Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 78-79   | 1 | 2 |

Section Interaction between symmetry and decomposition

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Interaction between symmetry and decomposition

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Suppose that the space-group action is reducible, i.e. that for each [g \in G][{\bf R}_{g} = \pmatrix{{\bf R}'_{g} &{\bf 0}\cr {\bf 0} &{\bf R}''_{g}\cr},\qquad {\bf t}_{g} = \openup 2pt\pmatrix{{\bf t}'_{g}\cr {\bf t}''_{g}\cr}\hbox{\semi}]by Schur's lemma, the decimation matrix must then be of the form[{\bf N} = \pmatrix{{\bf N}' &{\bf 0}\cr {\bf 0} &{\bf N}''\cr}]if it is to commute with all the [{\bf R}_{g}].

Putting[{\bf x} = \pmatrix{{\bf x}'\cr {\bf x}''\cr} \quad{\rm and}\quad {\bf h} = \pmatrix{{\bf h}'\cr {\bf h}''\cr},]we may define[\eqalign{S'_{g} ({\bf x}') &= {\bf R}'_{g} {\bf x}' + {\bf t}'_{g},\cr S''_{g} ({\bf x}'') &= {\bf R}''_{g} {\bf x}'' + {\bf t}''_{g},}]and write [S_{g} = S'_{g} \oplus S''_{g}] (direct sum) as a shorthand for[S_{g} ({\bf x}) = \openup2pt\pmatrix{S'_{g} ({\bf x}')\cr S''_{g} ({\bf x}'')\cr}. ]

We may also define the representation operators [S_{g}^{'\#}] and [S_{g}^{''\#}] acting on functions of [{\bf x}'] and [{\bf x}''], respectively (as in Section[link]), and the operators [S_{g}^{'*}] and [S_{g}^{''*}] acting on functions of [{\bf h}'] and [{\bf h}''], respectively (as in Section[link]). Then we may write[S_{g}^{\#} = (S'_{g})^{\#} \oplus (S''_{g})^{\#}]and[S_{g}^{*} = (S'_{g})^{*} \oplus (S''_{g})^{*}]in the sense that g acts on [f({\bf x}) \equiv f({\bf x}', {\bf x}'')] by[(S_{g}^{\#} f)({\bf x}', {\bf x}'') = f[(S'_{g})^{-1} ({\bf x}'), (S''_{g})^{-1} ({\bf x}'')]]and on [\Phi ({\bf h}) \equiv \Phi ({\bf h}', {\bf h}'')] by[\eqalign{(S_{g}^{*} \Phi)({\bf h}', {\bf h}'') &= \exp (2\pi i{\bf h}' \cdot {\bf t}'_{g}) \exp (2\pi i{\bf h}'' \cdot {\bf t}''_{g})\cr &\quad \times \Phi [{\bf R}_{g}^{'T} {\bf h}', {\bf R}_{g}^{''T} {\bf h}''].}]

Thus equipped we may now derive concisely a general identity describing the symmetry properties of intermediate quantities of the form[\eqalign{ T ({\bf x}', {\bf h}'') &= {\sum\limits_{{\bf h}^\prime}}\, F({\bf h}', {\bf h}'') \exp (-2\pi i{\bf h}' \cdot {\bf x}')\cr &= {1 \over |\!\det {\bf N}'|} {\sum\limits_{{\bf x}''}}\, \rho\llap{$-\!$} ({\bf x}', {\bf x}'') \exp (+2\pi i{\bf h}'' \cdot {\bf x}''),}]which arise through partial transformation of F on [{\bf h}'] or of [\rho\llap{$-\!$}] on [{\bf x}'']. The action of [g \in G] on these quantities will be

  • (i) through [(S'_{g})^{\#}] on the function [{\bf x}'\,\longmapsto\, T ({\bf x}', {\bf h}'')],

  • (ii) through [(S''_{g})^{*}] on the function [{\bf h}'' \,\longmapsto\, T ({\bf x}', {\bf h}'')],

and hence the symmetry properties of T are expressed by the identity[T = [(S'_{g})^{\#} \oplus (S''_{g})^{*}] T.]Applying this relation not to T but to [[(S'_{g^{-1}})^{\#} \oplus (S''_{e})^{*}] T] gives[[(S'_{g^{-1}})^{\#} \oplus (S''_{e})^{*}] T = [(S'_{e})^{\#} \oplus (S''_{g})^{*}] T,]i.e. [Scheme scheme2]

If the unique [F({\bf h}) \equiv F({\bf h}', {\bf h}'')] were initially indexed by[(\hbox{all } {\bf h}') \times (\hbox{unique } {\bf h}'')](see Section[link]), this formula allows the reindexing of the intermediate results [T ({\bf x}', {\bf h}'')] from the initial form[(\hbox{all } {\bf x}') \times (\hbox{unique } {\bf h}'')]to the final form[(\hbox{unique } {\bf x}') \times (\hbox{all } {\bf h}''),]on which the second transform (on [{\bf h}'']) may now be performed, giving the final results [\rho\llap{$-\!$} ({\bf x}', {\bf x}'')] indexed by[(\hbox{unique } {\bf x}') \times (\hbox{all } {\bf x}''),]which is an asymmetric unit. An analogous interpretation holds if one is going from [\rho\llap{$-\!$}] to F.

The above formula solves the general problem of transposing from one invariant subspace to another, and is the main device for decomposing the CDFT. Particular instances of this formula were derived and used by Ten Eyck (1973)[link]; it is useful for orthorhombic groups, and for dihedral groups containing screw axes [n_{m}] with g.c.d. [(m, n) = 1]. For comparison with later uses of orbit exchange, it should be noted that the type of intermediate results just dealt with is obtained after transforming on all factors in one summand.

A central piece of information for driving such a decomposition is the definition of the full asymmetric unit in terms of the asymmetric units in the invariant subspaces. As indicated at the end of Section[link], this is straightforward when G acts without fixed points, but becomes more involved if fixed points do exist. To this day, no systematic `calculus of asymmetric units' exists which can automatically generate a complete description of the asymmetric unit of an arbitrary space group in a form suitable for directing the orbit exchange process, although Shenefelt (1988)[link] has outlined a procedure for dealing with space group P622 and its subgroups. The asymmetric unit definitions given in Volume A of International Tables are incomplete in this respect, in that they do not specify the possible residual symmetries which may exist on the boundaries of the domains.


Shenefelt, M. (1988). Group Invariant Finite Fourier Transforms. PhD thesis, Graduate Centre of the City University of New York.
Ten Eyck, L. F. (1973). Crystallographic fast Fourier transforms. Acta Cryst. A29, 183–191.

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