Note on E representations in
C18 character table



16 irreducible representations of point group C18 have complex values. 8 two-dimensional real-valued representations E can be constructed as direct sum of the 8 pairs complex plus conjugate complex irreducible representation.

E1 = E1a ⊕ E1b
E2 = E2a ⊕ E2b
E3 = E3a ⊕ E3b
E4 = E4a ⊕ E4b
E5 = E5a ⊕ E5b
E6 = E6a ⊕ E6b
E7 = E7a ⊕ E7b
E8 = E8a ⊕ E8b


ε=exp(2πi/18)
C18 E C18 C9 C6 (C9)2 (C18)5 C3 (C18)7 (C9)4 C2 (C9)5 (C18)11 (C3)2 (C18)13 (C9)7 (C6)5 (C9)8 (C18)17
A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
E1 E1a
E1b
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
4*
4*
3*
3*
2*
2*
*
*
-1
-1
*
*
2*
2*
3*
3*
4*
4*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
E2 E2a
E2b
1
1
ε2*
ε2*
ε4*
ε4*
3*
3*
*
*
*
*
3*
3*
ε4*
ε4*
ε2*
ε2*
1
1
ε2*
ε2*
ε4*
ε4*
3*
3*
*
*
*
*
3*
3*
ε4*
ε4*
ε2*
ε2*
E3 E3a
E3b
1
1
ε3*
ε3*
3*
3*
-1
-1
3*
3*
ε3*
ε3*
1
1
ε3*
ε3*
3*
3*
-1
-1
3*
3*
ε3*
ε3*
1
1
ε3*
ε3*
3*
3*
-1
-1
3*
3*
ε3*
ε3*
E4 E4a
E4b
1
1
ε4*
ε4*
*
*
3*
3*
ε2*
ε2*
ε2*
ε2*
3*
3*
*
*
ε4*
ε4*
1
1
ε4*
ε4*
*
*
3*
3*
ε2*
ε2*
ε2*
ε2*
3*
3*
*
*
ε4*
ε4*
E5 E5a
E5b
1
1
4*
4*
*
*
ε3*
ε3*
ε2*
ε2*
2*
2*
3*
3*
ε*
ε*
ε4*
ε4*
-1
-1
ε4*
ε4*
ε*
ε*
3*
3*
2*
2*
ε2*
ε2*
ε3*
ε3*
*
*
4*
4*
E6 E6a
E6b
1
1
3*
3*
3*
3*
1
1
3*
3*
3*
3*
1
1
3*
3*
3*
3*
1
1
3*
3*
3*
3*
1
1
3*
3*
3*
3*
1
1
3*
3*
3*
3*
E7 E7a
E7b
1
1
2*
2*
ε4*
ε4*
ε3*
ε3*
*
*
ε*
ε*
3*
3*
4*
4*
ε2*
ε2*
-1
-1
ε2*
ε2*
4*
4*
3*
3*
ε*
ε*
*
*
ε3*
ε3*
ε4*
ε4*
2*
2*
E8 E8a
E8b
1
1
*
*
ε2*
ε2*
3*
3*
ε4*
ε4*
ε4*
ε4*
3*
3*
ε2*
ε2*
*
*
1
1
*
*
ε2*
ε2*
3*
3*
ε4*
ε4*
ε4*
ε4*
3*
3*
ε2*
ε2*
*
*


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