Note on E representations in
S10 character table



8 irreducible representations of point group S10 have complex values. 4 two-dimensional real-valued representations E can be constructed as direct sum of the 4 pairs complex plus conjugate complex irreducible representation.

E1g = E1g,a ⊕ E1g,b
E2g = E2g,a ⊕ E2g,b
E1u = E1u,a ⊕ E1u,b
E2u = E2u,a ⊕ E2u,b


ε=exp(2πi/5)
S10 E C5 (C5)2 (C5)3 (C5)4 i (S10)7 (S10)9 S10 (S10)3
Ag 1 1 1 1 1 1 1 1 1 1
E1g E1g,a
E1g,b
1
1
ε*
ε*
ε2*
ε2*
ε2*
ε2*
ε*
ε*
1
1
ε*
ε*
ε2*
ε2*
ε2*
ε2*
ε*
ε*
E2g E2g,a
E2g,b
1
1
ε2*
ε2*
ε*
ε*
ε*
ε*
ε2*
ε2*
1
1
ε2*
ε2*
ε*
ε*
ε*
ε*
ε2*
ε2*
Au 1 1 1 1 1 -1 -1 -1 -1 -1
E1u E1u,a
E1u,b
1
1
ε*
ε*
ε2*
ε2*
ε2*
ε2*
ε*
ε*
-1
-1
*
*
2*
2*
2*
2*
*
*
E2u E2u,a
E2u,b
1
1
ε2*
ε2*
ε*
ε*
ε*
ε*
ε2*
ε2*
-1
-1
2*
2*
*
*
*
*
2*
2*


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