ε=exp(2πi/8)
S8 |
E |
S8 |
C4 |
(S8)3 |
C2 |
(S8)5 |
(C4)3 |
(S8)7 |
|
A |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
|
B |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
E1 |
E1a E1b |
1 1 |
ε* ε* |
i -i |
-ε* -ε* |
-1 -1 |
-ε* -ε* |
-i i |
ε* ε* |
E2 |
E2a E2b |
1 1 |
i -i |
-1 -1 |
-i i |
1 1 |
i -i |
-1 -1 |
-i i |
E3 |
E3a E3b |
1 1 |
-ε* -ε* |
-i i |
ε* ε* |
-1 -1 |
ε* ε* |
i -i |
-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:
- The number of irreducible representations is usually equal the number of classes. For point group S8 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