International
Tables for Crystallography Volume A Space-group symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 |
International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 29-31
Section 1.3.3.2. Coset decomposition with respect to the translation subgroup
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Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands |
The translation subgroup of a space group can be used to distribute the operations of into different classes by grouping together all operations that differ only by a translation. This results in the decomposition of into cosets with respect to (see Section 1.1.4 for details of cosets).
Definition
Let be a space group with translation subgroup .
If the translation subgroup is a subgroup of index [i] in , a set of coset representatives for relative to consists of [i] operations , where is assumed to be the identity element of . The cosets of relative to can be imagined as columns of an infinite array with [i] columns, labelled by the coset representatives, as displayed in Table 1.3.3.3.
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Remark : We can assume some enumeration of the operations in because the translation vectors form a lattice. For example, with respect to a primitive basis, the coordinate vectors of the translations in are simply columns with integral components . A straightforward enumeration of these columns would start with
Writing out the matrix–column pairs, the coset consists of the operations of the form with running over the lattice translations of . This means that the operations of a coset with respect to the translation subgroup all have the same linear part, which is also evident from a listing of the cosets as columns of an infinite array, as in the example above.
The one-to-one correspondence between the point-group operations and the cosets relative to explicitly displays the isomorphism between the point group of and the factor group . This correspondence is also exploited in the listing of the general-position coordinates. What is given there are the coordinate triplets for coset representatives of relative to , which correspond to the first row of the array in Table 1.3.3.3. As just explained, the other operations in can be obtained from these coset representatives by adding a lattice translation to the translational part.
Furthermore, the correspondence between the point group and the coset decomposition relative to makes it easy to find a system of coset representatives of relative to . What is required is that the linear parts of the are precisely the operations in the point group of . If are the different operations in the point group of , then a system of coset representatives is obtained by choosing for every linear part a translation part such that is an operation in .
It is customary to choose the translation parts of the coset representatives such that their coordinates lie between 0 and 1, excluding 1. In particular, if the translation part of a coset representative is a lattice vector, it is usually chosen as the zero vector .
Note that due to the fact that is a normal subgroup of , a system of coset representatives for the right cosets is at the same time a system of coset representatives for the left cosets.