Note on E representations inTh character table

4 irreducible representations of point group Th have complex values. 2 two-dimensional real-valued representations E can be constructed as direct sum of the 2 pairs complex plus conjugate complex irreducible representation.

Eg = Eg,a ⊕ Eg,b
Eu = Eu,a ⊕ Eu,b

ε=exp(2πi/3)
Th E 4C3 4(C3)2 3C2 i 4(S6)5 4S6 h
Ag 1 1 1 1 1 1 1 1
Eg Eg,a
Eg,b
1
1
ε*
ε*
ε*
ε*
1
1
1
1
ε*
ε*
ε*
ε*
1
1
Tg 3 0 0 -1 3 0 0 -1
Au 1 1 1 1 -1 -1 -1 -1
Eu Eu,a
Eu,b
1
1
ε*
ε*
ε*
ε*
1
1
-1
-1
*
*
*
*
-1
-1
Tu 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 real-valued or pseudo irreducible representations. 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 Th 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 August, 12th 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement