Note on E representations inS16 character table

14 irreducible representations of point group S16 have complex values. 7 two-dimensional real-valued representations E can be constructed as direct sum of the 7 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

ε=exp(2πi/16)
S16 E S16 C8 (S16)3 C4 (S16)5 (C8)3 (S16)7 C2 (S16)9 (C8)5 (S16)11 (C4)3 (S16)13 (C8)7 (S16)15
A 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
E1 E1a
E1b
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
i
-i
3*
3*
2*
2*
*
*
-1
-1
*
*
2*
2*
3*
3*
-i
i
ε3*
ε3*
ε2*
ε2*
ε*
ε*
E2 E2a
E2b
1
1
ε2*
ε2*
i
-i
2*
2*
-1
-1
2*
2*
-i
i
ε2*
ε2*
1
1
ε2*
ε2*
i
-i
2*
2*
-1
-1
2*
2*
-i
i
ε2*
ε2*
E3 E3a
E3b
1
1
ε3*
ε3*
2*
2*
*
*
-i
i
ε*
ε*
ε2*
ε2*
3*
3*
-1
-1
3*
3*
ε2*
ε2*
ε*
ε*
i
-i
*
*
2*
2*
ε3*
ε3*
E4 E4a
E4b
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
E5 E5a
E5b
1
1
3*
3*
2*
2*
ε*
ε*
i
-i
*
*
ε2*
ε2*
ε3*
ε3*
-1
-1
ε3*
ε3*
ε2*
ε2*
*
*
-i
i
ε*
ε*
2*
2*
3*
3*
E6 E6a
E6b
1
1
2*
2*
-i
i
ε2*
ε2*
-1
-1
ε2*
ε2*
i
-i
2*
2*
1
1
2*
2*
-i
i
ε2*
ε2*
-1
-1
ε2*
ε2*
i
-i
2*
2*
E7 E7a
E7b
1
1
*
*
ε2*
ε2*
3*
3*
-i
i
ε3*
ε3*
2*
2*
ε*
ε*
-1
-1
ε*
ε*
2*
2*
ε3*
ε3*
i
-i
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 S16 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