## Note on E representations inC20h character table

36 irreducible representations of point group C20h have complex values. 18 two-dimensional real-valued representations E can be constructed as direct sum of the 18 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
E6g = E6g,a ⊕ E6g,b
E7g = E7g,a ⊕ E7g,b
E8g = E8g,a ⊕ E8g,b
E9g = E9g,a ⊕ E9g,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
E6u = E6u,a ⊕ E6u,b
E7u = E7u,a ⊕ E7u,b
E8u = E8u,a ⊕ E8u,b
E9u = E9u,a ⊕ E9u,b

ε=exp(2πi/20)
C20h 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 i (S20)11 (S5)3 (S20)13 (S10)7 (S4)3 (S5)9 (S20)17 (S10)9 (S20)19 σh S20 S10 (S20)3 S5 S4 (S10)3 (S20)7 (S5)7 (S20)9
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 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 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*
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*
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*
E2g E2g,a
E2g,b
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*
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*
E3g E3g,a
E3g,b
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*
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*
E4g E4g,a
E4g,b
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*
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*
E5g E5g,a
E5g,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
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
E6g E6g,a
E6g,b
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*
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*
E7g E7g,a
E7g,b
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*
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*
E8g E8g,a
E8g,b
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*
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*
E9g E9g,a
E9g,b
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*
*
*
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*
*
*
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 -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 -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*
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*
-1
-1
*
*
2*
2*
3*
3*
4*
4*
-i
i
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
i
-i
4*
4*
3*
3*
2*
2*
*
*
E2u E2u,a
E2u,b
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*
-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*
1
1
ε2*
ε2*
ε4*
ε4*
4*
4*
2*
2*
E3u E3u,a
E3u,b
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*
-1
-1
3*
3*
ε4*
ε4*
ε*
ε*
ε2*
ε2*
i
-i
2*
2*
*
*
4*
4*
ε3*
ε3*
1
1
ε3*
ε3*
4*
4*
*
*
2*
2*
-i
i
ε2*
ε2*
ε*
ε*
ε4*
ε4*
3*
3*
E4u E4u,a
E4u,b
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*
-1
-1
4*
4*
ε2*
ε2*
ε2*
ε2*
4*
4*
-1
-1
4*
4*
ε2*
ε2*
ε2*
ε2*
4*
4*
-1
-1
4*
4*
ε2*
ε2*
ε2*
ε2*
4*
4*
-1
-1
4*
4*
ε2*
ε2*
ε2*
ε2*
4*
4*
E5u E5u,a
E5u,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
-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
E6u E6u,a
E6u,b
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*
-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*
1
1
4*
4*
2*
2*
ε2*
ε2*
ε4*
ε4*
E7u E7u,a
E7u,b
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*
-1
-1
ε3*
ε3*
ε4*
ε4*
*
*
ε2*
ε2*
i
-i
2*
2*
ε*
ε*
4*
4*
3*
3*
1
1
3*
3*
4*
4*
ε*
ε*
2*
2*
-i
i
ε2*
ε2*
*
*
ε4*
ε4*
ε3*
ε3*
E8u E8u,a
E8u,b
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*
-1
-1
ε2*
ε2*
4*
4*
4*
4*
ε2*
ε2*
-1
-1
ε2*
ε2*
4*
4*
4*
4*
ε2*
ε2*
-1
-1
ε2*
ε2*
4*
4*
4*
4*
ε2*
ε2*
-1
-1
ε2*
ε2*
4*
4*
4*
4*
ε2*
ε2*
E9u E9u,a
E9u,b
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*
*
*
-1
-1
ε*
ε*
2*
2*
ε3*
ε3*
4*
4*
-i
i
ε4*
ε4*
3*
3*
ε2*
ε2*
*
*
1
1
*
*
ε2*
ε2*
3*
3*
ε4*
ε4*
i
-i
4*
4*
ε3*
ε3*
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:
• The number of irreducible representations is usually equal the number of classes. For point group C20h 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