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

International Tables for Crystallography (2013). Vol. D, ch. 3.1, pp. 392-394

Section 3.1.6. Group informatics and tensor calculus

V. Kopskýe* and P. Bočeka

3.1.6. Group informatics and tensor calculus

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We shall briefly describe here the intentions and contents of the accompanying software package GI[\star]KoBo-1 (Group Informatics, first two letters of authors names, release 1). A more detailed description is contained in the manual; the user may consult this file on the screen, but we recommend that it is printed out and that the printout is followed in order to become familiar with the theoretical background as well as with more detailed instructions for the use of the software.

The main purpose of this software is to describe the changes of tensor properties of crystalline materials during ferroic phase transitions, including basic information about domain states. The software provides powerful information in a standardized manner and it is based on a few advanced methodical points that are not yet available in textbooks. These points are:

  • (i) The introduction of typical variables, which was inspired by the symbolic method of the old invariant theory (Weitzenböck, 1923[link]).

  • (ii) The method of Clebsch–Gordan products (Kopský, 1976a[link],b[link]). The name stems from Clebsch–Gordan coefficients, known in quantum mechanics as coefficients of momentum addition. In this case, the coefficients are connected with the orthogonal group [{\cal O}(3)]; analogous coefficients were later introduced and calculated for crystal point groups (Koster et al., 1963[link]). They appear in Clebsch–Gordan products, which represent a better adaptation of results for our purposes.

  • (iii) Tables of tensorial covariants (Kopský, 1979a[link],b[link]). The name covariant may sound rather unusual now, but it was originally used by Weyl (1946[link]); it is equivalent to symmetry-adapted bases (form-invariant bases and other terms are also used). The term covariant is classical and its semantical use is easier.

  • (iv) Tables of fine structures of domain states (Kopský, 1982[link]). These are contained in a booklet which is practically unknown though, together with tables of tensorial covariants, it contains all answers concerning changes of tensor properties at ferroic phase transitions.

    Remark. The original term fine domain structure corresponds to the term degenerate domain structure in Chapter 3.4.

  • (v) Extended integrity bases (Patera et al., 1978[link]; Kopský, 1979c[link]). These represent finite sets of polynomial invariants and covariants suitable for the calculation of all types of interactions in symmetric systems.

  • (vi) Lattices of subgroups (Ascher, 1968[link]; Kopský, 1982[link]). Subgroups of a group constitute a partially ordered set of special properties called a lattice. The unfortunate coincidence of the term (in English) with crystallographic lattices should be disregarded; it is always possible to see from the context what we mean by this term.

These methods provide good ammunition for all types of group-theoretical considerations where work with characters is insufficient and knowledge of the explicit bases of irreducible representations is necessary. This is exactly the case for the theory of structural phase transitions, and the consideration of domain states, pairs of domain states and domain walls or twin boundaries. The main results of the software are contained in tables of symmetry descents [G \Downarrow H] and/or [G \Downarrow F_{i}], where G is the parent point group, H its normal subgroup and [F_{i}] is the set of conjugate subgroups. These tables provide information about changes of tensors at ferroic phase transitions as well as basic information about interactions, and they are also supplemented by tables of equitranslational subgroups of space groups.

To make this exposition quite clear, we begin in the manual from the beginning with a brief review of elementary group-theoretical concepts used in the software. Relevant elementary tables (listed below in Section A[link]) are followed by more advanced information proceeding towards the central goal of providing information for all symmetry descents (Section B[link]). To achieve this goal, it was also necessary to introduce our own standard notation for specifically oriented groups, for their elements and for irreducible representations (ireps). The reasons for the introduction of these standards are twofold: (i) There is no unique and commonly accepted notation in the literature. The recent book by Altmann & Herzig (1994[link]) contains slight inconsistencies (different symbols for elements in a group and in its subgroup) and is also not compatible with another prominent source (Bradley & Cracknell, 1972[link]). (ii) We need a strict specification of groups and their subgroups with reference to a Cartesian coordinate system and a strict specification of matrix ireps; neither is available in the literature. This does not mean that we introduce brand-new symbols; we simply adapt those that are already in use and we take extreme care that every symbol has a unique meaning.

It is recommended that users follow the manual when first using the software. The tabular content is as follows:

  • A. General information

    On opening the program, a panel Crystallographic appears on the screen. In the left-hand part are listed crystallographic geometric classes in a tree form wrapped to crystallographic systems (families). A click on a system brings to the screen a list of its crystallographic geometric classes as either Hermann–Mauguin, Schoenflies or Shubnikov symbols (the latter are used only on this level).

    The default choice for the notation is set in a table Options under the pull-down menu File. Hermann–Mauguin or Schoenflies symbols are offered for groups, standard or spectroscopic notation for group elements, classes of ireps and classes of conjugate elements. The temporary notation, under the pull-down menu Notation, also includes the choice of Shubnikov symbols.

    When the user clicks on a geometric class, all specifically oriented groups of this class used in the software appear in the right-hand part of the panel. At the top appear the groups in standard orientation that are further used as the standard parent groups, all remaining groups appearing below. For some groups two or three standard orientations are available, for reasons explained in the manual. The option of using Hermann–Mauguin or Schoenflies symbols for groups is available throughout; up to a certain point there are options for using either standard symbols of group elements and of ireps (as defined here) or spectroscopic symbols. A specific group is picked using the left mouse button, then a click with the right mouse button opens a pull-down menu. The following information is available under the titles:

    • (1) Basic info: This contains information relevant to each group of the geometric class, such as the number of elements, generators, isomorphism type, normalizer, number of conjugacy classes and the standard orientations.

    • (2) Group elements: Activates the group calculator, the keys of which list elements of the group. Performs calculation of products (strings) of up to ten elements by left or right multiplication.

    • (3) Correlation stnd./spectro: Symbols of groups, elements and classes of ireps for standard and spectroscopic notation are correlated.

    • (4) Class structure: Symbols of classes of conjugate elements are defined and elements of classes are listed.

    • (5) Class multiplication table: Displays class multiplication formulae.

    • (6) Character table: Standard and spectroscopic symbols of classes of ireps are specified and kernels of ireps are presented.

    • (7) Kronecker products: Tables of Kronecker products of classes of ireps are displayed. Up to this point, both standard and spectroscopic notation are used.

    • (8) Ireps and standard variables: Irreducible matrix representations (ireps) are explicitly defined. These also define the standard symbols of typical variables.

      Brief: Brief tables specify matrices of ireps for generators of the group.

      Full: Full tables provide these matrices for each group element. Kernels of ireps are presented. From this point onwards only the standard notation is used.

    • (9) Clebsch–Gordan products: Up to orthorhombic groups, one Clebsch–Gordan product table is given. For groups of higher systems, there are two options: complex Clebsch–Gordan tables, where variables [(\xi_{\alpha}, \eta_{\alpha})] are used, and real Clebsch–Gordan tables, with variables [(x_{\alpha}, y_{\alpha})] for two-dimensional real ireps.

    • (10) Tensorial covariants: Decomposition of tensors up to fourth rank into their covariant components is displayed.

  • B. Ferroic phase transitions

    • (11) Subgroups: Choice of this item displays a panel with the lattice of subgroups of the originally chosen point group G either in Schoenflies or in Hermann–Mauguin symbols. The pull-down menu Graph in the upper bar enables handling of the lattice: each item can be picked and moved to another place, the rearranged lattice can be fixed as default or reset, and the lattice can be sent to a printer. Notice that sets of conjugate subgroups are stacked like a pile of sheets of paper and can be unstacked. Consecutive clicks activate individual subgroups of the set. In the upper part of the panel, ireps of the parent group G in spectroscopic symbols are listed; these are followed by boxes in which it is possible to scroll for special vectors of the respective carrier space in terms of typical variables. Clicking on a vector marks its stabilizer (epikernel of the irep) in the lattice and activates it at the same time. With each irep is associated a table of the extended integrity basis; the table is called either by the option Integrity bases under the pull-down menu View or directly by using the right mouse button in the box of this irep. If the Ctrl key is pressed after the choice of one vector, another vector may be chosen; as a result, an intersection of the respective epikernels is marked in the lattice.

      Each subgroup of the lattice may also be activated independently. Lattices of subgroups serve themselves as menus for consideration of specific symmetry descents. Clicking on a subgroup H or on one of the stack of conjugate subgroups [F_{i}] activates the following information about symmetry descent [G \Downarrow H] or [G \Downarrow F_{i}]:

      • (1) Domain: This option brings to the screen the main table that describes changes of tensor properties in chosen ferroic descent. The option is available under the pull-down menu View or directly at the subgroup in the case of normal subgroups. The change of tensor properties is given with reference to the first domain state and hence to the group [F_{1}] from the set of conjugate subgroups. It is given for all tensors listed in Section[link] in a slightly different manner than in Table[link]. Namely, principal and secondary tensor parameters of the transition are distinguished. In other words, the onsetting tensor components are distinguished according to the ireps to which they belong. From this information, one can deduce the fine structure of domain states with all possible crossovers when some tensor properties are identical in several domains. In tables for those symmetry descents where the subgroup is not an epikernel of a certain irep, expressions of the subgroup as possible intersection of epikernels are presented.

        To each table is attached information about G-invariant forms of interactions. This consists of:

        • Integrity basis: of invariants in the primary order parameter.

        • Faint interactions: These are those interactions of the primary order parameter with secondary (faint) parameters that are responsible for the occurrence of faint parameters.

        • Electric switching interactions: The interactions of all order parameters with an external electric field.

        • Elastic switching interactions: The interactions of all order parameters with an external stress field.

      • (2) Integrity bases: Needs activation of the box of ireps as explained above. Displays the extended integrity basis of polynomials in variables of the chosen irep. This consists of the integrity basis of polynomial invariants and of the linear bases of polynomial covariants.

      • (3) Twinning group: This option works for the first group of the set of conjugate subgroups only. It displays a table that contains consecutive normalizers of the set of conjugate subgroups, left, right and double coset resolutions of the parent group G with respect to the subgroup [F_{1}], and the twinning groups assigned to double cosets. This is the basic information concerning pairs of domain states.

    Lattices of equitranslational subgroups of the space groups. The importance of these lattices was realized by Ascher (1968[link]), who prepared the first tables. However, his tables do not contain full information about subgroups; neither the parent group nor the subgroups are completely specified. The current version gives the full information about subgroups including their settings and origins. The pull-down menu Groups contains two options: Point and Space. The choice of the second option brings to the screen another panel, in the right-hand part of which are listed space groups of the geometric class G through Hermann–Mauguin symbols corresponding to all settings and cell choices where applicable. The number of the space-group type, the Schoenflies symbol, the setting and the cell choice are shown in the left-hand part of the panel when you click on one of these Hermann–Mauguin symbols. At the same time, the symbols of the point groups in the lattice change to Schoenflies symbols of oriented space-group types. As you click on any of these subgroups, the Hermann–Mauguin symbol that specifies the subgroup completely appears in the lower bar of the panel, reserved for this information. Though the embellished lattice symbols used in this presentation are self-explanatory, consultation of the manual is recommended.

    The option Point returns the lattice to its original form of the lattice of point groups.

The following is a list of tabular appendices contained in the manual:

  • Appendix A: correlation of various notations and Jones' faithful representation symbols;

  • Appendix B: Schoenflies and Hermann–Mauguin symbols of groups in standard orientations and of their subgroups;

  • Appendix C: isomorphisms used for defining irreducible representations;

  • Appendix D: standard polynomials;

  • Appendix E: labelling of covariants and conversion equations;

  • Appendix F: list of symmetry descents;

  • Appendix G: nonstandard lattice letters.

Our symbols for point-symmetry operations are compared with other sources in Appendix A. Symbols of all groups used in the software are given in Appendix B and isomorphisms in Appendix C. Standard polynomials in Appendix D are abbreviated symbols for more complicated polynomials that appear in the main tables. Appendix E is of primary importance for consideration of the relationship between tensor parameters and their contribution to Cartesian tensor components as already indicated in the text explaining Table[link]. In Appendix F are listed and classified all symmetry descents considered in the main table. Consultation of Appendix G is strongly recommended to all users who want to use the lattices of equitranslational subgroups of the space groups.


Altmann, S. L. & Herzig, P. (1994). Point-group theory tables. Oxford: Clarendon Press.
Ascher, E. (1968). Lattices of equi-translation subgroups of the space groups. Geneva: Battelle.
Bradley, C. J. & Cracknell, A. P. (1972). The mathematical theory of symmetry in solids. Representation theory for point groups and space groups. Oxford: Clarendon Press.
Kopský, V. (1976a). The use of the Clebsch–Gordan reduction of the Kronecker square of the typical representation in symmetry problems of crystal physics. I. Theoretical foundations. J. Phys. C: Solid State Phys. 9, 3391–3405.
Kopský, V. (1976b). The use of the Clebsch–Gordan reduction of the Kronecker square of the typical representation in symmetry problems of crystal physics. II. Tabulation of Clebsch–Gordan products for classical and magnetic crystal point groups. J. Phys. C: Solid State Phys. 9, 3405–3420.
Kopský, V. (1979a). Tensorial covariants of the 32 crystal point groups. Acta Cryst. A35, 83–95.
Kopský, V. (1979b). A simplified calculation and tabulation of tensorial covariants for magnetic point groups belonging to the same Laue class. Acta Cryst. A35, 95–101.
Kopský, V. (1979c). Extended integrity bases of irreducible matrix groups. The crystal point groups. J. Phys. A: Math. Gen. 12, 943–957.
Kopský, V. (1982). Group lattices, subduction of bases and fine domain structures for magnetic crystal point groups. Prague: Academia.
Koster, G. F., Dimmock, J. O., Wheeler, R. E. & Statz, H. (1963). Properties of the 32 groups. Cambridge: MIT Press.
Patera, J., Sharp, R. T. & Winternitz, P. (1978). Polynomial irreducible tensors for point groups. J. Math. Phys. 19, 2362–2376.
Weitzenböck, R. (1923). Invariantentheorie. Groningen: Noordhof.
Weyl, H. (1946). The classical groups. Princeton: UP.

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