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

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

V. Janovec,a* Th. Hahnb and H. Klapperc

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, 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

3.2.3.3.6. Orbits of ordered pairs and double cosets

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An ordered pair [({\bf S}_i,{\bf S}_k)] is formed by two objects [{\bf S}_i, {\bf S}_k] from the orbit [G{\bf S}_1]. Let [{\sf P}] denote the set of all ordered pairs that can be formed from the objects of the orbit [G{\bf S}_1]. The group action [\varphi] of group G on the set [{\sf P}] is defined by the following relation:[\displaylines{\varphi:g({\bf S}_i,{\bf S}_k)=(g{\bf S}_i,g{\bf S}_k)=({\bf S}_r,{\bf S}_s), \cr \hfill \hfill g\in G,\quad ({\bf S}_i,{\bf S}_k), ({\bf S}_r, {\bf S}_s)\in {\sf P}.\hfill(3.2.3.97)}]The requirements (3.2.3.47)[link] to (3.2.3.49)[link] are fulfilled, mapping (3.2.3.97)[link] defines an action of group G on the set [\sf P].

The group action (3.2.3.97)[link] introduces the G-equivalence of ordered pairs: Two ordered pairs [({\bf S}_i,{\bf S}_k)] and [({\bf S}_r,{\bf S}_s)] are crystallographically equivalent (with respect to the group G), [({\bf S}_i,{\bf S}_k)\,\lower2pt\hbox{${\buildrel{G}\over{\sim}}$}\,({\bf S}_r,{\bf S}_s)], if there exists an operation [g\in G] that transforms [({\bf S}_i,{\bf S}_k)] into [({\bf S}_r,{\bf S}_s)], [g \in G\,\, (g{\bf S}_i,g{\bf S}_k)=({\bf S}_r,{\bf S}_s), \quad ({\bf S}_i,{\bf S}_k), ({\bf S}_r, {\bf S}_s)\in {\sf P}.\eqno(3.2.3.98)]

An orbit of ordered pairs [G({\bf S}_i,{\bf S}_k)] comprises all ordered pairs crystallographically equivalent with [({\bf S}_i,{\bf S}_k)]. One can choose as a representative of the orbit [G({\bf S}_i,{\bf S}_k)] an ordered pair [({\bf S}_1,{\bf S}_j)] with the first member [{\bf S}_1] since there is always an operation [g_{1i}\in G] such that [g_{i1}{\bf S}_i={\bf S}_1]. The orbit [F_1({\bf S}_1,{\bf S}_j)] assembles all ordered pairs with the first member [{\bf S}_1]. This orbit can be expressed as[\eqalignno{F_1({\bf S}_1,{\bf S}_j)&=(F_1{\bf S}_1,F_1{\bf S}_j)=({\bf S}_1,F_1(g_j{\bf S}_1))&\cr&=({\bf S}_1,(F_1g_j)(F_1{\bf S}_1))=({\bf S}_1,(F_1g_jF_1){\bf S}_1),&\cr&&(3.2.3.99)}]where the identity [F_1{\bf S}_1={\bf S}_1] [see relation (3.2.3.70)[link]] has been used.

Thus the double coset [F_1g_jF_1] contains all operations from G that produce all ordered pairs with the first member [{\bf S}_1] that are G-equivalent with [({\bf S}_1,{\bf S}_j=g_j{\bf S}_1)]. If one chooses [g_r\in G] that is not contained in the double coset [F_1g_jF_1], then the ordered pair [({\bf S}_1,{\bf S}_r=g_r{\bf S}_1)] must belong to another orbit [G({\bf S}_1,{\bf S}_r)] [\neq] [G({\bf S}_1,{\bf S}_j)]. Hence to distinct double cosets there correspond distinct classes of ordered pairs with the first member [{\bf S}_1], i.e. distinct orbits of ordered pairs. Since the group G can be decomposed into disjoint double cosets [see (3.2.3.36)[link]], one gets

Proposition 3.2.3.35 . Let G be a group and [{\sf P}] a set of all ordered pairs that can be formed from the objects of the orbit [G{\bf S}_1]. There is a one-to-one correspondence between the G orbits of ordered pairs of the set [{\sf P}] and the double cosets of the decomposition [\displaylines{G=F_{1} \cup F_{1}g_2F_{1} \cup\ldots\cup F_{1}g_jF_{1}\cup\ldots\cup F_{1}g_qF_{1},\cr\hfill\hfill j=1,2,\ldots q. \hfill(3.2.3.100)\cr\hfill\hfill G({\bf S}_1,{\bf S}_j)\leftrightarrow F_{1}g_jF_{1} \hbox{ where } {\bf S}_j=g_j{\bf S}_1.\hfill(3.2.3.101)}%fd3.2.3.101]This bijection allows one to express the partition of the set [\sf P] of all ordered pairs into G orbits, [{\sf P} = G({\bf S}_1,{\bf S}_1) \cup G({\bf S}_1,g_2{\bf S}_1) \cup\ldots \cup ({\bf S}_1,{\bf S}_j) \cup\ldots\cup G({\bf S}_1,g_q{\bf S}_1), \eqno(3.2.3.102)]where [\{g_1=e, g_2,\ldots g_j,\ldots g_q\}] is the set of representatives of double cosets in the decomposition (3.2.3.100)[link] (Janovec, 1972[link]).

Proposition 3.2.3.35[link] 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[link] ).

For more details and other applications of group action see e.g. Kopský (1983[link]), Lang (1965[link]), Michel (1980[link]), Opechowski (1986[link]), Robinson (1982[link]), and especially Kerber (1991[link], 1999[link]).

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.








































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