do
if ForAll(trans,IsInt)
then
index:=1;
stillinvertices:=true;
while index<=newimagesize and stillinvertices
do
newimage[index]:=image[index]+trans;
if not newimage[index] in allvertices
then
stillinvertices:=false;
fi;
index:=index+1;
od;
if stillinvertices
then
Add(orbit,Set(newimage));
fi;
fi;
od;
od;
return Set(orbit);
end);
#############################################################################
##
#O OrbitPartInFacesStandardSpaceGroup
##
# And a special function not only for polyhedrae:
# this takes a set of vertices (vectors) and returns the part of the
# part of the orrbit lying inside a set of sets of vectors.
# This works like OnSets, but does only return a part of the orbit.
# <vertexset> has to be an element of <faceset>
#
# This doesn't even use that the vertices are all in [-1,1].
# It just works for arbitray polyhedra.
#
# works better for small <vertexset> than for large.
#
InstallMethod(OrbitPartInFacesStandardSpaceGroup,
[IsGroup,IsDenseList,IsDenseList],
function(group,vertexset,faceset)
local allvertices, currentfirst, facesbyfirstvertex,
currentfirstposition, face, i, dim, orbit,
pointGroupReps, setaff, newimagesize, g, image,
point, targetbatch, trans, index, thereisaface,
candidatefaces, newimage, newentry;
if not IsStandardSpaceGroup(group) or not IsAffineCrystGroupOnRight(group)
then
TryNextMethod();
fi;
if not (IsSet(vertexset)
and IsSet(faceset)
and ForAll(faceset,IsSet)
and vertexset in faceset
and IsMatrix(vertexset))
then
Error("<vertexset> must be an element of <faceset> which must be a set of sets of vectors");
fi;
# all vertices that turn up in a given position in any face:
# and all faces partitioned by there first element.
#
allvertices:=List([1..Maximum(List(faceset,Size))],i->[]);
currentfirst:=[];
facesbyfirstvertex:=[];
currentfirstposition:=0;
for face in faceset
do
for i in [1..Size(face)]
do
Add(allvertices[i],face[i]);
if i=1
then
if currentfirst=face[1]
then
Add(facesbyfirstvertex[currentfirstposition],face);
else
currentfirst:=face[1];
currentfirstposition:=currentfirstposition+1;
facesbyfirstvertex[currentfirstposition]:=[face];
fi;
fi;
od;
od;
Apply(allvertices,Set);
MakeImmutable(allvertices);
dim:=Size(faceset[1][1]);
orbit:=[];
pointGroupReps:=Set(PointGroupRepresentatives(group));
setaff:=List(vertexset,x->Concatenation(x,[1]));
newimagesize:=Size(setaff);
for g in pointGroupReps
do
image:=Set(setaff*g,i->i{[1..dim]});
point:=image[1];
#<point>+trans has to be the smallest vector in the image set:
for targetbatch in facesbyfirstvertex
do
trans:=targetbatch[1][1]-point;
if ForAll(trans,IsInt)
then
index:=1;
thereisaface:=true;
candidatefaces:=targetbatch;
newimage:=[targetbatch[1][1]];
while index<=newimagesize and thereisaface
do
newentry:=image[index]+trans;
newimage[index]:=newentry;
if not newentry in allvertices[index]
then
thereisaface:=false;
else
candidatefaces:=Filtered(candidatefaces,c->c[index]=newentry);
index:=index+1;
fi;
if candidatefaces=[]
then
thereisaface:=false;
fi;
od;
if thereisaface
then
Add(orbit,newimage);
fi;
fi;
od;
od;
return Set(orbit);
end);
#############################################################################
##
#O OrbitPartAndRepresentativesInFacesStandardSpaceGroup
##
# And a special function not only for polyhedrae:
# this takes a set of vertices (vectors) and returns the part of the
# part of the orrbit lying inside a set of sets of vectors.
# This works like OnSets, but does only return a part of the orbit.
# <vertexset> has to be an element of <faceset>.
#
# This doesn't even use that the vertices are all in [-1,1].
# It just works for arbitray polyhedra.
#
# works better for small <vertexset> than for large.
#
# the output is a list of pairs, the first entry of each pair is an orbit
# element and the second one a representative taking <vertexset> to that
# orbit element.
#
InstallMethod(OrbitPartAndRepresentativesInFacesStandardSpaceGroup,
[IsGroup,IsDenseList,IsDenseList],
function(group,vertexset,faceset)
local allvertices, currentfirst, facesbyfirstvertex,
currentfirstposition, face, i, dim, orbit, reps,
pointGroupReps, setaff, newimagesize, g, image,
point, targetbatch, trans, index, thereisaface,
candidatefaces, newimage, newentry, representative;
if not IsStandardSpaceGroup(group) or not IsAffineCrystGroupOnRight(group)
then
TryNextMethod();
fi;
if not (IsSet(vertexset)
and IsSet(faceset)
and ForAll(faceset,IsSet)
and vertexset in faceset
and IsMatrix(vertexset))
then
Error("<vertexset> must be an element of <faceset> which must be a set of sets of vectors");
fi;
# all vertices that turn up in a given position in any face:
# and all faces partitioned by there first element.
#
allvertices:=List([1..Maximum(List(faceset,Size))],i->[]);
currentfirst:=[];
facesbyfirstvertex:=[];
currentfirstposition:=0;
for face in faceset
do
for i in [1..Size(face)]
do
Add(allvertices[i],face[i]);
if i=1
then
if currentfirst=face[1]
then
Add(facesbyfirstvertex[currentfirstposition],face);
else
currentfirst:=face[1];
currentfirstposition:=currentfirstposition+1;
facesbyfirstvertex[currentfirstposition]:=[face];
fi;
fi;
od;
od;
Apply(allvertices,Set);
MakeImmutable(allvertices);
dim:=Size(faceset[1][1]);
orbit:=[];
reps:=[];
pointGroupReps:=Set(PointGroupRepresentatives(group));
setaff:=List(vertexset,x->Concatenation(x,[1]));
newimagesize:=Size(setaff);
for g in pointGroupReps
do
image:=Set(setaff*g,i->i{[1..dim]});
point:=image[1];
#<point>+trans has to be the smallest vector in the image set:
for targetbatch in facesbyfirstvertex
do
trans:=targetbatch[1][1]-point;
if ForAll(trans,IsInt)
then
index:=2;
thereisaface:=true;
candidatefaces:=targetbatch;
newimage:=[targetbatch[1][1]];
while index<=newimagesize and thereisaface
do
newentry:=image[index]+trans;
newimage[index]:=newentry;
if not newentry in allvertices[index]
then
thereisaface:=false;
else
candidatefaces:=Filtered(candidatefaces,c->c[index]=newentry);
index:=index+1;
fi;
if candidatefaces=[]
then
thereisaface:=false;
fi;
od;
if thereisaface
then
if not newimage in orbit
then
representative:=ShallowCopy(g);
representative[dim+1]:=representative[dim+1]+trans;
Add(orbit,newimage);
Add(reps,representative);
SortParallel(orbit,reps);
fi;
fi;
fi;
od;
od;
return List([1..Size(orbit)],i->[orbit[i],reps[i]]);
end);
#############################################################################
##
#O StabilizerOnSetsStandardSpaceGroup
##
##
InstallMethod(StabilizerOnSetsStandardSpaceGroup,
[IsGroup,IsDenseList],
function(group,set)
local pointGroupReps, dim, stabgens, setaff, representative,
alpha, image, trans, addel;
if not (IsStandardSpaceGroup(group) and IsAffineCrystGroupOnRight(group))
then
TryNextMethod();
fi;
pointGroupReps:=PointGroupRepresentatives(group);
if not IsSet(set) and IsMatrix(set)
then
Error("<set> must be a set of vectors");
fi;
dim:=Size(set[1]);
if not dim=DimensionOfMatrixGroup(group)-1
then
Error("group and vectors don't fit");
fi;
stabgens:=[];
setaff:=List(set,x->Concatenation(x,[1]));
representative:=setaff[1];
for alpha in pointGroupReps
do
image:=Set(setaff,i->i*alpha);
trans:=image[1]-representative;
if ForAll(trans,IsInt)
then
if image=setaff+trans
then
addel:=MutableCopyMat(alpha);
addel[dim+1]:=addel[dim+1]-trans;
Add(stabgens,addel);
fi;
fi;
od;
return Group(Set(stabgens));
end);
#############################################################################
##
#O RepresentativeActionOnRightOnSets(group,set,imageset) for StandardSpaceGroups
##
InstallMethod(RepresentativeActionOnRightOnSets,"for crystallographic groups on right",
[IsGroup,IsDenseList,IsDenseList],
function(group,set, imageset)
local dim, pointgroupreps, setaff, imagesetaff, imagepoint,
g, setaffimage, point, difference, candidate;
if not IsStandardSpaceGroup(group) and IsAffineCrystGroupOnRight(group)
then
Error("group must be a StandardSpaceGroup acting on right");
fi;
dim:=DimensionOfMatrixGroup(group)-1;
if not ForAll([set,imageset],IsSet) or set=[] or imageset=[] or Size(set)<>Size(imageset)
then
Error("set and imageset must be propper non empty sets of the same size");
fi;
if not Set(set,Size)=[dim]
or not Size(set)=Size(imageset)
or not Set(imageset,Size)=[dim]
or not ForAll(set,IsVector)
or not ForAll(imageset,IsVector)
then
Error("<set> and <imageset> must be sets of vectors of the right length");
fi;
pointgroupreps:=PointGroupRepresentatives(group);
setaff:=List(set,x->Concatenation(x,[1]));
imagesetaff:=List(imageset,x->Concatenation(x,[1]));
imagepoint:=imagesetaff[1];
for g in pointgroupreps
do
setaffimage:=List(setaff,s->s*g);
difference:=imagepoint-Minimum(setaffimage);
if ForAll(difference,IsInt)
then
if ForAll(setaffimage,i->i+difference in imagesetaff)
then
candidate:=ShallowCopy(g);
candidate[dim+1]:=candidate[dim+1]+difference;
return candidate;
fi;
fi;
od;
return fail;
end);
#############################################################################
##
#O GramianOfAverageScalarProductFiniteMatrixGroup
##
# Suppose $G$ is a finite group of matrices. Suppose further, that you
# know they are orthogonal with respect to (some) euclidian scalar product.
# This method finds a Gramian matrix describing a positive definite symmetric
# bilinear form.
# This form isn't necessrily normed. But the gramian matrix has only
# rational entries.
#
# The returned gramian matrix is that of the average scalar product induced
# by $G$.
#
InstallMethod(GramianOfAverageScalarProductFromFiniteMatrixGroup,
[IsGroup],
function(group)
local dim, basis, summands, g, gramian, i, j, denom;
if not IsFinite(group)
then
Error("this only works for finite groups");
fi;
dim:=Size(Representative(group));
basis:=IdentityMat(dim);
if ForAll(GeneratorsOfGroup(group),g->TransposedMat(g)=Inverse(g))
then
gramian:=basis;
else
summands:=Set(group,i->i*TransposedMat(i));
gramian:=Sum(summands);
fi;
return gramian;
end);
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