Coset decomposition with respect to the translation subgroup

Souvignier, B.

doi:10.1107/97809553602060000921

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

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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

B. Souvignier^a ^*

^aRadboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands
Correspondence e-mail: souvi@math.ru.nl

1.3.3.2. Coset decomposition with respect to the translation subgroup

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The translation subgroup $[{\cal T}]$ of a space group $[{\cal G} ]$ can be used to distribute the operations of $[{\cal G}]$ into different classes by grouping together all operations that differ only by a translation. This results in the decomposition of $[{\cal G}]$ into cosets with respect to $[{\cal T}]$ (see Section 1.1.4 for details of cosets).

Definition

Let $[{\cal G}]$ be a space group with translation subgroup $[{\cal T} ]$ .

(i) The right coset $[\ispecialfonts{\cal T} {\sfi W}]$ of an operation $[\ispecialfonts{\sfi W} \in {\cal G} ]$ with respect to $[{\cal T}]$ is the set $[\ispecialfonts\{ {\sfi t} {\sfi W} \mid {\sfi t} \in {\cal T} \} ]$ .

Analogously, the set $[\ispecialfonts{\sfi W} {\cal T} = \{ {\sfi W} {\sfi t} \mid {\sfi t} \in {\cal T} \} ]$ is called the left coset of $[\ispecialfonts{\sfi W}]$ with respect to $[{\cal T}]$ .
(ii) A set $[\ispecialfonts\{ {\sfi W}_1, \ldots, {\sfi W}_m \} ]$ of operations in $[{\cal G}]$ is called a system of coset representatives relative to $[{\cal T}]$ if every operation $[\ispecialfonts{\sfi W}]$ in $[{\cal G}]$ is contained in exactly one coset $[\ispecialfonts{\cal T} {\sfi W}_i ]$ .
(iii) Writing $[{\cal G}]$ as the disjoint union $[ \ispecialfonts{\cal G} = {\cal T} {\sfi W}_1 \cup \ldots \cup {\cal T} {\sfi W}_m ]$ is called the coset decomposition of $[{\cal G}]$ relative to $[{\cal T}]$ .

If the translation subgroup $[{\cal T}]$ is a subgroup of index [i] in $[{\cal G}]$ , a set of coset representatives for $[{\cal G} ]$ relative to $[{\cal T}]$ consists of [i] operations $[\ispecialfonts{\sfi W}_1, {\sfi W}_2, \ldots, {\sfi W}_{[i]}]$ , where $[\ispecialfonts{\sfi W}_1 ]$ is assumed to be the identity element $[\ispecialfonts{\sfi e}]$ of $[{\cal G} ]$ . The cosets of $[{\cal G}]$ relative to $[{\cal T}]$ 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.

Table 1.3.3.3| top | pdf |
Right-coset decomposition of $[{\cal G}]$ relative to $[{\cal T}]$

$[\ispecialfonts{\sfi W}_1 = {\sfi e}]$	$[\ispecialfonts{\sfi W}_2]$	$[ \ispecialfonts{\sfi W}_3]$	$[\ldots]$	$[\ispecialfonts{\sfi W}_{[i]}]$
$[\ispecialfonts{\sfi t}_1]$	$[\ispecialfonts{\sfi t}_1 {\sfi W}_2]$	$[\ispecialfonts{\sfi t}_1 {\sfi W}_3]$	$[\ldots ]$	$[\ispecialfonts{\sfi t}_1 {\sfi W}_{[i]}]$
$[\ispecialfonts{\sfi t}_2]$	$[\ispecialfonts{\sfi t}_2 {\sfi W}_2]$	$[\ispecialfonts{\sfi t}_2 {\sfi W}_3]$	$[\ldots ]$	$[\ispecialfonts{\sfi t}_2 {\sfi W}_{[i]}]$
$[\ispecialfonts{\sfi t}_3]$	$[\ispecialfonts{\sfi t}_3 {\sfi W}_2]$	$[\ispecialfonts{\sfi t}_3 {\sfi W}_3]$	$[\ldots ]$	$[\ispecialfonts{\sfi t}_3 {\sfi W}_{[i]}]$
$[\ispecialfonts{\sfi t}_4]$	$[\ispecialfonts{\sfi t}_4 {\sfi W}_2]$	$[\ispecialfonts{\sfi t}_4 {\sfi W}_3]$	$[\ldots ]$	$[\ispecialfonts{\sfi t}_4 {\sfi W}_{[i]}]$
$[\vdots]$	$[\vdots]$	$[\vdots]$		$[\vdots]$

Remark: We can assume some enumeration $[\ispecialfonts{\sfi t}_1, {\sfi t}_2, {\sfi t}_3, \ldots ]$ of the operations in $[{\cal T}]$ because the translation vectors form a lattice. For example, with respect to a primitive basis, the coordinate vectors of the translations in $[{\cal G}]$ are simply columns $[\pmatrix{ l \cr m \cr n }]$ with integral components [l,m,n] . A straightforward enumeration of these columns would start with $[\eqalign{&\pmatrix{ 0 \cr 0 \cr 0 }, \, \pmatrix{ 1 \cr 0 \cr 0 }, \, \pmatrix{ 0 \cr 1 \cr 0 }, \, \pmatrix{ 0 \cr 0 \cr 1 }, \, \pmatrix{ \bar{1} \cr 0 \cr 0 }, \, \pmatrix{ 0 \cr \bar{1} \cr 0 }, \, \pmatrix{ 0 \cr 0 \cr \bar{1} }, \cr & \pmatrix{ 1 \cr 1 \cr 0 }, \, \pmatrix{ 1 \cr 0 \cr 1 }, \, \pmatrix{ 0 \cr 1 \cr 1 } \ldots }]$

Writing out the matrix–column pairs, the coset $[{\cal T} ({\bi W}, {\bi w}) ]$ consists of the operations of the form $[({\bi I}, {\bi t}) ({\bi W}, {\bi w}) = ({\bi W}, {\bi w} + {\bi t}) ]$ with $[{\bi t}]$ running over the lattice translations of $[{\cal T}]$ . 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.

Proposition

Let $[\ispecialfonts{\sfi W} = ({\bi W}, {\bi w})]$ and $[\ispecialfonts{\sfi W}' = ({\bi W}', {\bi w}') ]$ be two operations of a space group $[{\cal G}]$ with translation subgroup $[{\cal T}]$ .

(1) If $[{\bi W} \neq {\bi W}']$ , then the cosets $[\ispecialfonts{\cal T} {\sfi W} ]$ and $[\ispecialfonts{\cal T} {\sfi W}']$ are disjoint, i.e. their intersection is empty.
(2) If $[{\bi W} = {\bi W}']$ , then the cosets $[\ispecialfonts{\cal T} {\sfi W}]$ and $[\ispecialfonts{\cal T} {\sfi W}']$ are equal, because $[\ispecialfonts{\sfi W} {\sfi W}'^{-1}]$ has linear part $[{\bi I}]$ and is thus an operation contained in $[{\cal T}]$ .

The one-to-one correspondence between the point-group operations and the cosets relative to $[{\cal T}]$ explicitly displays the isomorphism between the point group $[{\cal P}]$ of $[{\cal G}]$ and the factor group $[{\cal G}/{\cal T}]$ . 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 $[{\cal G}]$ relative to $[{\cal T} ]$ , which correspond to the first row of the array in Table 1.3.3.3. As just explained, the other operations in $[{\cal G} ]$ 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 $[{\cal T}]$ makes it easy to find a system of coset representatives $[\ispecialfonts\{ {\sfi W}_1, \ldots, {\sfi W}_m \}]$ of $[{\cal G}]$ relative to $[{\cal T}]$ . What is required is that the linear parts of the $[\ispecialfonts{\sfi W}_i]$ are precisely the operations in the point group of $[{\cal G}]$ . If $[{\bi W}_1, \ldots, {\bi W}_m]$ are the different operations in the point group $[{\cal P}]$ of $[{\cal G}]$ , then a system of coset representatives is obtained by choosing for every linear part $[{\bi W}_i]$ a translation part $[{\bi w}_i]$ such that $[\ispecialfonts{\sfi W}_i = ({\bi W}_i, {\bi w}_i)]$ is an operation in $[{\cal G} ]$ .

It is customary to choose the translation parts $[{\bi w}_i]$ 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 $[{\bi o}]$ .

Note that due to the fact that $[{\cal T}]$ is a normal subgroup of $[{\cal G}]$ , a system of coset representatives for the right cosets is at the same time a system of coset representatives for the left cosets.

References

International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 29-31