Halving subgroups and dichromatic (black-and-white) groups

Janovec, V.; Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000643

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 3.2, p. 384

Section 3.2.3.2.7. Halving subgroups and dichromatic (black-and-white) groups

V. Janovec,^a ^* Th. Hahn^b and H. Klapper^c

^a Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic,^bInstitut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and ^cMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail: janovec@fzu.cz

3.2.3.2.7. Halving subgroups and dichromatic (black-and-white) groups

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Any subgroup H of a group G of index 2, called a halving subgroup, is a normal subgroup. The decomposition of G into left cosets of H consists of two left cosets, $[G=H\cup gH . \eqno(3.2.3.33)]$ Sometimes it is convenient to distinguish elements of the coset [gH] from elements of the halving subgroup H. This can be achieved by attaching a sign (usually written as a superscript) to all elements of the coset. We shall use for this purpose the sign $[^\spadesuit]$ . To aid understanding, we shall also mark for a while the elements of the group H with another sign, $[^\heartsuit]$ . The multiplication law for these `decorated elements' can be written in the following form: $[g_1^\heartsuit g_2^\heartsuit=g_3^\heartsuit,\quad g_4^\heartsuit g_5^\spadesuit=g_6^\spadesuit,\quad g_7^\spadesuit g_8^\heartsuit=g_9^\spadesuit,\quad g_{10}^\spadesuit g_{11}^\spadesuit=g_{12}^\heartsuit. \eqno(3.2.3.34)]$ Now we replace the label $[^\heartsuit]$ by a dummy `no mark' sign (i.e. we remove $[^\heartsuit]$ ), but we still keep in mind the multiplication rules (3.2.3.34). Then the decomposition (3.2.3.33) becomes $[G=H\cup g^\spadesuit H,\eqno(3.2.3.33a)]$ since the coset $[g^\spadesuit H]$ assembles all marked elements and H consists of all bare elements of the group G.

The sign $[^\spadesuit]$ can carry useful additional information, e.g. the application of labelled operations $[g^\spadesuit]$ is connected with some changes or new effects, whereas the application of a bare operation brings about no such changes or effects.

The label $[^\spadesuit]$ can be replaced by various signs which can have different meanings. Thus in Chapter 3.3 a prime ['] signifies a nontrivial twinning operation, in Chapter 1.5 it is associated with time inversion in magnetic structures, and in black-and-white patterns or structures a prime denotes an operation which exchanges black and white `colours' (the qualifier `black-and-white' concerns group operations, but not the black-and-white pattern itself). In Chapter 3.4 , a star $[^\star]$ denotes a transposing operation which exchanges two domain states, while underlining signifies an operation exchanging two sides of an interface and underlined operations with a star signify twinning operations of a domain twin. Various interpretations of the label attached to the symbol of an operation have given rise to several designations of groups with partition (3.2.3.34): black-and-white, dichromatic, magnetic, anti-symmetry, Shubnikov or Heech–Shubnikov and other groups. For more details see Opechowski (1986).

References

Opechowski, W. (1986). Crystallographic and metacrystallographic groups. Amsterdam: North-Holland.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.2, p. 384