The CrystGap package provides functions for the computation with affine
crystallographic groups, in particular space groups. Also provided are
some functions dealing with related linear matrix groups, such as point
groups. For the definition of the standard crystallographic notions we
refer to the International Tables Hah95, in particular the chapter
by Wondratschek Won95, and to the introductory chapter in
BBNWZ78. Some material can also be found in the chapters The
Crystallographic Groups Library and Irreducible Maximal Finite Integral
Matrix Groups. The principal algorithms used in this package are
described in egn97b, a preprint of which in included in the doc
directory of this package.
CrystGap is implemented in the GAP3 language, and runs on any system
supporting GAP3 3.4.4. The function WyckoffLattice, however, requires
the share package \sf XGap, which in turn runs only under Unix. The
functions described in this chapter can be used only after loading
CrystGap with the command
gap> RequirePackage( "cryst" );
CrystGap has been developed by
Bettina.Eick@math.RWTH-Aachen.de
gaehler@pth.polytechnique.fr
werner@dcs.st-and.ac.uk
Please send bug reports, suggestions and other comments to any of these e-mail addresses.
The first and third authors acknowledge financial support from the Graduiertenkolleg Analyse und Konstruktion in der Mathematik. The second author was supported by the Swiss Bundesamt für Bildung und Wissenschaft in the framework of the HCM programme of the European Community. This collaboration was in part made possible by financial support from the HCM project Computational Group Theory.
An affine crystallographic group G is a subgroup of the group of all Euclidean motions of d-dimensional space, with the property that its subgroup T of all pure translations is a freely abelian, normal subgroup of G, which has rank at most equal to d, and which has finite index in G.
In this package, the term CrystGroup always refers to such an affine crystallographic group. Linear matrix groups, whether crystallographic or not, will carry different designations (see below). CrystGroups are represented as special matrix groups, whose elements are affine matrices of the form
[ M 0 ]
[ t 1 ]
acting on row vectors (x,1) from the right. Note that this is different
from the crystallographic convention, where matrices usually act from the
left on column vectors (see also The Crystallographic Groups Library).
We have adopted this convention to maintain compatibility with the rest
of GAP3.
The ``linear" parts M of the elements of a CrystGroup G generate the point group P of G, which is isomorphic to the quotient G/T. There is a natural homomorphism from G to P, whose kernel is T. The translation vectors of the elements of T generate a free Z-module L, called the translation lattice of G. CrystGroups can be defined with respect to any basis of Euclidean space, but internally most computations will be done in a basis which contains a basis of L (see More about Crystallographic Groups).
CrystGroups carry a special operations record CrystGroupOps, and are
identified with a tag isCrystGroup. CrystGroups must be constructed
with a call to CrystGroup (see CrystGroup) which sets the tag
isCrystGroup to true, and sets the operations record to
CrystGroupOps.
Warning\: The groups in GAP3' s crystallographic groups library (see
The Crystallographic Groups Library), whether they are extracted with
SpaceGroup or TransposedSpaceGroup, are not CrystGroups in the
sense of this package, because CrystGroups have different record entries
and a different operations record. However, a group extracted with
TransposedSpaceGroup from that library can be converted to a CrystGroup
by a call to CrystGroup (see CrystGroup).
A CrystGroup which has a translation subgroup of full rank is called a space group. Certain functions are available only for space groups, and not for general CrystGroups, notably all functions dealing with Wyckoff positions (see Wyckoff Positions).
Space groups which are equivalent under conjugation in the affine group (shortly: affine equivalent space groups) are said to belong to the same space group type. As is well known, in three dimensions there are 219 such space group types (if only conjugation by transformations with positive determinant is allowed, there are 230).
Representatives of all space group types in dimensions 2, 3 and 4 can be
obtained from the crystallographic groups library contained in GAP3
(see The Crystallographic Groups Library). They must be extracted with
the function CrystGroup, and not with the usual extraction functions
SpaceGroup and TransposedSpaceGroup of that library, as these latter
functions return groups which do not have an operations record that would
allow to compute with them. CrystGroup accepts exactly the same
arguments as SpaceGroup and TransposedSpaceGroup. It returns the same
group as TransposedSpaceGroup, but equipped with a working operations
record.
Space group types (and thus space groups) are classified into ℤ-classes and ℚ-classes. Two space groups belong to the same ℤ-class if their point groups, expressed in a basis of their respective translation lattices, are conjugate as subgroups of GL(d,ℤ). If the point groups are conjugate as subgoups of GL(d,ℚ), the two space groups are said to be in the same ℚ-class. This provides also a classification of point groups (expressed in a lattice basis, i.e., integral point groups) into ℤ-classes and ℚ-classes.
For a given finite integral matrix group P, representing a point group
expressed in a lattice basis, a set of representative space groups for
each space group type in the ℤ-class of P can be obtained with
SpaceGroupsPointGroup (see SpaceGroupsPointGroup). If, moreover, the
normalizer of P in GL(d,ℤ) is known (see NormalizerGL), exactly
one representative is obtained for each space group type.
Representatives of all ℤ-classes of maximal irreducible finite point
groups are contained in a GAP3 library (see Irreducible Maximal Finite
Integral Matrix Groups) in all dimensions up to 11, and in prime
dimensions up to 23. For some other dimensions, at least ℚ-class
representatives are available.
Important information about a space group is contained in its affine normalizer (see AffineNormalizer), which is the normlizer of the space group in the affine group. In a way, the affine normalizer can be regarded as the symmetry of the space group.
Warning\: Groups which are called space groups in this manual should
not be confused with groups extracted with SpaceGroup from the
crystallographic groups library (see The Crystallographic Groups
Library). The latter are not CrystGroups in the sense of this package.
61.3 More about Crystallographic Groups
In this section we describe how a CrystGroup G is represented
internally. The casual user can skip this section in a first reading.
Although the generators of a CrystGroup can be specified with respect to
any basis, most computations are done internally in a special, standard
basis, which is stored in G.internalBasis. The results are translated
into the user-specified basis only afterwards. G.internalBasis consists
of a (standard) basis of the translation lattice of G, complemented, if
necessary, with suitable standard basis vectors. The standard basis of
the translation lattice is stored in G.translations.
As soon as G.internalBasis has been determined, both the CrystGroup G
and its point group P obtain a component internalGenerators. For the
point group P, the component P.internalGenerators contains a set of
generators of P, expressed with respect to the internalBasis of G,
whereas for the CrystGroup G the component G.internalGenerators
contains a set of homomorphic preimages of P.internalGenerators in G,
also expressed in the internalBasis of G. Thus G.internalGenerators
does not contain any translation generators. These are easy to add,
however: With respect to the internal basis, the translations are
generated by the first k standard basis vectors, where k is the rank
of the translation lattice.
Note that the internalGenerators of both a point group P and a
CrystGroup G may be changed by some functions, notably by FpGroup.
Thus they need not have any obvious connection to P.generators and
G.generators, respectively. Internal record entries of a CrystGroup
should never be changed by the user.
CrystGroup( matgroup )
CrystGroup( generating matrices )
CrystGroup( list of generators, identity )
CrystGroup( integers )
CrystGroup( string )
CrystGroup accepts as arguments either a group of affine matrices, or a
list of generating affine matrices, or an argument identifying a space
group from the crystallographic groups library, i.e., a list of two or
five integers, or a string containing a Hermann-Mauguin symbol, and
converts it into a CrystGroup in the sense of this package. CrystGroup
tests whether the generators are indeed affine matrices.
IsCrystGroup( G )
tests whether G.isCrystGroup is present and true. G.isCrystGroup is
set by CrystGroup.
PointGroup( G )
extracts the point group P of a space group G, binds it to
G.pointGroup, and returns it. It also determines the homomorphism from
G to P, and binds it to G.pointHomom. A point group P has always
a component P.isPointGroup set to true, and a component
P.crystGroup containing the CrystGroup from which it was constructed.
TranslationsCrystGroup( G )
determines a basis of the translation lattice of G, binds it to
G.translations, and returns it. Note that this translation lattice is
always invariant under the point group P of G. If G.translations
is not yet present, a finite presentation of P needs to be determined.
A basis of the translation lattice can also be added by the user, with
AddTranslationsCrystGroup (see AddTranslationsCrystGroup).
Warning\: The component G.translations must never be set by hand.
The functions TranslationsCrystGroup and AddTranslationsCrystGroups
have important (and wanted) side effects.
61.8 AddTranslationsCrystGroup
AddTranslationsCrystGroup( G, basis )
Since TranslationsCrystGroup (see TranslationsCrystGroup) needs a
presentation of the point group, the computation of G.translations can
be rather time consuming. If a basis of the translation lattice is known,
AddTranslationsCrystGroup can be used to add this knowledge to a
CrystGroup. If G.translations is already known, its value is kept
without further notice. It is the responsibility of the user that the
basis handed over to AddTranslationsCrystGroup is a correct basis of
the translation lattice. In case of doubt, the function
CheckTranslations (see CheckTranslations) can be used to check
whether the basis added was indeed correct.
Warning\: The component G.translations must never be set by hand.
The functions TranslationsCrystGroup and AddTranslationsCrystGroups
have important (and wanted) side effects.
CheckTranslations( G )
checks whether G.translations is indeed correct. If G.translations is
incorrect, a warning message is printed, otherwise GAP3 remains silent.
In the case of an incorrect translation basis a new CrystGroup must be
created, and the computations must be started afresh, because the wrong
translation basis may have produced wrong information components.
CheckTranslations is useful if a basis has been added with
AddTranslationsCrystGroup, and doubts arise later whether the basis
added was correct.
ConjugatedCrystGroup( G, c )
returns a new CrystGroup which is a conjugate of G. The conjugator c
can either be a d-dimensional linear matrix (which then is complemented
with the zero translation), or a (d+1)-dimensional affine matrix. The
generators are conjugated as gc = c*g*c-1. Some components which
are bound in G are copied and translated to the new basis, in
particular G.generators, G.translations, G.internalBasis, and
G.wyckoffPositons. If G.internalBasis is bound,
ConjugatedCrystGroup( G, G.internalBasis )
returns a CrystGroup whose translation lattice (of rank k) is generated
by the first k rows of the identity matrix. ConjugatedCrystGroup
allows as input only a parent CrystGroup.
61.11 FpGroup for point groups
FpGroup( P )
computes a finite presentation of the point group P, and binds it to
P.fpGroup. If P (and thus its CrystGroup G := P.crystGroup) is
solvable, a power-commutator presentation is returned.
Warning\: If P is solvable, the abstract generators are not
necessarily isomorphic images of P.generators (see More about
Crystallographic Groups).
FpGroup( G )
computes a finite presentation of the CrystGroup G, and binds it to
G.fpGroup. If the point group (and thus G) is solvable, a
power-commutator presentation is returned. The presentation is always
an extension of the presentation of the point group (which is computed
if necessary).
Warning\: The abstract generators of the presentation are not
necessarily isomorphic images of G.generators (see More about
Crystallographic Groups).
61.13 MaximalSubgroupsRepresentatives
MaximalSubgroupsRepresentatives( S, "translationEqual", [, ind ] )
MaximalSubgroupsRepresentatives( S, "classEqual", ind )
MaximalSubgroupsRepresentatives( S, ind )
returns a list of conjugacy class representatives of maximal subgroups of the CrystGroup S. If ind is present, which must be a prime or a list of primes, only those subgroups are returned whose index is a power of a prime contained in or equal to ind. If the flag ``translationEqual" is present, only those subgroups are returned which are translation-equal (translationengleich) with S. If the flag ``classEqual" is present, only those subgroups are return which are class-equal (klassengleich) with S. ind is optional only if the flag ``latticeEqual" is present. In all other cases, ind is required.
IsSpaceGroup( S )
determines whether the CrystGroup S is a space group (see Crystallographic Groups).
IsSymmorphicSpaceGroup( S )
determines whether the space group S is symmorphic. A space group is called symmorphic if it is equivalent to a semidirect product of its point group with its translation subgroup.
SpaceGroupsPointGroup( P )
SpaceGroupsPointGroup( P, normalizer elements )
where P is any finite subgroup of GL(d,Z), returns a list of all space groups with point group P, up to conjugacy in the full translation group of Euclidean space. All these space groups are returned as CrystGroups in standard representation. If a second argument is present, which must be a list of elements of the normalizer of P in GL(d,Z), only space groups inequivalent under conjugation with these elements are returned. If these normalizer elements, together with P, generate the full normalizer of P in GL(d,Z), then exactly one representative of each space group type is obtained.
A Wyckoff position of a space group G is an equivalence class of points in Euclidean space, having stabilizers which are conjugate subgroups of G. Apart from a subset of lower dimension, which contains points with even bigger stabilizers, a Wyckoff position consists of a G-orbit of some affine subspace A. A Wyckoff position W therefore can be specified by a representative affine subspace A and its stabilizer subgroup. In CrystGap, a Wyckoff position W is represented as a record with the following components:
W.basis:WyckoffBasis( W ).
W.translation:W.translation is such that A = L + W.translation. WyckoffTranslation( W ).
W.stabilizer:WyckoffStabilizer( W ).
W.class:WyckoffPosClass( W ).
W.spaceGroup:WyckoffSpaceGroup( W ).
W.isWyckoffPosition:IsWyckoffPosition( W ).
W.operations:
WyckoffPositions( G )
returns the list of all Wyckoff positions of the space group G.
61.19 WyckoffPositionsByStabilizer
WyckoffPositionsByStabilizer( G, U ),
where G is a space group and U a subgroup of the point group or a list of such subgroups, determines only the Wyckoff positions (see WyckoffPositions) having a representative affine subspace whose stabilizer has a point group equal to the subgroup U or contained in the list U, respectively.
WyckoffPositionsQClass( G, S )
For space groups with larger point groups, most of the time in the
computation of Wyckoff positions (see WyckoffPositions) is spent
computing the subgroup lattice of the point group. If Wyckoff positions
are needed for several space groups which are in the same Q class, and
therefore have isomorphic point groups, one can avoid recomputing the
same subgroup lattice for each of them as follows. For the computation
of the Wyckoff positions of the first space group S one uses a call to
WyckoffPositions. For the remaining space groups, S is then passed
as a second argument to WyckoffPositionsQClass( G, S ), which uses
some of the results already obtained for S.
WyckoffOrbit( W )
takes a Wyckoff position W (see Wyckoff Positions) and returns a list of Wyckoff positions which are different representations of W, such that the representative affine subspaces of these representations form an orbit under the space group G of W, modulo lattice translations.
WyckoffLattice( G )
If a point x in a Wyckoff position W1 has a stabilizer which is a
subgroup of the stabilizer of some point y in a Wyckoff position W2,
then the closure of W1 will contain W2. These incidence relations
are best represented in a graph. WyckoffLattice( G ) determines and
displays this graph using XGAP (note that XGAP runs only under Unix plus
the X Window System). Each Wyckoff position is represented by a
vertex. If W1 contains W2, its vertex is placed below that of W2
(i.e., Wyckoff positions with bigger stabilizers are placed higher up),
and the two are connected, either directly (if there is no other Wyckoff
position in between) or indirectly. With the left mouse button and with
the XGAP CleanUp menu it is possible to change the layout of the graph
(see the XGAP manual). When clicking with the right mouse button on a
vertex, a pop up menu appears, which allows to obtain the following
information about the representative affine subspace of the Wyckoff
position:
StabDim::
StabSize::
ClassSize::
IsAbelian, IsCyclic, IsNilpotent, IsPerfect, IsSimple,
IsSolvable::
Isomorphism::
ConjClassInfo::
Translation::
Basis::
NormalizerGL( G ),
where G is a finite subgroup of GL(d,Z), returns the normalizer of G in GL(d,Z). At present, this function is available only for groups which are the point group of a CrystGroup extracted from the space group library.
CentralizerGL( G ),
where G is a finite subgroup of GL(d,Z), returns the centralizer of G in GL(d,Z). At present, this function is available only for groups which are the point group of a CrystGroup extracted from the space group library.
PointGroupsBravaisClass( B )
PointGroupsBravaisClass( B [, norm ] )
where B is a finite integral matrix group, returns a list of representatives of those conjugacy classes of subgroups of B which are in the same Bravais class as B. These representatives are returned as parent groups, not subgroups. If B is a Bravais group, the list contains a representative of each point group in the Bravais class of B. If a second argument is present, which must be a list of elements of the normalizer of B in GL(d,ℤ), only subgroups inequivalent under conjugation with these elements are returned.
TranslationNormalizer( S )
returns the normalizer of the space group S in the full translation
group. At present, this function is implemented only for space groups,
not for general CrystGroups. The translation normalizer TN of S may
contain a continuous subgroup C. A basis of the space of such
continuous translations is bound in TN.continuousTranslations. Since
this subgroup is not finitely generated, it is not contained in the
group generated by TN.generators. Properly speaking, the translation
normalizer is the span of TN and C together.
AffineNormalizer( S )
returns the affine normalizer of the space group S. The affine
normalizer contains the translation normalizer as a subgroup. Similarly
as with TranslationNormalizer, the subgroup C of continuous
translations, which is not finitely generated, is not part of the group
that is returned. However, a basis of the space of continuous
translations is bound in the component continuousTranslations.
At present, this function is available only for for space groups, not for
general CrystGroups. Moreover, the NormalizerGL (see NormalizerGL)
of the point group of S must be know, which currently is the case only
for CrystGroups extracted from the space group library.
61.28 AffineInequivalentSubgroups
AffineInequivalentSubgroups( sub )
takes as input a list of subgroups with common parent space group S, and returns a sublist of those which are affine inequivalent. For this, the affine normalizer of S is required, which currently is available only if S is a space group extracted from the space groups library.
61.29 Other functions for CrystGroups
In the operations record of a CrystGroup many of the usual GAP3
functions are replaced with a CrystGroup specific implementation. For
other functions the default implementation can be used. Since
CrystGroups are matrix groups, all functions which work for a finite
matrix group should work also for a finite CrystGroup (i.e., one which
contains no pure translations). Of course, functions which require a
finite group as input will work only for finite CrystGroups.
Following is a (probably not exhaustive) list of functions that are known
to work for also for infinite CrystGroups.
in
Parent, IsParent, Group, IsGroup
Subgroup, IsSubgroup, AsSubgroup, Index
Centralizer, Centre, Normalizer
Closure, NormalClosure
Intersection, NormalIntersection
ConjugacyClassSubgroups, ConjugateSubgroups
DerivedSubgroup, CommutatorSubgroup, Core
DerivedSeries, SubnormalSeries
FactorGroup, CommutatorFactorGroup
ConjugateSubgroup, TrivialSubgroup
IsAbelian, IsCentral, IsTrivial
IsNormal, IsSubnormal, IsPerfect, IsSolvable
The following functions work for CrystGroups provided the subgroup
H has finite index in G. The elements of the resulting domain are
given in ascending order (with respect to an ad hoc, but fixed ordering).
Cosets( G, H )
RightCosets( G, H )
LeftCosets( G, H )
The following functions dealing with group operations work for
CrystGroups provided the orbits of the action are finite. Since
CrystGroups are not finite in general, this is a non-trivial
requirement, and so some care is needed.
Orbit( G, d, opr )
Orbits( G, D, opr )
OrbitLengths( G, D, opr )
Stabilizer( G, d, opr )
RepresentativeOperation( G, d, e, opr )
RepresentativesOperation( G, d, opr )
The following functions have a CrystGroup specific implementation, but
work for finite CrystGroups only:
Elements( G )
ConjugacyClasses( G )
PermGroup( G )
SylowSubgroup( G, p )
Elements of a color group C are colored in the following way. The
elements having the same color as C.identity form a subgroup H, which
has finite index n in C. H is called the ColorSubgroup of C.
Elements of C have the same color if and only if they are in the same
right coset of H in C. A fixed list of right cosets of H in C,
called ColorCosets, therefore determines a labelling of the colors,
which runs from 1 to n. Elements of H by definition have color 1,
i.e., the coset with representative C.identity is always the first
element of ColorCosets. Right multiplication by a fixed element g of
C induces a permutation p(g) of the colors of the parent of C.
This defines a natural homomorphism of C into the permutation group of
degree n. The image of this homomorphism is called the
ColorPermGroup of C, and the homomorphism to it is called the
ColorHomomorphism of C.
A color group is constructed with
ColorGroup( G, H ),
which returns a colored copy of G, with color subgroup H. G must be a parent group, and H must be a finite index subgroup of G. Color subgroups must be constructed as subgroups of color parent groups, and not by coloring uncolored subgroups. Subgroups of color groups will inherit the coloring of their parent, including the labelling of the colors.
Color groups are identified with a tag isColorGroup. They always have
a component colorSubgroup. Color parent groups moreover always have a
component colorCosets, which fixes a labelling of the colors.
Groups which may be colored include, in particular, CrystGroups, but
coloring of any finite group, such as a finite matrix group or
permutation group, should work as well.
IsColorGroup( G )
checks whether G.isColorGroup is bound and true.
ColorSubgroup( G )
returns the color subgroup of G.
ColorCosets( G )
returns the color cosets of G.
ColorOfElement( G, elem )
returns the color of an element.
ColorPermGroup( G )
returns the ColorPermGroup of G, which is the permutation group induced by G acting on the colors of the parent of G.
ColorHomomorphism( G )
returns the homomomorphism from G to its ColorPermGroup.
61.38 Subgroup for color groups
If C is a color group,
Subgroup( C, [elems] )
returns a colored subgroup of C, whereas
C.operations.UncoloredSubgroup( C, [elems] )
returns an ordinary, uncolored subgroup.
61.39 PointGroup for color CrystGroups
If C is a color CrystGroup whose color subgroup is lattice-equal (or
translationengleich) with C, the point group of C can consistently be
colored. In that case,
PointGroup( C )
returns a colored point group. Otherwise, the point group will be uncolored. An uncolored point group can always be obtained with
C.operations.UncoloredPointGroup( C )
61.40 Inequivalent colorings of space groups
Two colorings of a space group S are equivalent if the two
ColorSubgroups are conjugate in the affine normalizer of S.
AffineInequivalentSubgroups( L )
where L is a list of sub space groups with a common parent space group S, returns a list of affine inequivalent subgroups from L. At present, this routine is supported only for CrystGroups constructed from the space group library.
A list of prime index p subgroups of S (actually, a list of conjugacy class representatives of such subgroups) can be obtained with
Filtered( MaximalSubgroupsRepresentatives( S, p ), U -> U.index = p )
These two routines together therefore allow to determine all inequivalent colorings of S with p colors.
gap3-jm