## Note on E representations inC15h character table

28 irreducible representations of point group C15h have complex values. 14 two-dimensional real-valued representations E can be constructed as direct sum of the 14 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'5 = E'5a ⊕ E'5b
E'6 = E'6a ⊕ E'6b
E'7 = E'7a ⊕ E'7b
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''5 = E''5a ⊕ E''5b
E''6 = E''6a ⊕ E''6b
E''7 = E''7a ⊕ E''7b

ε=exp(2πi/15)
C15h E C15 (C15)2 C5 (C15)4 C3 (C5)2 (C15)7 (C15)8 (C5)3 (C3)2 (C15)11 (C5)4 (C15)13 (C15)14 σh S15 (S15)17 S5 (S15)19 S3 (S5)7 (S15)7 (S15)23 (S5)3 (S3)5 (S15)11 (S5)9 (S15)13 (S15)29
A' 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
E'1 E'1a
E'1b
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε5*
ε5*
ε6*
ε6*
ε7*
ε7*
ε7*
ε7*
ε6*
ε6*
ε5*
ε5*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε5*
ε5*
ε6*
ε6*
ε7*
ε7*
ε7*
ε7*
ε6*
ε6*
ε5*
ε5*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
E'2 E'2a
E'2b
1
1
ε2*
ε2*
ε4*
ε4*
ε6*
ε6*
ε7*
ε7*
ε5*
ε5*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε5*
ε5*
ε7*
ε7*
ε6*
ε6*
ε4*
ε4*
ε2*
ε2*
1
1
ε2*
ε2*
ε4*
ε4*
ε6*
ε6*
ε7*
ε7*
ε5*
ε5*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε5*
ε5*
ε7*
ε7*
ε6*
ε6*
ε4*
ε4*
ε2*
ε2*
E'3 E'3a
E'3b
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
E'4 E'4a
E'4b
1
1
ε4*
ε4*
ε7*
ε7*
ε3*
ε3*
ε*
ε*
ε5*
ε5*
ε6*
ε6*
ε2*
ε2*
ε2*
ε2*
ε6*
ε6*
ε5*
ε5*
ε*
ε*
ε3*
ε3*
ε7*
ε7*
ε4*
ε4*
1
1
ε4*
ε4*
ε7*
ε7*
ε3*
ε3*
ε*
ε*
ε5*
ε5*
ε6*
ε6*
ε2*
ε2*
ε2*
ε2*
ε6*
ε6*
ε5*
ε5*
ε*
ε*
ε3*
ε3*
ε7*
ε7*
ε4*
ε4*
E'5 E'5a
E'5b
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
E'6 E'6a
E'6b
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
E'7 E'7a
E'7b
1
1
ε7*
ε7*
ε*
ε*
ε6*
ε6*
ε2*
ε2*
ε5*
ε5*
ε3*
ε3*
ε4*
ε4*
ε4*
ε4*
ε3*
ε3*
ε5*
ε5*
ε2*
ε2*
ε6*
ε6*
ε*
ε*
ε7*
ε7*
1
1
ε7*
ε7*
ε*
ε*
ε6*
ε6*
ε2*
ε2*
ε5*
ε5*
ε3*
ε3*
ε4*
ε4*
ε4*
ε4*
ε3*
ε3*
ε5*
ε5*
ε2*
ε2*
ε6*
ε6*
ε*
ε*
ε7*
ε7*
A'' 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
E''1 E''1a
E''1b
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε5*
ε5*
ε6*
ε6*
ε7*
ε7*
ε7*
ε7*
ε6*
ε6*
ε5*
ε5*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
-1
-1
*
*
2*
2*
3*
3*
4*
4*
5*
5*
6*
6*
7*
7*
7*
7*
6*
6*
5*
5*
4*
4*
3*
3*
2*
2*
*
*
E''2 E''2a
E''2b
1
1
ε2*
ε2*
ε4*
ε4*
ε6*
ε6*
ε7*
ε7*
ε5*
ε5*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε5*
ε5*
ε7*
ε7*
ε6*
ε6*
ε4*
ε4*
ε2*
ε2*
-1
-1
2*
2*
4*
4*
6*
6*
7*
7*
5*
5*
3*
3*
*
*
*
*
3*
3*
5*
5*
7*
7*
6*
6*
4*
4*
2*
2*
E''3 E''3a
E''3b
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε6*
ε6*
ε3*
ε3*
-1
-1
3*
3*
6*
6*
6*
6*
3*
3*
-1
-1
3*
3*
6*
6*
6*
6*
3*
3*
-1
-1
3*
3*
6*
6*
6*
6*
3*
3*
E''4 E''4a
E''4b
1
1
ε4*
ε4*
ε7*
ε7*
ε3*
ε3*
ε*
ε*
ε5*
ε5*
ε6*
ε6*
ε2*
ε2*
ε2*
ε2*
ε6*
ε6*
ε5*
ε5*
ε*
ε*
ε3*
ε3*
ε7*
ε7*
ε4*
ε4*
-1
-1
4*
4*
7*
7*
3*
3*
*
*
5*
5*
6*
6*
2*
2*
2*
2*
6*
6*
5*
5*
*
*
3*
3*
7*
7*
4*
4*
E''5 E''5a
E''5b
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
1
1
ε5*
ε5*
ε5*
ε5*
-1
-1
5*
5*
5*
5*
-1
-1
5*
5*
5*
5*
-1
-1
5*
5*
5*
5*
-1
-1
5*
5*
5*
5*
-1
-1
5*
5*
5*
5*
E''6 E''6a
E''6b
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
1
1
ε6*
ε6*
ε3*
ε3*
ε3*
ε3*
ε6*
ε6*
-1
-1
6*
6*
3*
3*
3*
3*
6*
6*
-1
-1
6*
6*
3*
3*
3*
3*
6*
6*
-1
-1
6*
6*
3*
3*
3*
3*
6*
6*
E''7 E''7a
E''7b
1
1
ε7*
ε7*
ε*
ε*
ε6*
ε6*
ε2*
ε2*
ε5*
ε5*
ε3*
ε3*
ε4*
ε4*
ε4*
ε4*
ε3*
ε3*
ε5*
ε5*
ε2*
ε2*
ε6*
ε6*
ε*
ε*
ε7*
ε7*
-1
-1
7*
7*
*
*
6*
6*
2*
2*
5*
5*
3*
3*
4*
4*
4*
4*
3*
3*
5*
5*
2*
2*
6*
6*
*
*
7*
7*

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 C15h 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