Note on E representations in
C18h character table



32 irreducible representations of point group C18h have complex values. 16 two-dimensional real-valued representations E can be constructed as direct sum of the 16 pairs complex plus conjugate complex irreducible representation.

E1g = E1g,a ⊕ E1g,b
E2g = E2g,a ⊕ E2g,b
E3g = E3g,a ⊕ E3g,b
E4g = E4g,a ⊕ E4g,b
E5g = E5g,a ⊕ E5g,b
E6g = E6g,a ⊕ E6g,b
E7g = E7g,a ⊕ E7g,b
E8g = E8g,a ⊕ E8g,b
E1u = E1u,a ⊕ E1u,b
E2u = E2u,a ⊕ E2u,b
E3u = E3u,a ⊕ E3u,b
E4u = E4u,a ⊕ E4u,b
E5u = E5u,a ⊕ E5u,b
E6u = E6u,a ⊕ E6u,b
E7u = E7u,a ⊕ E7u,b
E8u = E8u,a ⊕ E8u,b


ε=exp(2πi/18)
C18h 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 i (S9)5 (S18)11 (S3)5 (S18)13 (S9)7 (S6)5 (S9)17 (S18)17 σh S18 S9 S6 (S9)11 (S18)5 S3 (S18)7 (S9)13
Ag 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Bg 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
E1g E1g,a
E1g,b
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*
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*
E2g E2g,a
E2g,b
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*
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*
E3g E3g,a
E3g,b
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*
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*
E4g E4g,a
E4g,b
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*
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*
E5g E5g,a
E5g,b
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*
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*
E6g E6g,a
E6g,b
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*
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*
E7g E7g,a
E7g,b
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*
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*
E8g E8g,a
E8g,b
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*
*
*
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*
*
*
Au 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
Bu 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
E1u E1u,a
E1u,b
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*
-1
-1
*
*
2*
2*
3*
3*
4*
4*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
4*
4*
3*
3*
2*
2*
*
*
E2u E2u,a
E2u,b
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*
-1
-1
2*
2*
4*
4*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
4*
4*
2*
2*
-1
-1
2*
2*
4*
4*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
4*
4*
2*
2*
E3u E3u,a
E3u,b
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*
-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*
1
1
ε3*
ε3*
3*
3*
E4u E4u,a
E4u,b
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*
-1
-1
4*
4*
ε*
ε*
ε3*
ε3*
2*
2*
2*
2*
ε3*
ε3*
ε*
ε*
4*
4*
-1
-1
4*
4*
ε*
ε*
ε3*
ε3*
2*
2*
2*
2*
ε3*
ε3*
ε*
ε*
4*
4*
E5u E5u,a
E5u,b
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*
-1
-1
ε4*
ε4*
ε*
ε*
3*
3*
2*
2*
ε2*
ε2*
ε3*
ε3*
*
*
4*
4*
1
1
4*
4*
*
*
ε3*
ε3*
ε2*
ε2*
2*
2*
3*
3*
ε*
ε*
ε4*
ε4*
E6u E6u,a
E6u,b
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*
-1
-1
ε3*
ε3*
ε3*
ε3*
-1
-1
ε3*
ε3*
ε3*
ε3*
-1
-1
ε3*
ε3*
ε3*
ε3*
-1
-1
ε3*
ε3*
ε3*
ε3*
-1
-1
ε3*
ε3*
ε3*
ε3*
-1
-1
ε3*
ε3*
ε3*
ε3*
E7u E7u,a
E7u,b
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*
-1
-1
ε2*
ε2*
4*
4*
3*
3*
ε*
ε*
*
*
ε3*
ε3*
ε4*
ε4*
2*
2*
1
1
2*
2*
ε4*
ε4*
ε3*
ε3*
*
*
ε*
ε*
3*
3*
4*
4*
ε2*
ε2*
E8u E8u,a
E8u,b
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*
*
*
-1
-1
ε*
ε*
2*
2*
ε3*
ε3*
4*
4*
4*
4*
ε3*
ε3*
2*
2*
ε*
ε*
-1
-1
ε*
ε*
2*
2*
ε3*
ε3*
4*
4*
4*
4*
ε3*
ε3*
2*
2*
ε*
ε*


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