Note on E representations in
C12h character table



20 irreducible representations of point group C12h have complex values. 10 two-dimensional real-valued representations E can be constructed as direct sum of the 10 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
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


ε=exp(2πi/12)
C12h E C12 C6 C4 C3 (C12)5 C2 (C12)7 (C3)2 (C4)3 (C6)5 (C12)11 i (S12)7 (S3)5 (S4)3 (S6)5 (S12)11 σh S12 S6 S4 S3 (S12)5
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
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
E1g E1g,a
E1g,b
1
1
ε*
ε*
ε2*
ε2*
i
-i
2*
2*
*
*
-1
-1
*
*
2*
2*
-i
i
ε2*
ε2*
ε*
ε*
1
1
ε*
ε*
ε2*
ε2*
i
-i
2*
2*
*
*
-1
-1
*
*
2*
2*
-i
i
ε2*
ε2*
ε*
ε*
E2g E2g,a
E2g,b
1
1
ε2*
ε2*
2*
2*
-1
-1
2*
2*
ε2*
ε2*
1
1
ε2*
ε2*
2*
2*
-1
-1
2*
2*
ε2*
ε2*
1
1
ε2*
ε2*
2*
2*
-1
-1
2*
2*
ε2*
ε2*
1
1
ε2*
ε2*
2*
2*
-1
-1
2*
2*
ε2*
ε2*
E3g E3g,a
E3g,b
1
1
i
-i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i
E4g E4g,a
E4g,b
1
1
2*
2*
2*
2*
1
1
2*
2*
2*
2*
1
1
2*
2*
2*
2*
1
1
2*
2*
2*
2*
1
1
2*
2*
2*
2*
1
1
2*
2*
2*
2*
1
1
2*
2*
2*
2*
1
1
2*
2*
2*
2*
E5g E5g,a
E5g,b
1
1
*
*
ε2*
ε2*
i
-i
2*
2*
ε*
ε*
-1
-1
ε*
ε*
2*
2*
-i
i
ε2*
ε2*
*
*
1
1
*
*
ε2*
ε2*
i
-i
2*
2*
ε*
ε*
-1
-1
ε*
ε*
2*
2*
-i
i
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
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
E1u E1u,a
E1u,b
1
1
ε*
ε*
ε2*
ε2*
i
-i
2*
2*
*
*
-1
-1
*
*
2*
2*
-i
i
ε2*
ε2*
ε*
ε*
-1
-1
*
*
2*
2*
-i
i
ε2*
ε2*
ε*
ε*
1
1
ε*
ε*
ε2*
ε2*
i
-i
2*
2*
*
*
E2u E2u,a
E2u,b
1
1
ε2*
ε2*
2*
2*
-1
-1
2*
2*
ε2*
ε2*
1
1
ε2*
ε2*
2*
2*
-1
-1
2*
2*
ε2*
ε2*
-1
-1
2*
2*
ε2*
ε2*
1
1
ε2*
ε2*
2*
2*
-1
-1
2*
2*
ε2*
ε2*
1
1
ε2*
ε2*
2*
2*
E3u E3u,a
E3u,b
1
1
i
-i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i
1
1
i
-i
E4u E4u,a
E4u,b
1
1
2*
2*
2*
2*
1
1
2*
2*
2*
2*
1
1
2*
2*
2*
2*
1
1
2*
2*
2*
2*
-1
-1
ε2*
ε2*
ε2*
ε2*
-1
-1
ε2*
ε2*
ε2*
ε2*
-1
-1
ε2*
ε2*
ε2*
ε2*
-1
-1
ε2*
ε2*
ε2*
ε2*
E5u E5u,a
E5u,b
1
1
*
*
ε2*
ε2*
i
-i
2*
2*
ε*
ε*
-1
-1
ε*
ε*
2*
2*
-i
i
ε2*
ε2*
*
*
-1
-1
ε*
ε*
2*
2*
-i
i
ε2*
ε2*
*
*
1
1
*
*
ε2*
ε2*
i
-i
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