Intermediate subgroups and partitions of an orbit into suborbits

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. 388-390

Section 3.2.3.3.5. Intermediate subgroups and partitions of an orbit into suborbits

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

3.2.3.3.5. Intermediate subgroups and partitions of an orbit into suborbits

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Proposition 3.2.3.30 . Let $[G{\bf S}_1]$ be a G orbit from Proposition 3.2.3.23 and [L_1] an intermediate group, $[F_1 \subset L_1 \subset G. \eqno(3.2.3.78)]$ A successive decomposition of G into left cosets of [L_1] and [L_1] into left cosets of [F_1] [see (3.2.3.25)] introduces a two-indices relabelling of the objects of a G orbit defined by the one-to-one correspondence $[h_jp_kF_1 \leftrightarrow {\bf S}_{jk}, \quad j=1,2,\ldots m, \quad k=1,2,\ldots, d, \eqno(3.2.3.79)]$ where $[\{h_1,h_2,\ldots,h_m\}]$ are the representatives of the decompositions of G into left cosets of [L_1] , $[G=h_1L_{1} \cup h_2L_{1} \cup\ldots\cup h_jL_{1}\cup\ldots \cup h_mL_{1}, \quad m=[G:L_1], \eqno(3.2.3.80)]$ and $[\{p_1,p_2,\ldots,p_d\}]$ are the representatives of the decompositions of [L_1] into left cosets of [F_1] , $[L_{1}=p_1F_{1} \cup p_2F_{1} \cup\ldots \cup p_kF_{1}\cup\ldots \cup p_dF_{1}, \quad d=[L_1:F_1]. \eqno(3.2.3.81)]$

The index n of $[F_{1}]$ in G can be expressed as a product of indices m and d [see (3.2.3.26)], $[n=[G:F_1]=[G:L_{1}][L_{1}:F_{1}]=md. \eqno(3.2.3.82)]$ If G is a finite group, then the index n can be expressed in terms of orders of groups G, [F_1] and [L_1] : $[n=|G|:|F_1|=(|G|:|L_{1}|)(|L_{1}|:|F_{1}|)=md. \eqno(3.2.3.83)]$

When one chooses $[{\bf S}_{1} ={\bf S}_{11}]$ , then the members of the orbit $[G{\bf S}_{11}]$ can be arranged into an $[m\times d]$ array, $[\matrix{{\bf S}_{11} & {\bf S}_{12} &\ldots &{\bf S}_{1k} &\ldots&{\bf S}_{1d} \cr {\bf S}_{21} & {\bf S}_{22} &\ldots &{\bf S}_{2k} &\ldots &{\bf S}_{2d} \cr\vdots &\vdots&\ddots &\vdots &\ddots &\vdots\cr{ \bf S}_{j1} &{ \bf S}_{j2} &\ldots &{\bf S}_{jk} &\ldots &{\bf S}_{jd} \cr\vdots &\vdots &\ddots &\vdots &\ddots &\vdots\cr {\bf S}_{m1} & {\bf S}_{m2} &\ldots &{\bf S}_{mk} &\ldots &{\bf S}_{md}} \eqno(3.2.3.84)]$

The set of objects of the jth row of this array forms an [L_j] orbit with the representative $[{\bf S}_{j 1}]$ , $[\eqalignno{ &\{ {\bf S}_{j1}, {\bf S}_{j2},\ldots, {\bf S}_{jk},\ldots, {\bf S}_{jd}\}&\cr&\quad = \{h_jp_1{\bf S}_{11}, h_jp_2{\bf S}_{11},\ldots, h_jp_k{\bf S}_{11},\ldots, h_jp_d{\bf S}_{11}\}&\cr&\quad= L_j{\bf S}_{j 1},&(3.2.3.85)}]$ where $[L_j=h_jL_1h_j^{-1}, \quad {\bf S}_{j 1}=h_j{\bf S}_{11}, \quad j=1,2,\ldots, m. \eqno(3.2.3.86)]$ The intermediate group [L_1] thus induces a splitting of the orbit $[G{\bf S}_{11}]$ into m suborbits $[L_j{\bf S}_{j 1}]$ , $[j=1,2,\ldots,m]$ : $[G{\bf S}_{11}=L_1{\bf S}_{11} \cup L_2{\bf S}_{21} \cup\ldots\cup L_j{\bf S}_{j1}\cup\ldots\cup L_m{\bf S}_{m1}, \quad m=[G:L_1]. \eqno(3.2.3.87)]$ Aizu (1972) denotes this partitioning factorization of species.

The relation (3.2.3.79) is just the application of the correspondence (3.2.3.69) of Proposition 3.2.3.23 on the successive decomposition (3.2.3.25). Derivation of the second part of Proposition 3.2.3.30 can be sketched in the following way: $[\eqalignno{\{ {\bf S}_{j1}, {\bf S}_{j2},\ldots, {\bf S}_{jd}\}&= h_j\{p_1{\bf S}_{11},p_2{\bf S}_{11},\ldots, p_d{\bf S}_{11}\}&\cr&=h_j\{p_1, p_2,\ldots, p_d\}F_1{\bf S}_{11}&\cr &= h_jL_1{\bf S}_{11} = h_jL_1h_j^{-1}{\bf S}_{j1} =L_j{\bf S}_{j1}, &\cr j&=1,2,\ldots, m, &(3.2.3.88)}]$ where the relation (3.2.3.70) is used.

We note that the described partitioning of an orbit into suborbits depends on the choice of representative of the first suborbit $[{\bf S}_{11}]$ and that the number of conjugate subgroups [L_j] may be equal to or smaller than the number m of suborbits (see Example [oS] 3.2.3.34).

Each intermediate group [L_1] in Proposition 3.2.3.30 can usually be associated with a certain attribute, e.g. a secondary order parameter , which specifies the suborbits.

Example [oP] 3.2.3.31 . Let G be a point group and [X_1] a point of general position [(I_G(X_1)=e)] in the point space. A symmetry descent to a subgroup $[L_1\subset G]$ is accompanied by a splitting of the orbit [GX_1] of [|G|] equivalent points into [m=|G|:|L_1|] suborbits each consisting of [|L_1|] equivalent points. The first suborbit is [L_1X_1] , the others are [L_jX_j] , $[L_j=h_jL_1h_j^{-1}]$ , [X_j=h_jX_1] , $[j=1,2,\ldots,m]$ , where [h_j] are representatives of left cosets of [L_1] in the decomposition of G [see (3.2.3.80)].

Splitting of orbits of points of general position is a special case in which [I_L(X_1)=I_G(X_1)] . Splitting of orbits of points of special position is more complicated if $[I_L(X_1)\subset I_G(X_1)]$ (see Wondratschek, 1995).

Example [oC] 3.2.3.32 . Let us consider a phase transition accompanied by a lowering of space-group symmetry from a parent space group $[\cal G]$ with translation subgroup T and point group G to a low-symmetry space group $[\cal F]$ with translation subgroup U and point group F. There exists a unique intermediate group $[\cal M]$ , called the group of Hermann, which has translation subgroup T and point group [M=F] (see e.g. Hahn & Wondratschek, 1994; Wadhawan, 2000; Wondratschek & Aroyo, 2001).

The decomposition of $[{\cal G}]$ into left cosets of $[{\cal M}]$ , corresponding to (3.2.3.80), is in a one-to-one correspondence with the decomposition of G into left cosets of F, since $[{\cal G}]$ and $[{\cal M}]$ have the same translation subgroup $[\bf{T}]$ and $[{\cal M}]$ and $[{\cal F}]$ have the same point group. Therefore, the index $[n\equiv [{\cal G}:{\cal M}]=[G:F]=|G|:|F|]$ .

Since $[{\cal M}]$ and $[{\cal F}]$ have the same point group F, the decomposition of $[{\cal M}]$ into left cosets of $[{\cal F}]$ , corresponding to (3.2.3.81), is in a one-to-one correspondence with the decomposition of $[\bf{T}]$ into left cosets of $[\bf{U}]$ , $[{\bf T}={\bf t}_1{\bf U}+{\bf t}_2{\bf U}+\ldots+ {\bf t}_d{\bf U}. \eqno(3.2.3.89)]$ Representatives $[{\bf t}_1, {\bf t}_2,\ldots {\bf t}_d]$ are translations. The corresponding vectors lead from the origin of a `superlattice' primitive unit cell of the low-symmetry phase to lattice points of $[\bf{T}]$ located within or on the side faces of this `superlattice' primitive unit cell (Van Tendeloo & Amelinckx, 1974). The number [d_t] of these vectors is equal to the ratio $[v_{\cal F}:v_{\cal G}=Z_{\cal F}:Z_{\cal G}]$ , where $[v_{\cal F}]$ and $[v_{\cal G}]$ are the volumes of the primitive unit cell of the low-symmetry phase and the parent phase , respectively, and $[Z_{\cal F}]$ and $[Z_{\cal G}]$ are the number of chemical formula units in the primitive unit cell of the low-symmetry phase and the parent phase, respectively.

There is another useful formula for expressing $[d_t=[{\bf T}:{\bf U}]]$ . The primitive basis vectors $[{\bf b}_1, {\bf b}_2, {\bf b}_3]$ of $[\bf U]$ are related to the primitive basis vectors $[{\bf a}_1, {\bf a}_2, {\bf a}_3]$ of $[\bf{T}]$ by a linear relation, $[{\bf b}_i=\textstyle\sum\limits_{j=1}^{3}{\bf a}_jm_{ji}, \quad i=1,2,3, \eqno(3.2.3.90)]$ where $[m_{ji}]$ are integers. The volumes of primitive unit cells are $[v_{\cal G}={\bf a}_1({\bf a}_2\times {\bf a}_3)]$ and $[v_{\cal F}={\bf b}_1({\bf b}_2\times {\bf b}_3)]$ . Using (3.2.3.90), one gets $[v_{\cal F}={\rm det}(m_{ij})v_{\cal G}]$ , where $[ {\rm det}(m_{ij})]$ is the determinant of the $[(3\times 3)]$ matrix of the coefficients $[m_{ij}]$ . Hence the index $[d_t=(v_{\cal F}:v_{\cal G})={\rm det}(m_{ij})]$ .

Thus we get for the index N of $[{\cal F}]$ in $[{\cal G}]$ $[\eqalignno{N&= [{\cal G}:{\cal F}]= [G:F][{\bf T}:{\bf U}]&\cr &= (|G|:|F|)(v_{\cal F}:v_{\cal G})=(|G|:|F|)(Z_{\cal F}:Z_{\cal G})&\cr&= (|G|:|F|) {\rm det}(m_{ij})=nd_t. &(3.2.3.91)}]$

Each suborbit, represented by a row in the array (3.2.3.84), contains all basic (microscopic) domain states that are related by pure translations. These domain states exhibit the same tensor properties, i.e. they belong to the same ferroic domain state.

Example [sT] 3.2.3.33 . Let us consider a phase transition with a symmetry descent $[G \supset F_1]$ with an orbit $[G{\bf S}_{11}]$ of domain states. Let [L_1] be an intermediate group, $[F_1 \subset L_1 \subset G]$ , and $[{\lambda}^{(1)}]$ the principal order parameter associated with the symmetry descent $[G \supset L_1]$ [cf. (3.2.3.58)], $[I_G({\lambda}^{(1)})=L_1]$ . Since [L_1] is an intermediate group, the quantity $[{\lambda}^{(1)}]$ represents a secondary order parameter of the symmetry descent $[G \supset F_1]$ . The G orbit of $[{\lambda}^{(1)}]$ is $[G{\lambda}^{(1)} =\{{\lambda}^{(1)}, {\lambda}^{(2)},\ldots, {\lambda}^{(m)}\}, \quad m=[G:L_1]. \eqno(3.2.3.92)]$ As in Example [oT] 3.2.3.28, there is a bijection between left cosets of the decomposition of G into left cosets of [L_1] [see (3.2.3.80)] and the G orbit of secondary order parameters (3.2.3.92). One can, therefore, associate with the suborbit $[L_j{\bf S}_{j1}]$ the value $[{\lambda}^{(j)}]$ of the secondary order parameter $[\lambda]$ , $[L_j{\bf S}_{j1} \leftrightarrow {\lambda}^{(j)}, \quad j=1,2,\ldots, m. \eqno(3.2.3.93)]$ A suborbit $[L_j{\bf S}_{j1}]$ is thus comprised of objects of the orbit $[G{\bf S}_{11}]$ with the same value of the secondary order parameter $[\lambda^{(j)}]$ .

Example [oS] 3.2.3.34 . Let us choose for the intermediate group [L_1] the normalizer [N_G(F_1)] . Then the suborbits equal $[\displaylines{N_G(F_j){\bf S}_{j1}=\{h_j{\bf S}_{11},h_jp_{2}{\bf S}_{11},\ldots, h_jp_{d}{\bf S}_{11}\},\cr\hfill\hfill j=1,2,\ldots, m=[G:N_G(F_1)], \hfill(3.2.3.94)}]$ where $[p_{1}=e,p_{2},\ldots, p_{d}]$ are representatives of left cosets $[p_{k}F_1]$ in the decomposition of [N_G(F_1)] , $[N_G(F_{1})=p_1F_{1} \cup p_2F_{1} \cup\ldots \cup p_dF_{1}, \quad d=[N_G(F_1):F_1], \eqno(3.2.3.95)]$ and [h_j] are representatives of the decomposition (3.2.3.77). The suborbit $[ F_j{\bf S}_{j1}]$ consists of all objects with the same stabilizer [F_j] , $[\displaylines{I_G({\bf S}_{j1})=I_G({\bf S}_{j2})=\ldots= I_G({\bf S}_{jd})=F_j, \cr j=1,2,\ldots, m=[G:N_G(F_1)]. \cr\hfill(3.2.3.96)}]$

Propositions 3.2.3.23 and 3.2.3.30 are examples of structures that a group action induces from a group G on a G-set. Another important example is a permutation representation of the group G which associates operations of G with permutations of the objects of the orbit $[G{\bf S}_i]$ [see e.g. Kerber (1991, 1999); for application of the permutation representation in domain-structure analysis and domain engineering, see e.g. Fuksa & Janovec (1995, 2002)].

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International Tables for Crystallography (2006). Vol. D. ch. 3.2, pp. 388-390