Note on E representations in
C15 character table



14 irreducible representations of point group C15 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/15)
C15 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
A 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*
ε4*
ε4*
ε5*
ε5*
ε6*
ε6*
ε7*
ε7*
ε7*
ε7*
ε6*
ε6*
ε5*
ε5*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
E2 E2a
E2b
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*
E3 E3a
E3b
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*
E4 E4a
E4b
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*
E5 E5a
E5b
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*
E6 E6a
E6b
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*
E7 E7a
E7b
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*


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