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. 1.2, pp. 6667

Calculations with characters of representations of point groups can be done in the character module of the program. It is selected in the main window by clicking `character'. A selection window opens in which a point group may be selected just as in the tensor module. The point groups are organized according to dimension and geometric crystal class. Selection of a point group leads to the display of the character table if one asks for it by selecting `view character table'.
The character table consists of a square array of (complex) numbers. The number of rows is the number of nonequivalent irreducible representations and is equal to the number of columns, which is the number of conjugacy classes of the group. For crystallographic groups, the complex numbers that form the entries of the character table are cyclotomic numbers. These are linear combinations with fractions as coefficients of complex numbers of the form For example, the square root of (i) can be written as A real number like can be written as Another example isHowever, many entries for the threedimensional point groups are simply integers.
The program provides the following information as rows above the characters of the irreducible representation:
Below the character table, the following information is displayed:
As an example, the generalized character table for the threedimensional point group is given in Table 1.2.7.1.

The data connected with a character table can be seen by choosing `view character table'. The characters of the irreducible representations, the determinant representation and the vector representation are shown in the main window after selection of `accept character table'. From the character of these representations, characters of other representations may be calculated. The results are added as rows to the table, which is shown after each calculation.
Calculations using rows from the table may have one or more arguments. Operations with one argument will produce, for example, the decomposition into irreducible components, the character of the pth power, the symmetrized or antisymmetrized square, or the character of the corresponding physical (real) representation. Operations with two or more arguments yield products and sums of characters. The arguments of a unitary, binary or multiple operation are selected by clicking on the button in front of the corresponding characters. If the result is a new character (e.g. the product of two characters), it is added as a row to the list of characters. If the result is not a character (e.g. the decomposition into irreducible components), the result is given on the worksheet.
Suppose one wants to determine the number of elastic constants for a material with cubic 432 symmetry. After selecting the character table for the group 432, one clicks on the button in front of `vector representation' in the character table. This yields the character of the threedimensional vector representation of the group. The character of the symmetrized square is obtained by selecting `symmetrized square'. This gives the character of a sixdimensional representation. Determining the number of times the trivial representation occurs by selecting `decompose' gives the number of free parameters in the metric tensor, i.e. 1. Clicking on `symmetrized square' for the character of the sixdimensional representation gives the character of a 21dimensional representation. Decomposition yields the multiplicity 3 for the trivial representation, which means that there are three independent tensor elements for a tensor of symmetry type , which in turn means that there are three elastic constants for the group 432 (see Table 1.2.6.9). For the explicit determination of the independent tensor elements, the tensor module of the program should be used.
Of course, many kinds of calculations unrelated to tensors can be carried out using the character module. Examples include the calculation of selection rules in spectroscopy or the splitting of energy levels under a symmetrybreaking perturbation.