Note on E representations in
C9h character table



16 irreducible representations of point group C9h 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.

E'1 = E'1a ⊕ E'1b
E'2 = E'2a ⊕ E'2b
E'3 = E'3a ⊕ E'3b
E'4 = E'4a ⊕ E'4b
E''1 = E''1a ⊕ E''1b
E''2 = E''2a ⊕ E''2b
E''3 = E''3a ⊕ E''3b
E''4 = E''4a ⊕ E''4b


ε=exp(2πi/9)
C9h E C9 (C9)2 C3 (C9)4 (C9)5 (C3)2 (C9)7 (C9)8 σh S9 (S9)11 S3 (S9)13 (S9)5 (S3)5 (S9)7 (S9)17
A' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
E'1 E'1a
E'1b
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
E'2 E'2a
E'2b
1
1
ε2*
ε2*
ε4*
ε4*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε4*
ε4*
ε2*
ε2*
1
1
ε2*
ε2*
ε4*
ε4*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε4*
ε4*
ε2*
ε2*
E'3 E'3a
E'3b
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*
E'4 E'4a
E'4b
1
1
ε4*
ε4*
ε*
ε*
ε3*
ε3*
ε2*
ε2*
ε2*
ε2*
ε3*
ε3*
ε*
ε*
ε4*
ε4*
1
1
ε4*
ε4*
ε*
ε*
ε3*
ε3*
ε2*
ε2*
ε2*
ε2*
ε3*
ε3*
ε*
ε*
ε4*
ε4*
A'' 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1
E''1 E''1a
E''1b
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
-1
-1
*
*
2*
2*
3*
3*
4*
4*
4*
4*
3*
3*
2*
2*
*
*
E''2 E''2a
E''2b
1
1
ε2*
ε2*
ε4*
ε4*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε4*
ε4*
ε2*
ε2*
-1
-1
2*
2*
4*
4*
3*
3*
*
*
*
*
3*
3*
4*
4*
2*
2*
E''3 E''3a
E''3b
1
1
ε3*
ε3*
ε3*
ε3*
1
1
ε3*
ε3*
ε3*
ε3*
1
1
ε3*
ε3*
ε3*
ε3*
-1
-1
3*
3*
3*
3*
-1
-1
3*
3*
3*
3*
-1
-1
3*
3*
3*
3*
E''4 E''4a
E''4b
1
1
ε4*
ε4*
ε*
ε*
ε3*
ε3*
ε2*
ε2*
ε2*
ε2*
ε3*
ε3*
ε*
ε*
ε4*
ε4*
-1
-1
4*
4*
*
*
3*
3*
2*
2*
2*
2*
3*
3*
*
*
4*
4*


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