Obviously the E representation is reducible. Nevertheless the E representation is treated often as irreducible representation and is called real-valued or pseudo irreducible representation. One should keep in mind that general statements for character tables fail for real-valued representations. For example:
- The number of irreducible representations is usually equal the number of classes. For point group this statement is true for the complex irreducible representations. The number of real-valued irreducible representations is less than the number of classes.
- The sum of the squared characters of the neutral symmetry element over all irreducible representations is equal to the total number of symmetry elements
h = ∑ Χi(E) Χi(E)
This statement is true for the complex irreducible representations but not for the real-valued irreducible representations - Reduction formula: The occurence of i-th irreducible representation in an reducible representation is given by the well known formula
ci = 1/h ∑R nR χi(R)irred χ(R)red
This statement is true for the complex irreducible representations. The norm of the two-dimensional real-valued irreducible representations is 2h (instead of h) meaning that the reduction formula has to be modified:ci(E) = 1/(2h) ∑R nR χi(R)irred χ(R)red = (∑R nR χi(R)irred χ(R)red) / (∑R nR χi(R)irred χi(R)irred)
Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement