Note on E representations inC20 character table

18 irreducible representations of point group C20 have complex values. 9 two-dimensional real-valued representations E can be constructed as direct sum of the 9 pairs complex plus conjugate complex irreducible representation.

E1 = E1a ⊕ E1b
E2 = E2a ⊕ E2b
E3 = E3a ⊕ E3b
E4 = E4a ⊕ E4b
E5 = E5a ⊕ E5b
E6 = E6a ⊕ E6b
E7 = E7a ⊕ E7b
E8 = E8a ⊕ E8b
E9 = E9a ⊕ E9b

ε=exp(2πi/20)
C20 E C20 C10 (C20)3 C5 C4 (C10)3 (C20)7 (C5)2 (C20)9 C2 (C20)11 (C5)3 (C20)13 (C10)7 (C4)3 (C5)4 (C20)17 (C10)9 (C20)19
A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
E1 E1a
E1b
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
i
-i
4*
4*
3*
3*
2*
2*
*
*
-1
-1
*
*
2*
2*
3*
3*
4*
4*
-i
i
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
E2 E2a
E2b
1
1
ε2*
ε2*
ε4*
ε4*
4*
4*
2*
2*
-1
-1
2*
2*
4*
4*
ε4*
ε4*
ε2*
ε2*
1
1
ε2*
ε2*
ε4*
ε4*
4*
4*
2*
2*
-1
-1
2*
2*
4*
4*
ε4*
ε4*
ε2*
ε2*
E3 E3a
E3b
1
1
ε3*
ε3*
4*
4*
*
*
2*
2*
-i
i
ε2*
ε2*
ε*
ε*
ε4*
ε4*
3*
3*
-1
-1
3*
3*
ε4*
ε4*
ε*
ε*
ε2*
ε2*
i
-i
2*
2*
*
*
4*
4*
ε3*
ε3*
E4 E4a
E4b
1
1
ε4*
ε4*
2*
2*
2*
2*
ε4*
ε4*
1
1
ε4*
ε4*
2*
2*
2*
2*
ε4*
ε4*
1
1
ε4*
ε4*
2*
2*
2*
2*
ε4*
ε4*
1
1
ε4*
ε4*
2*
2*
2*
2*
ε4*
ε4*
E5 E5a
E5b
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
E6 E6a
E6b
1
1
4*
4*
2*
2*
ε2*
ε2*
ε4*
ε4*
-1
-1
ε4*
ε4*
ε2*
ε2*
2*
2*
4*
4*
1
1
4*
4*
2*
2*
ε2*
ε2*
ε4*
ε4*
-1
-1
ε4*
ε4*
ε2*
ε2*
2*
2*
4*
4*
E7 E7a
E7b
1
1
3*
3*
4*
4*
ε*
ε*
2*
2*
-i
i
ε2*
ε2*
*
*
ε4*
ε4*
ε3*
ε3*
-1
-1
ε3*
ε3*
ε4*
ε4*
*
*
ε2*
ε2*
i
-i
2*
2*
ε*
ε*
4*
4*
3*
3*
E8 E8a
E8b
1
1
2*
2*
ε4*
ε4*
ε4*
ε4*
2*
2*
1
1
2*
2*
ε4*
ε4*
ε4*
ε4*
2*
2*
1
1
2*
2*
ε4*
ε4*
ε4*
ε4*
2*
2*
1
1
2*
2*
ε4*
ε4*
ε4*
ε4*
2*
2*
E9 E9a
E9b
1
1
*
*
ε2*
ε2*
3*
3*
ε4*
ε4*
i
-i
4*
4*
ε3*
ε3*
2*
2*
ε*
ε*
-1
-1
ε*
ε*
2*
2*
ε3*
ε3*
4*
4*
-i
i
ε4*
ε4*
3*
3*
ε2*
ε2*
*
*

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 C20 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 August, 12th 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement