Note on E representations in
C3h character table



4 irreducible representations of point group C3h 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.

E' = E'a ⊕ E'b
E'' = E''a ⊕ E''b


ε=exp(2πi/3)
C3h E C3 (C3)2 σh S3 (S3)5
A' 1 1 1 1 1 1
E' E'a
E'b
1
1
ε*
ε*
ε*
ε*
1
1
ε*
ε*
ε*
ε*
A'' 1 1 1 -1 -1 -1
E'' E''a
E''b
1
1
ε*
ε*
ε*
ε*
-1
-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