Note on E representations in
S6 character table



4 irreducible representations of point group S6 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)
S6 E C3 (C3)2 i (S6)5 S6
Ag 1 1 1 1 1 1
Eg Eg,a
Eg,b
1
1
ε*
ε*
ε*
ε*
1
1
ε*
ε*
ε*
ε*
Au 1 1 1 -1 -1 -1
Eu Eu,a
Eu,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