International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 3.2, p. 410
Section 3.2.3.3.6. Orbits of ordered pairs and double cosets^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic,^{b}Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and ^{c}Mineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany |
An ordered pair is formed by two objects from the orbit . Let denote the set of all ordered pairs that can be formed from the objects of the orbit . The group action of group G on the set is defined by the following relation:The requirements (3.2.3.47) to (3.2.3.49) are fulfilled, mapping (3.2.3.97) defines an action of group G on the set .
The group action (3.2.3.97) introduces the G-equivalence of ordered pairs: Two ordered pairs and are crystallographically equivalent (with respect to the group G), , if there exists an operation that transforms into ,
An orbit of ordered pairs comprises all ordered pairs crystallographically equivalent with . One can choose as a representative of the orbit an ordered pair with the first member since there is always an operation such that . The orbit assembles all ordered pairs with the first member . This orbit can be expressed aswhere the identity [see relation (3.2.3.70)] has been used.
Thus the double coset contains all operations from G that produce all ordered pairs with the first member that are G-equivalent with . If one chooses that is not contained in the double coset , then the ordered pair must belong to another orbit . Hence to distinct double cosets there correspond distinct classes of ordered pairs with the first member , i.e. distinct orbits of ordered pairs. Since the group G can be decomposed into disjoint double cosets [see (3.2.3.36)], one gets
Proposition 3.2.3.35 . Let G be a group and a set of all ordered pairs that can be formed from the objects of the orbit . There is a one-to-one correspondence between the G orbits of ordered pairs of the set and the double cosets of the decomposition This bijection allows one to express the partition of the set of all ordered pairs into G orbits, where is the set of representatives of double cosets in the decomposition (3.2.3.100) (Janovec, 1972).
Proposition 3.2.3.35 applies directly to pairs of domain states (domain pairs) and allows one to find twin laws that can appear in the low-symmetry phase (see Section 3.4.3 ).
For more details and other applications of group action see e.g. Kopský (1983), Lang (1965), Michel (1980), Opechowski (1986), Robinson (1982), and especially Kerber (1991, 1999).
References
Janovec, V. (1972). Group analysis of domains and domain pairs. Czech. J. Phys. B, 22, 974–994.Kerber, A. (1991). Algebraic combinatorics via finite group action. Mannheim: B. I. Wissenschaftsverlag.
Kerber, A. (1999). Applied finite group actions. Berlin: Springer.
Kopský, V. (1983). Algebraic investigations in Landau model of structural phase transitions, I, II, III. Czech. J. Phys. B, 33, 485–509, 720–744, 845–869.
Lang, S. (1965). Algebra. Reading, MA: Addison-Wesley.
Michel, L. (1980). Symmetry defects and broken symmetry. Configurations. Hidden symmetry. Rev. Mod. Phys. 52, 617–651.
Opechowski, W. (1986). Crystallographic and metacrystallographic groups. Amsterdam: North-Holland.
Robinson, D. J. S. (1982). A course in the theory of groups. New York: Springer.