Note on E representations in
C17 character table



16 irreducible representations of point group C17 have complex values. 8 two-dimensional real-valued representations E can be constructed as direct sum of the 8 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
E8 = E8a ⊕ E8b


ε=exp(2πi/17)
C17 E C17 (C17)2 (C17)3 (C17)4 (C17)5 (C17)6 (C17)7 (C17)8 (C17)9 (C17)10 (C17)11 (C17)12 (C17)13 (C17)14 (C17)15 (C17)16
A 1 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*
ε4*
ε4*
ε5*
ε5*
ε6*
ε6*
ε7*
ε7*
ε8*
ε8*
ε8*
ε8*
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*
ε8*
ε8*
ε7*
ε7*
ε5*
ε5*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε5*
ε5*
ε7*
ε7*
ε8*
ε8*
ε6*
ε6*
ε4*
ε4*
ε2*
ε2*
E3 E3a
E3b
1
1
ε3*
ε3*
ε6*
ε6*
ε8*
ε8*
ε5*
ε5*
ε2*
ε2*
ε*
ε*
ε4*
ε4*
ε7*
ε7*
ε7*
ε7*
ε4*
ε4*
ε*
ε*
ε2*
ε2*
ε5*
ε5*
ε8*
ε8*
ε6*
ε6*
ε3*
ε3*
E4 E4a
E4b
1
1
ε4*
ε4*
ε8*
ε8*
ε5*
ε5*
ε*
ε*
ε3*
ε3*
ε7*
ε7*
ε6*
ε6*
ε2*
ε2*
ε2*
ε2*
ε6*
ε6*
ε7*
ε7*
ε3*
ε3*
ε*
ε*
ε5*
ε5*
ε8*
ε8*
ε4*
ε4*
E5 E5a
E5b
1
1
ε5*
ε5*
ε7*
ε7*
ε2*
ε2*
ε3*
ε3*
ε8*
ε8*
ε4*
ε4*
ε*
ε*
ε6*
ε6*
ε6*
ε6*
ε*
ε*
ε4*
ε4*
ε8*
ε8*
ε3*
ε3*
ε2*
ε2*
ε7*
ε7*
ε5*
ε5*
E6 E6a
E6b
1
1
ε6*
ε6*
ε5*
ε5*
ε*
ε*
ε7*
ε7*
ε4*
ε4*
ε2*
ε2*
ε8*
ε8*
ε3*
ε3*
ε3*
ε3*
ε8*
ε8*
ε2*
ε2*
ε4*
ε4*
ε7*
ε7*
ε*
ε*
ε5*
ε5*
ε6*
ε6*
E7 E7a
E7b
1
1
ε7*
ε7*
ε3*
ε3*
ε4*
ε4*
ε6*
ε6*
ε*
ε*
ε8*
ε8*
ε2*
ε2*
ε5*
ε5*
ε5*
ε5*
ε2*
ε2*
ε8*
ε8*
ε*
ε*
ε6*
ε6*
ε4*
ε4*
ε3*
ε3*
ε7*
ε7*
E8 E8a
E8b
1
1
ε8*
ε8*
ε*
ε*
ε7*
ε7*
ε2*
ε2*
ε6*
ε6*
ε3*
ε3*
ε5*
ε5*
ε4*
ε4*
ε4*
ε4*
ε5*
ε5*
ε3*
ε3*
ε6*
ε6*
ε2*
ε2*
ε7*
ε7*
ε*
ε*
ε8*
ε8*


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