Note on E representations in
C5h character table



8 irreducible representations of point group C5h have complex values. 4 two-dimensional real-valued representations E can be constructed as direct sum of the 4 pairs complex plus conjugate complex irreducible representation.

E'1 = E'1a ⊕ E'1b
E'2 = E'2a ⊕ E'2b
E''1 = E''1a ⊕ E''1b
E''2 = E''2a ⊕ E''2b


ε=exp(2πi/5)
C5h E C5 (C5)2 (C5)3 (C5)4 σh S5 (S5)7 (S5)3 (S5)9
A' 1 1 1 1 1 1 1 1 1 1
E'1 E'1a
E'1b
1
1
ε*
ε*
ε2*
ε2*
ε2*
ε2*
ε*
ε*
1
1
ε*
ε*
ε2*
ε2*
ε2*
ε2*
ε*
ε*
E'2 E'2a
E'2b
1
1
ε2*
ε2*
ε*
ε*
ε*
ε*
ε2*
ε2*
1
1
ε2*
ε2*
ε*
ε*
ε*
ε*
ε2*
ε2*
A'' 1 1 1 1 1 -1 -1 -1 -1 -1
E''1 E''1a
E''1b
1
1
ε*
ε*
ε2*
ε2*
ε2*
ε2*
ε*
ε*
-1
-1
*
*
2*
2*
2*
2*
*
*
E''2 E''2a
E''2b
1
1
ε2*
ε2*
ε*
ε*
ε*
ε*
ε2*
ε2*
-1
-1
2*
2*
*
*
*
*
2*
2*


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