Note on E representations in
C4h character table



4 irreducible representations of point group C4h 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


C4h E C4 C2 (C4)3 i (S4)3 σh S4
Ag 1 1 1 1 1 1 1 1
Bg 1 -1 1 -1 1 -1 1 -1
Eg Eg,a
Eg,b
1
1
i
-i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i
Au 1 1 1 1 -1 -1 -1 -1
Bu 1 -1 1 -1 -1 1 -1 1
Eu Eu,a
Eu,b
1
1
i
-i
-1
-1
-i
i
-1
-1
-i
i
1
1
i
-i


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