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. 3.4, pp. 511512
Section 3.4.3.4. Classes of equivalent domain pairs and their classifications^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 Prague 8, Czech Republic, and ^{b}Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic 
Two domain pairs that are crystallographically equivalent, [see equation (3.4.3.5)], have different orientations in space but their inherent properties are the same. It is, therefore, useful to divide all domain pairs of a ferroic phase into classes of equivalent domain pairs. All domain pairs that are equivalent (in G) with a given domain pair, say , can be obtained by applying to all operations of G, i.e. by forming a Gorbit .
One can always find in this orbit a domain pair that has in the first place the first domain state . We shall call such a pair a representative domain pair of the orbit. The initial orbit and the orbit are identical:
The set of ordered pairs (including trivial ones) that can be formed from n domain states can be divided into Gorbits (classes of equivalent domain pairs):
Similarly, as there is a onetoone correspondence between domain states and left cosets of the stabilizer (symmetry group) of the first domain state [see equation (3.4.2.9)], there is an analogous relation between Gorbits of domain pairs and socalled double cosets of .
A double coset of is a set of left cosets that can be expressed as , where runs over all operations of (see Section 3.2.3.2.8 ). A group G can be decomposed into disjoint double cosets of : where is the set of representatives of double cosets.
There is a onetoone correspondence between double cosets of the decomposition (3.4.3.35) and Gorbits of domain pairs (3.4.3.34) (see Section 3.2.3.3.6 , Proposition 3.2.3.35 ):
We see that the representatives of the double cosets in decomposition (3.4.3.35) define domain pairs which represent all different Gorbits of domain pairs. Just as different left cosets specify all domain states, different double cosets determine all classes of equivalent domain pairs (Gorbits of domain pairs).
The properties of double cosets are reflected in the properties of corresponding domain pairs and provide a natural classification of domain pairs. A specific property of a double coset is that it is either identical or disjoint with its inverse. A double coset that is identical with its inverse, is called an invertible (ambivalent) double coset. The corresponding class of domain pairs consists of transposable (ambivalent) domain pairs.
A double coset that is disjoint with its inverse, is a noninvertible (polar) double coset ( denotes an empty set) and the corresponding class of domain pairs comprises nontransposable (polar) domain pairs. A double coset and its inverse are called complementary double cosets. Corresponding classes called complementary classes of equivalent domain pairs consist of transposed domain pairs that are nonequivalent.
Another attribute of a double coset is the number of left cosets which it comprises. If an invertible double coset consists of one left coset, then the domain pairs in the Gorbit are completely transposable. An invertible double coset comprising several left cosets is associated with a Gorbit consisting of partially transposable domain pairs. Noninvertible double cosets can be divided into simple nontransposable double cosets (complementary double cosets consist of one left coset each) and multiple nontransposable double cosets (complementary double cosets comprise more than one left coset each).
Thus there are four types of double cosets (see Table 3.2.3.1 in Section 3.2.3.2 ) to which there correspond the four basic types of domain pairs presented in Table 3.4.3.2.

These results can be illustrated using the example of a phase transition with G = with four domain states (see Fig. 3.4.2.2). The corresponding four left cosets of are given in Table 3.4.2.1. Any operation from the first left coset (identical with ) transforms the second left coset into itself, i.e. this left coset is a double coset. Since it consists of an operation of order two, it is a simple invertible double coset. The corresponding representative domain pair is . By applying operations of on , one gets the class of equivalent domain pairs (Gorbit): . These domain pairs can be labelled as `180° pairs' according to the angle between the spontaneous polarization in the two domain states.
When one applies operations from the first left coset on the third left coset, one gets the fourth left coset, therefore a double coset consists of these two left cosets. An inverse of any operation of this double coset belongs to this double coset, hence it is a multiple invertible double coset. Corresponding domain pairs are partially transposable ones. A representative pair is, for example, which is indeed a partially transposable domain pair [cf. (3.4.3.19) and (3.4.3.20)]. The class of equivalent ordered domain pairs is . These are `90° domain pairs'.
An example of noninvertible double cosets is provided by the decomposition of the group into left and double cosets of displayed in Table 3.4.3.3. The inverse of the second left coset (second line) is equal to the third left coset (third line) and vice versa. Each of these two left cosets thus corresponds to a double coset and these double cosets are complementary double cosets. Corresponding representative simple nontransposable domain pairs are and , and are depicted in Fig. 3.4.3.2.

We conclude that double cosets determine classes of equivalent domain pairs that can appear in the ferroic phase resulting from a phase transition with a symmetry descent . Left coset and double coset decompositions for all crystallographic pointgroup descents are available in the software GIKoBo1, path: Subgroups\View\Twinning groups.
A double coset can be specified by any operation belonging to it. This representation is not very convenient, since it does not reflect the properties of corresponding domain pairs and there are many operations that can be chosen as representatives of a double coset. It turns out that in a continuum description the twinning group can represent classes of equivalent domain pairs with two exceptions:
Bearing in mind these two exceptions, one can, in the continuum description, represent Gorbits of domain pairs by twinning groups .
We have used this correspondence in synoptic Table 3.4.2.7 of symmetry descents at ferroic phase transitions. For each symmetry descent , the twinning groups given in column specify possible Gorbits of domain pairs that can appear in the domain structure of the ferroic phase (Litvin & Janovec, 1999). We divide all orbits of domain pairs (represented by corresponding twinning groups ) that appear in Table 3.4.2.7 into classes of nonferroelastic and ferroelastic domain pairs and present them with further details in the three synoptic Tables 3.4.3.4, 3.4.3.6 and 3.4.3.7 described in Sections 3.4.3.5 and 3.4.3.6.
As we have seen, a classification of domain pairs according to their internal symmetry (summarized in Table 3.4.3.2) introduces a partition of all domain pairs that can be formed from domain states of the Gorbit into equivalence classes of pairs with the same internal symmetry. Similarly, any inherent physical property of domain pairs induces a partition of all domain pairs into corresponding equivalence classes. Thus, for example, the classification of domain pairs, based on tensor distinction or switching of domain states (see Table 3.4.3.1, columns two and three), introduces a division of domain pairs into corresponding equivalence classes.
References
Litvin, D. B. & Janovec, V. (1999). Classification of domain pairs and tensor distinction. Ferroelectrics, 222, 87–93.