Note on E representations in
C9 character table



8 irreducible representations of point group C9 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.

E1 = E1a ⊕ E1b
E2 = E2a ⊕ E2b
E3 = E3a ⊕ E3b
E4 = E4a ⊕ E4b


ε=exp(2πi/9)
C9 E C9 (C9)2 C3 (C9)4 (C9)5 (C3)2 (C9)7 (C9)8
A 1 1 1 1 1 1 1 1 1
E1 E1a
E1b
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
E2 E2a
E2b
1
1
ε2*
ε2*
ε4*
ε4*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε4*
ε4*
ε2*
ε2*
E3 E3a
E3b
1
1
ε3*
ε3*
ε3*
ε3*
1
1
ε3*
ε3*
ε3*
ε3*
1
1
ε3*
ε3*
ε3*
ε3*
E4 E4a
E4b
1
1
ε4*
ε4*
ε*
ε*
ε3*
ε3*
ε2*
ε2*
ε2*
ε2*
ε3*
ε3*
ε*
ε*
ε4*
ε4*


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