Топтық теориясындағы мінездеме кестелері

In group theory, a branch of abstract algebra, a character table is a two dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters, the traces of the matrices representing group elements of the column's class in the given row's group representation. In chemistry, crystallography, and spectroscopy, character tables of point groups are used to classify e. g. molecular vibrations according to their symmetry, and to predict whether a transition between two states is forbidden for symmetry reasons. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry devote a chapter to the use of symmetry group character tables.

The outer automorphism group acts on the character table by permuting columns (conjugacy classes) and accordingly rows, which gives another symmetry to the table. For example, abelian groups have the outer automorphism , which is non trivial except for elementary abelian 2 groups, and outer because abelian groups are precisely those for which conjugation (inner automorphisms) acts trivially. In the example of above, this map sends and accordingly switches and (switching their values of and ). Note that this particular automorphism (negative in abelian groups) agrees with complex conjugation. Formally, if is an automorphism of G and is a representation, then is a representation. If is an inner automorphism (conjugation by some element a), then it acts trivially on representations, because representations are class functions (conjugation does not change their value). Thus a given class of outer automorphisms, it acts on the characters – because inner automorphisms act trivially, the action of the automorphism group descends to the quotient
This relation can be used both ways: given an outer automorphism, one can produce new representations (if the representation is not equal on conjugacy classes that are interchanged by the outer automorphism), and conversely, one can restrict possible outer automorphisms based on the character table.

To find the total number of vibrational modes of a water molecule, the irreducible representation Γirreducible needs to calculate from the character table of a water molecule first.