Note on E representations in
S16 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:



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