Note on E representations in
T_{h} character table
4 irreducible representations of point group
T_{h} have complex values.
2 twodimensional realvalued representations E can be constructed as direct sum of the 2 pairs complex plus conjugate complex irreducible representation.
E_{g} = E_{g,a} ⊕ E_{g,b}
E_{u} = E_{u,a} ⊕ E_{u,b}
ε=exp(2πi/3)
T_{h} 
E 
4C_{3} 
4(C_{3})^{2} 
3C_{2} 
i 
4(S_{6})^{5} 
4S_{6} 
3σ_{h} 

A_{g} 
1 
1 
1 
1 
1 
1 
1 
1 
E_{g} 
E_{g,a} E_{g,b} 
1 1 
ε^{*} ε^{*} 
ε^{*} ε^{*} 
1 1 
1 1 
ε^{*} ε^{*} 
ε^{*} ε^{*} 
1 1 

T_{g} 
3 
0 
0 
1 
3 
0 
0 
1 

A_{u} 
1 
1 
1 
1 
1 
1 
1 
1 
E_{u} 
E_{u,a} E_{u,b} 
1 1 
ε^{*} ε^{*} 
ε^{*} ε^{*} 
1 1 
1 1 
ε^{*} ε^{*} 
ε^{*} ε^{*} 
1 1 

T_{u} 
3 
0 
0 
1 
3 
0 
0 
1 
Obviously the E representations are reducible. Nevertheless the E representations are treated often as irreducible representations and are called realvalued or pseudo irreducible representations. One should keep in mind that general statements for character tables fail for realvalued representations. For example:
 The number of irreducible representations is usually equal the number of classes. For point group T_{h} this statement is true for the complex irreducible representations. The number of realvalued 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 realvalued irreducible representations  Reduction formula: The occurence of ith 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 twodimensional realvalued 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