Stabilizers (isotropy groups)

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. 386-387

Section 3.2.3.3.2. Stabilizers (isotropy groups)

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

3.2.3.3.2. Stabilizers (isotropy groups)

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The concept of a stabilizer is closely connected with the notion of the symmetry group of an object. Under the symmetry group F of an object $[{\bf S}]$ one understands the set of all operations (isometries) that map the object onto itself, i.e. leave this object $[{\bf S}]$ invariant. In this approach, one usually `attaches' the symmetry elements to the object. Then the symmetry group F of the object is its inherent property which does not depend on the orientation and position of the object in space. The term eigensymmetry is used in Chapter 3.3 for symmetry groups defined in this way.

The notion of a stabilizer describes the symmetry properties of an object from another standpoint, in which the object and the group of isometries are decoupled and introduced independently. One chooses a reference coordinate system and a group G of isometries, the operations of which have a defined orientation in this reference system. Usually, it is convenient to choose as the reference system the standard coordinate system (crystallographic or crystallophysical) of the group G. The object $[{\bf S}_i]$ under consideration is specified not only per se but also by its orientation in the reference system. Those operations of G that map the object in this orientation onto itself form a group called the stabilizer of $[{\bf S}_i]$ in the group G. An algebraic definition is formulated in the following way:

Definition [s] 3.2.3.12 . The stabilizer (isotropy group) $[I_G({\bf S}_i)]$ of an object $[{\bf S}_i]$ of a G-set $[{\sf A}]$ in group G is that subgroup of G comprised of all operations of G that do not change $[{\bf S}_i]$ , $[I_G({\bf S}_i)=\{g \in G|g{\bf S}_i={\bf S}_i\}, \quad g\in G, \ {\bf S}_i\in {\sf A}. \eqno(3.2.3.54)]$

Unlike the `eigensymmetry', the stabilizer $[I_G({\bf S}_i)]$ depends on the group G, is generally a subgroup of G, $[I_G({\bf S}_i)\subseteq G]$ , and may change with the orientation of the object $[{\bf S}_i]$ .

There is an important relation between stabilizers of two objects from a G-set (see e.g. Aizu, 1970; Kerber, 1991):

Proposition 3.2.3.13 . Consider two objects $[{\bf S}_i, {\bf S}_k]$ from a G-set related by an operation g from the group G. The respective stabilizers $[I_G({\bf S}_i), I_G({\bf S}_k)]$ are conjugate by the same operation g, $[\hbox{if } {\bf S}_k=g{\bf S}_i, \hbox{ then } I_G({\bf S}_k)=gI_G({\bf S}_i)g^{-1}. \eqno(3.2.3.55)]$

Let us illustrate the meaning of stabilizers with four examples of group action considered above.

Example [sP] 3.2.3.14 . Let $[\cal G]$ be a crystallographic space group and X a point of the three-dimensional point space [E(3)] (see Example 3.2.3.7). The stabilizer $[{\cal I_G}(X)]$ , called the site-symmetry group of the point X in $[\cal G]$ , consists of all symmetry operations of $[\cal G]$ that leave the point X invariant. Consequently, the stabilizer $[{\cal I_G}(X)]$ is a crystallographic point group. If the stabilizer $[{\cal I_G}(X)]$ consists only of the identity operation, then the point X is called a point of general position. If $[{\cal I_G}(X)]$ is a non-trivial point group, X is called a point of special position (IT A , 2005).

Example [sC] 3.2.3.15 . The symmetry of domain states $[{\bf S}_i, {\bf S}_k,\ldots]$ , treated in Example [sP] 3.2.3.9, is adequately expressed by their stabilizers in the group G of the parent (high-symmetry) phase , $[I_G({\bf S}_i) = F_i]$ , $[I_G({\bf S}_k) = F_k,\ldots]$ . These groups are called symmetry groups of domain states. If domain states $[{\bf S}_i, {\bf S}_k]$ are related by an operation $[g\in G]$ , then their symmetry groups are, according to (3.2.3.55), conjugate by g, $[\hbox{if } {\bf S}_k=g{\bf S}_i \hbox{ then } F_k= gF_ig^{-1}. \eqno(3.2.3.56)]$

Symmetry characterization of domain states by their stabilizers properly reflects a difference between ferroelastic single domain states and ferroelastic disoriented domain states (see Sections 3.4.3 and 3.4.4 ).

Example [sT] 3.2.3.16 . The notion of the stabilizer enables one to formulate a basic relation between the symmetry group of the parent phase, the symmetry group of the first domain state $[{\bf S}_1]$ and order parameters of the transition. In a microscopic description, the symmetry of the parent phase is described by a space group $[\cal G]$ and the symmetry of the first basic (microscopic) single domain state $[{\sf S}_1]$ by the stabilizer $[{\cal I}_{\cal G}({\sf S}_1)={\cal F}_1]$ . The stabilizer of the primary order parameter $[{\eta}^{(1)}]$ must fulfil the condition $[I_{\cal G}({\eta}^{(1)})=I_{\cal G}({\sf S}_1)={\cal F}_1. \eqno(3.2.3.57)]$ The appearance of nonzero $[{\eta}^{(1)}]$ in the ferroic phase thus fully accounts for the symmetry descent $[{\cal G}\supset {\cal F}_1]$ at the transition.

In a continuum description, a role analogous to $[{\eta}^{(1)}]$ is played by a principal tensor parameter $[\mu^{(1)}]$ (see Section 3.1.3 ). Its stabilizer $[I_G({\mu}^{(1)})]$ in the parent point group G equals the point group [F_1] of the first single domain state $[{\sf S}_1]$ , $[I_G({\mu}^{(1)})=I_G({\bf S}_1)=F_1. \eqno(3.2.3.58)]$

This contrasts with the secondary order parameter $[\lambda^{(1)}]$ (secondary tensor parameter in a continuum description). Its stabilizer $[I_G({\lambda}^{(1)})=L_1\eqno(3.2.3.59)]$ is an intermediate group $[F_1\subset L_1\subset G]$ , i.e. the appearance of $[{\lambda}^{(1)}]$ would lead only to a partial symmetry descent $[G \supset L_1]$ with $[L_1 \supset F_1]$ .

Example [sS] 3.2.3.17 . The stabilizer of a subgroup $[F_i\subset G]$ from Example [aS] 3.2.3.11 is the normalizer [N_G(F_i)] defined in Section 3.2.3.2.5: $[I_G(F_i)=\{g\in G|gF_ig^{-1}=F_i\}=N_G(F_i).\eqno(3.2.3.60)]$

In general, a stabilizer, which is a subgroup of G, is an example of a structure which is induced by a group action on the group G. On the other hand, a group action exerts a partition of the set $[{\sf A}]$ into equivalence classes called orbits.

References

International Tables for Crystallography (2005). Vol. A. Space-group symmetry, 5th edition, edited by Th. Hahn. Heidelberg: Springer.Google Scholar

Aizu, K. (1970). Possible species of ferromagnetic, ferroelectric and ferroelastic crystals. Phys. Rev. B, 2, 754–772.Google Scholar

Kerber, A. (1991). Algebraic combinatorics via finite group action. Mannheim: B. I. Wissenschaftsverlag.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.2, pp. 386-387