Note on E representation in
S_{4} character table
2 irreducible representations of point group
S_{4} have complex values.
A two-dimensional real-valued representation E can be constructed as direct sum of the pair complex plus conjugate complex irreducible representation.
E = E_{a} ⊕ E_{b}
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 S_{4} 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 = ∑ Χ_{i}(E) Χ_{i}(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
c_{i} = 1/h ∑_{R} n_{R} χ_{i}(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:c_{i}(E) = 1/(2h) ∑_{R} n_{R} χ_{i}(R)^{irred} χ(R)^{red} = (∑_{R} n_{R} χ_{i}(R)^{irred} χ(R)^{red}) / (∑_{R} n_{R} χ_{i}(R)^{irred} χ_{i}(R)^{irred})
Last update August, 12^{th} 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement