Note on E representations in
C14 character table



12 irreducible representations of point group C14 have complex values. 6 two-dimensional real-valued representations E can be constructed as direct sum of the 6 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


ε=exp(2πi/14)
C14 E C14 C7 (C14)3 (C7)2 (C14)5 (C7)3 C2 (C7)4 (C14)9 (C7)5 (C14)11 (C7)6 (C14)13
A 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
E1 E1a
E1b
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
3*
3*
2*
2*
*
*
-1
-1
*
*
2*
2*
3*
3*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
E2 E2a
E2b
1
1
ε2*
ε2*
3*
3*
*
*
*
*
3*
3*
ε2*
ε2*
1
1
ε2*
ε2*
3*
3*
*
*
*
*
3*
3*
ε2*
ε2*
E3 E3a
E3b
1
1
ε3*
ε3*
*
*
2*
2*
ε2*
ε2*
ε*
ε*
3*
3*
-1
-1
3*
3*
ε*
ε*
ε2*
ε2*
2*
2*
*
*
ε3*
ε3*
E4 E4a
E4b
1
1
3*
3*
*
*
ε2*
ε2*
ε2*
ε2*
*
*
3*
3*
1
1
3*
3*
*
*
ε2*
ε2*
ε2*
ε2*
*
*
3*
3*
E5 E5a
E5b
1
1
2*
2*
3*
3*
ε*
ε*
*
*
ε3*
ε3*
ε2*
ε2*
-1
-1
ε2*
ε2*
ε3*
ε3*
*
*
ε*
ε*
3*
3*
2*
2*
E6 E6a
E6b
1
1
*
*
ε2*
ε2*
3*
3*
3*
3*
ε2*
ε2*
*
*
1
1
*
*
ε2*
ε2*
3*
3*
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