## Note on E representation inT character table

2 irreducible representations of point group T have complex values. A two-dimensional real-valued representation E can be constructed as direct sum of the pair complex plus conjugate complex irreducible representation.

E = Ea ⊕ Eb

ε=exp(2πi/3)
T E 4C3 4(C3)2 3C2
A 1 1 1 1
E Ea
Eb
1
1
ε*
ε*
ε*
ε*
1
1
T 3 0 0 -1

Obviously the E representation is reducible. Nevertheless the E representation is treated often as irreducible representation and is called real-valued or pseudo irreducible representation. One should keep in mind that general statements for character tables fail for real-valued representations. For example:
• The number of irreducible representations is usually equal the number of classes. For point group T this statement is true for the complex irreducible representations. The number of real-valued irreducible representations is less than the number of classes.
• The sum of the squared characters of the neutral symmetry element over all irreducible representations is equal to the total number of symmetry elements

#### h = ∑ Χi(E) Χi(E)

This statement is true for the complex irreducible representations but not for the real-valued irreducible representations
• Reduction formula: The occurence of i-th irreducible representation in an reducible representation is given by the well known formula

#### ci = 1/h ∑R nR χi(R)irred χ(R)red

This statement is true for the complex irreducible representations. The norm of the two-dimensional real-valued irreducible representations is 2h (instead of h) meaning that the reduction formula has to be modified:

#### ci(E) = 1/(2h) ∑R nR χi(R)irred χ(R)red = (∑R nR χi(R)irred χ(R)red) / (∑R nR χi(R)irred χi(R)irred)

Last update August, 12th 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement