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. 3.1, pp. 392394

We shall briefly describe here the intentions and contents of the accompanying software package GIKoBo1 (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:
These methods provide good ammunition for all types of grouptheoretical 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 and/or , where G is the parent point group, H its normal subgroup and 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 grouptheoretical concepts used in the software. Relevant elementary tables (listed below in Section A) are followed by more advanced information proceeding towards the central goal of providing information for all symmetry descents (Section B). 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) 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). (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 brandnew 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:
The following is a list of tabular appendices contained in the manual:

Our symbols for pointsymmetry 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 3.1.3.1. 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.
References
Altmann, S. L. & Herzig, P. (1994). Pointgroup theory tables. Oxford: Clarendon Press.Ascher, E. (1968). Lattices of equitranslation 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.
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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.
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