Note on E representations in
C8 character table



6 irreducible representations of point group C8 have complex values. 3 two-dimensional real-valued representations E can be constructed as direct sum of the 3 pairs complex plus conjugate complex irreducible representation.

E1 = E1a ⊕ E1b
E2 = E2a ⊕ E2b
E3 = E3a ⊕ E3b


ε=exp(2πi/8)
C8 E C8 C4 (C8)3 C2 (C8)5 (C4)3 (C8)7
A 1 1 1 1 1 1 1 1
B 1 -1 1 -1 1 -1 1 -1
E1 E1a
E1b
1
1
ε*
ε*
i
-i
*
*
-1
-1
*
*
-i
i
ε*
ε*
E2 E2a
E2b
1
1
i
-i
-1
-1
-i
i
1
1
i
-i
-1
-1
-i
i
E3 E3a
E3b
1
1
*
*
-i
i
ε*
ε*
-1
-1
ε*
ε*
i
-i
*
*


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