ε=exp(2πi/7)
C7h |
E |
C7 |
(C7)2 |
(C7)3 |
(C7)4 |
(C7)5 |
(C7)6 |
σh |
S7 |
(S7)9 |
(S7)3 |
(S7)11 |
(S7)5 |
(S7)13 |
|
A' |
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* |
ε3* ε3* |
ε2* ε2* |
ε* ε* |
1 1 |
ε* ε* |
ε2* ε2* |
ε3* ε3* |
ε3* ε3* |
ε2* ε2* |
ε* ε* |
E'2 |
E'2a E'2b |
1 1 |
ε2* ε2* |
ε3* ε3* |
ε* ε* |
ε* ε* |
ε3* ε3* |
ε2* ε2* |
1 1 |
ε2* ε2* |
ε3* ε3* |
ε* ε* |
ε* ε* |
ε3* ε3* |
ε2* ε2* |
E'3 |
E'3a E'3b |
1 1 |
ε3* ε3* |
ε* ε* |
ε2* ε2* |
ε2* ε2* |
ε* ε* |
ε3* ε3* |
1 1 |
ε3* ε3* |
ε* ε* |
ε2* ε2* |
ε2* ε2* |
ε* ε* |
ε3* ε3* |
|
A'' |
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* |
ε3* ε3* |
ε2* ε2* |
ε* ε* |
-1 -1 |
-ε* -ε* |
-ε2* -ε2* |
-ε3* -ε3* |
-ε3* -ε3* |
-ε2* -ε2* |
-ε* -ε* |
E''2 |
E''2a E''2b |
1 1 |
ε2* ε2* |
ε3* ε3* |
ε* ε* |
ε* ε* |
ε3* ε3* |
ε2* ε2* |
-1 -1 |
-ε2* -ε2* |
-ε3* -ε3* |
-ε* -ε* |
-ε* -ε* |
-ε3* -ε3* |
-ε2* -ε2* |
E''3 |
E''3a E''3b |
1 1 |
ε3* ε3* |
ε* ε* |
ε2* ε2* |
ε2* ε2* |
ε* ε* |
ε3* ε3* |
-1 -1 |
-ε3* -ε3* |
-ε* -ε* |
-ε2* -ε2* |
-ε2* -ε2* |
-ε* -ε* |
-ε3* -ε3* |
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:
- The number of irreducible representations is usually equal the number of classes. For point group C7h this statement is true for the complex irreducible representations. The number of real-valued irreducible representations is less than the number of classes.
- The sum of the squared characters of the neutral symmetry element over all irreducible representations is equal to the total number of symmetry elements
h = ∑ Χi(E) Χi(E)
This statement is true for the complex irreducible representations but not for the real-valued irreducible representations - Reduction formula: The occurence of i-th irreducible representation in an reducible representation is given by the well known formula
ci = 1/h ∑R nR χi(R)irred χ(R)red
This statement is true for the complex irreducible representations. The norm of the two-dimensional real-valued irreducible representations is 2h (instead of h) meaning that the reduction formula has to be modified:ci(E) = 1/(2h) ∑R nR χi(R)irred χ(R)red = (∑R nR χi(R)irred χ(R)red) / (∑R nR χi(R)irred χi(R)irred)
Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement