Double cosets

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, pp. 384-385

Section 3.2.3.2.8. Double cosets

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

3.2.3.2.8. Double cosets

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Let $[F_{1}]$ and $[H_{1}]$ be two proper subgroups of the group G. The set of all distinct products $[hg_{j}f]$ , where $[g_{j}]$ is a fixed element of the group G and f and h run over all elements of the subgroups $[F_{1}]$ and $[H_{1}]$ , respectively, is called a double coset of $[F_{1}]$ and $[H_{1}]$ in G. The symbol of this double coset is $[H_{1}g_{j}F_{1}]$ , $[\displaylines{F_1g_jH_1=\{fg_jh|\, \forall f\in F_1, \forall h\in H_1\}, \cr\hfill \hfill g_j\in G, \, F_1\subset G, \, H_1\subset G, \hfill(3.2.3.35)}]$ where the sign $[\forall]$ means `for all'.

In the symmetry analysis of domain structures, only double cosets with $[H_{1}=F_{1}]$ are used. We shall, therefore, formulate subsequent definitions and statements only for this special type of double coset.

The fixed element $[g_{j}]$ is called the representative of the double coset $[F_{1}g_{j}F_{1}]$ . Any element of a double coset can be chosen as its representative.

Two double cosets are either identical or disjoint.

Proposition 3.2.3.6. The union of all distinct double cosets constitutes a partition of G and is called the decomposition of the group G into double cosets of $[F_{1}]$ , since [F_1F_1=F_1] . If the set of double cosets of $[F_{1}]$ in G is finite, then the decomposition of G into the double cosets of $[F_{1}]$ can be written as $[G=F_{1}g_1F_{1} \cup F_{1}g_2F_{1} \cup\ldots\cup F_{1}g_qF_{1}. \eqno(3.2.3.36)]$ For the representative [g_1] of the first double coset $[F_{1}g_{j}F_{1}]$ the unit element e is usually chosen, [g_1=e] . Then the first double coset is identical with the subgroup $[F_{1}]$ .

A double coset $[F_{1}g_jF_{1}]$ consists of left cosets of the form $[fg_{j}F_{1}]$ , where $[f \in F_{1}]$ . The number r of left cosets of $[F_{1}]$ in the double coset $[F_{1}g_{j}F_{1}]$ is (Hall, 1959) $[ r=[F_{1}:F_{1j}], \eqno(3.2.3.37)]$ where $[F_{1j}=F_{1} \cap g_{j}F_{1}g_{j}^{-1}. \eqno(3.2.3.38)]$

The following definitions and statements are used in Chapter 3.4 for the double cosets $[F_{1}g_jF_{1}]$ [for derivations and more details, see Janovec (1972)].

The inverse $[(F_{1}g_jF_{1})^{-1}]$ of a double coset $[F_{1}g_jF_{1}]$ is a double coset $[F_{1}g_j^{-1}F_{1}]$ , which is either identical or disjoint with the double coset $[F_{1}g_jF_{1}]$ . The double coset that is its own inverse is called an invertible (self-inverse, ambivalent) double coset. The double coset that is disjoint with its inverse is called a non-invertible (polar) double coset and the double cosets $[F_{1}g_jF_{1}]$ and $[(F_{1}g_jF_{1})^{-1}= F_{1}g_j^{-1}F_{1}]$ are called complementary polar double cosets.

The inverse left coset $[(g_jF_{1})^{-1}]$ contains representatives of all left cosets of the double coset $[F_{1}g_j^{-1}F_{1}]$ . If a left coset $[g_jF_{1}]$ belongs to an invertible double coset, then $[(g_{j}F_{1})^{-1}]$ contains representatives of left cosets constituting the double coset $[F_{1}g_{j}F_{1}]$ . If a left coset $[g_jF_{1}]$ belongs to a non-invertible double coset, then $[(g_{j}F_{1})^{-1}]$ contains representatives of left cosets constituting the complementary double coset $[(F_{1}g_{j}F_{1})^{-1}]$ .

A double coset consisting of only one left coset, $[F_{1}g_{j}F_{1}=g_{j}F_{1}, \eqno(3.2.3.39)]$ is called a simple double coset. A double coset $[F_{1}g_{j}F_{1}]$ is simple if and only if the inverse $[(g_{j}F_{1}) ^{-1}]$ of the left coset $[g_{j}F_{1}]$ is again a left coset. For an invertible simple double coset $[g_{j}F_{1}=(g_{j}F_{1})^{-1}]$ .

The union of all simple double cosets $[F_{1}g_{j}F_{1}=g_{j}F_{1}]$ in the double coset decomposition of G (3.2.3.36) constitutes the normalizer $[N_{G}(F_{1})]$ (Speiser, 1927).

A double coset that comprises more than one left coset will be called a multiple double coset. Four types of double cosets [FgF] are displayed in Table 3.2.3.1. The double coset decompositions of all crystallographic point groups are available in the software GI $[\star]$ KoBo-1 under Subgroups\View\Twinning Group.

Table 3.2.3.1 | top | pdf |
Four types of double cosets

		$[FgF\neq gF ]$
$[FgF=(FgF)^{-1}]$	Invertible simple	Invertible multiple
$[FgF \cap (FgF)^{-1}=\emptyset]$	Non-invertible simple	Non-invertible multiple

Double cosets and the decomposition (3.2.3.36) of a group in double cosets are mathematical tools for partitioning a set of pairs of objects into equivalent classes (see Section 3.2.3.3.6). Such a division enables one to find possible twin laws and different types of domain walls that can appear in a domain structure resulting from a phase transition with a given symmetry descent (see Chapters 3.3 and 3.4 ).

More detailed introductions to group theory can be found in Budden (1972), Janssen (1973), Ledermann (1973), Rosen (1995), Shubnikov & Koptsik (1974), Vainshtein (1994) and Vainshtein et al. (1995). More advanced books on group theory are, for example, Bradley & Cracknell (1972), Hall (1959), Lang (1965), Opechowski (1986), Robinson (1982) and Speiser (1927). Parts of group theory relevant to phase transitions and tensor properties are treated in the manual of the software GI $[\star]$ KoBo-1. Representations of the crystallographic groups are presented in Chapter 1.2 of this volume and in the software GI $[\star]$ KoBo-1 (see the manual).

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International Tables for Crystallography (2006). Vol. D. ch. 3.2, pp. 384-385