Note on E representations in
Th 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:



Last update August, 12th 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement