Crystallographic group

A discrete group of motions of an -dimensional Euclidean space having a bounded fundamental domain. Two crystallographic groups are said to be equivalent if they are conjugate in the group of affine transformations of .

The origin of the theory of crystallographic groups is connected with the study of the symmetry of ornaments and the structure of crystals . A classification of all planar (two-dimensional) and spatial (three-dimensional) crystallographic groups was achieved at the end of the 19th century by E.S. Fedorov and somewhat later by A. Schoenflies (see [2], [3], and also [6], [7], [9]). Up to equivalence, there are 17 planar and 219 spatial crystallographic groups; if, however, the spatial groups are considered up to conjugacy with respect to orientation-preserving affine transformations, their number is 230. In 1910, L. Bieberbach investigated crystallographic groups of arbitrary dimension . In particular, he proved the following theorems:

1) Any -dimensional crystallographic group contains linearly independent parallel translations; the group of linear parts of the transformations from is finite. (For this was proved in [3].)

2) Two crystallographic groups are equivalent if and only if they are isomorphic as abstract groups.

3) For any , there are — up to equivalence — only finitely many -dimensional crystallographic groups (this is a solution to Hilbert's 18th problem).

Theorem 1 yields the following description of the structure of crystallographic groups as abstract groups. Let be the set of all parallel translations in a crystallographic group . Then is a normal subgroup of finite index, isomorphic to , and is its own centralizer in . The existence of a normal subgroup of an abstract group possessing these properties is a sufficient condition for to be isomorphic to a crystallographic group [7].

The group of linear parts of a crystallographic group preserves the lattice ; in other words, relative to a basis of the transformations in are represented by matrices with integer entries.

In order to specify a crystallographic group it is necessary to specify — in addition to and — a vector for each such that the transformation

belongs to . The vector is defined up to addition by a vector from . The mapping

is a one-dimensional cocycle on with values in , where is the vector space associated with .

Any triple , where is a finite linear group, is a -invariant lattice and a one-dimensional cocycle on with values in , corresponds as just described with some crystallographic group. Under this correspondence, two triples and , where and are cohomologous cocycles, correspond to equivalent crystallographic groups. To the zero cohomology class corresponds the split (or symmorphic) crystallographic group, which, relative to a suitable choice of an origin, is the set of all transformations

where .

In matrix interpretation, the description of all -dimensional crystallographic groups reduces to the description of all finite groups of square matrices of order with integer entries (up to conjugacy in the group ) and, for each such group , to the computation of the cohomology group .

Two cohomology classes define equivalent crystallographic groups if and only if they are transformed into one another by the normalizer of in . Bieberbach's theorem 2 and a result due to H. Zassenhaus [7] imply that the natural homomorphism

is an isomorphism. This is readily deduced from the exact cohomology sequence of .

Two crystallographic groups belong to the same class (arithmetical class) if their groups of linear parts are conjugate in (in ). For there are 32 classes and 73 arithmetical classes of crystallographic groups.

Among the finite groups of matrices with integer entries one can single out the lattice symmetry groups, i.e. the groups of all orthogonal transformations that preserve some fixed lattice in a vector space (and are presented relative to a basis of this lattice). In 1848, A. Bravais determined all possible -dimensional lattice symmetry groups and accordingly divided all -dimensional lattices into 14 types (known as Bravais types). The subgroups of that are lattice symmetry groups are called Bravais subgroups.

The Bravais subgroups may also be interpreted as stabilizer subgroups for the natural action of on the set of positive-definite quadratic forms in variables. They may therefore be determined using reduction theory (see [11] and Quadratic forms, reduction of). Every maximal finite subgroup of is a Bravais subgroup (but the converse is not true).

The following table lists the number of finite subgroups in (up to conjugacy).'

<tbody> </tbody> Number of maximal finite subgroups Number of Bravais subgroups Number of finite subgroups 1 1 1 2 2 2 5 13 3 4 14 73 4 9 64 710 5 17 ? ?

An intersection of Bravais subgroups is again a Bravais subgroup. The smallest Bravais subgroup containing the group of linear parts of a crystallographic group , up to conjugacy in (in ), is known as the geometrical (arithmetical) holohedry of . If is a crystallographic group in general position, in the sense that there is no affine transformation mapping it onto a crystallographic group whose lattice of parallel translations has lower symmetry, then is the lattice symmetry group of parallel translations of . Two crystallographic groups belong to the same syngony (Bravais type) if their geometrical (arithmetical) holohedries coincide. For there are 7 syngonies and 14 Bravais types of crystallographic groups.

Linear representations of crystallographic groups.

The irreducible finite-dimensional complex linear representations of a crystallographic group are described as follows. Let be some character (a homomorphism into the multiplicative group of complex numbers) of the group , put

and let be an irreducible representation of such that , where . Then the representation of induced by the representation of its subgroup (see Induced representation) is irreducible. All irreducible representations of are obtained in this way (see [9], [10]).

References

[1a]	A. Bravais, "Abhandlung über die systeme von regelmässig auf seiner Ebene oder Raum vertheilten Punkten" , Teubner (1897)
[1b]	A. Bravais, "Abhandlung über symmetrische Polyeder" , Teubner (1890)
[2]	E.S. Fedorov, "The symmetry and structure of crystals. Fundamental works" , Moscow (1949) pp. 111–255 (In Russian)
[3]	A. Schoenflies, "Kristallsysteme und Kristallstruktur" , Teubner (1891)
[4a]	L. Bieberbach, "Ueber die Bewegungsgruppen der Euklidischen Räume I" Math. Ann. , 70 (1911) pp. 297–336
[4b]	L. Bieberbach, "Ueber die Bewegungsgruppen der Euklidischen Räume II" Math. Ann. , 72 (1912) pp. 400–412
[5]	B. Delone, N. Padurov, A. Aleksandrov, "Mathematical foundation for the lattice analysis of crystals" , Moscow-Leningrad (1934) (In Russian)
[6]	A.V. Shubnikov, "An atlas of crystallographic symmetry groups" , Moscow-Leningrad (1946) (In Russian)
[7]	H. Zassenhaus, "Ueber ein Algorithmus zur Bestimmung der Raumgruppen" Comm. Math. Helv. , 21 (1948) pp. 117–141
[8]	A.I. Mal'tsev, "Classical algebra" , Selected works , 1 , Moscow (1976) pp. 371–375 (In Russian) Zbl 0422.01024
[9]	G.Ya. Lyubarskii, "Anwendung der Gruppentheorie in der Physik" , Deutsch. Verlag Wissenschaft. (1962) (Translated from Russian)
[10]	D.K. Faddeev, "Tables of the fundamental unitary representations of Fedorov groups" , Moscow-Leningrad (1961) (In Russian) MR158025
[11]	B.N. Delone, R.V. Galiulin, M.I. Shtrogrin, "On the Bravais types of lattices" J. Soviet Math. , 4 : 1 (1975) pp. 79–156 Itogi Nauki i Tekhn. Sovrem. Probl. Mat. , 2 (1973) pp. 119–254 Zbl 0334.50005

Comments

Let be the group of affine transformations of . The translations in form a normal subgroup and the quotient is the group of linear transformations. Let be a subgroup of . Then the translations in form a normal subgroup ; the quotient is often called the point group of ; this is called the group of linear parts above.

Fedorov obtained his classification results during 1885–1889, whereas Schoenflies obtained a classification around 1891. The correct list of 230 groups was found only after comparing the lists of Fedorov and Schoenflies (see [a1] for historical and other remarks). The book [a2] by L. Sohnke was known to both.

For maximal finite subgroups of for one may consult [a3].

References

[a1]	R.L.E. Schwartzenberger, "-dimensional crystallography" , Pitman (1980)
[a2]	L. Sohnke, "Entwicklung einer Theorie der Kristallstruktur" , Teubner (1879)
[a3]	W. Plesken, M. Post, "On maximal finite irreducible subgroups of I, II" Math. Computation , 31 (1977) pp. 536–551; 552–573
[a4]	M. Klemm, "Symmetrien von Ornamenten und Kristallen" , Springer (1982) MR0663004 Zbl 0482.20034
[a5]	J.J. Burkhardt, "Die Bewegungsgruppen der Kristallographie" , Birkhäuser (1947)