E Representations in Point Group S28

Note on E representations in
S₂₈ character table

26 irreducible representations of point group S₂₈ have complex values. 13 two-dimensional real-valued representations E can be constructed as direct sum of the 13 pairs complex plus conjugate complex irreducible representation.
E₁ = E_1a ⊕ E_1b
E₂ = E_2a ⊕ E_2b
E₃ = E_3a ⊕ E_3b
E₄ = E_4a ⊕ E_4b
E₅ = E_5a ⊕ E_5b
E₆ = E_6a ⊕ E_6b
E₇ = E_7a ⊕ E_7b
E₈ = E_8a ⊕ E_8b
E₉ = E_9a ⊕ E_9b
E₁₀ = E_10a ⊕ E_10b
E₁₁ = E_11a ⊕ E_11b
E₁₂ = E_12a ⊕ E_12b
E₁₃ = E_13a ⊕ E_13b

ε=exp(2πi/28)
S₂₈		E	S₂₈	C₁₄	(S₂₈)³	C₇	(S₂₈)⁵	(C₁₄)³	S₄	(C₇)²	(S₂₈)⁹	(C₁₄)⁵	(S₂₈)¹¹	(C₇)³	(S₂₈)¹³	C₂	(S₂₈)¹⁵	(C₇)⁴	(S₂₈)¹⁷	(C₁₄)⁹	(S₂₈)¹⁹	(C₇)⁵	(S₄)³	(C₁₄)¹¹	(S₂₈)²³	(C₇)⁶	(S₂₈)²⁵	(C₁₄)¹³	(S₂₈)²⁷
	A	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
	B	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1
E₁	E_1a E_1b	1 1	ε^* ε^*	ε^2* ε^2*	ε^3* ε^3*	ε^4* ε^4*	ε^5* ε^5*	ε^6* ε^6*	i -i	-ε^6* -ε^6*	-ε^5* -ε^5*	-ε^4* -ε^4*	-ε^3* -ε^3*	-ε^2* -ε^2*	-ε^* -ε^*	-1 -1	-ε^* -ε^*	-ε^2* -ε^2*	-ε^3* -ε^3*	-ε^4* -ε^4*	-ε^5* -ε^5*	-ε^6* -ε^6*	-i i	ε^6* ε^6*	ε^5* ε^5*	ε^4* ε^4*	ε^3* ε^3*	ε^2* ε^2*	ε^* ε^*
E₂	E_2a E_2b	1 1	ε^2* ε^2*	ε^4* ε^4*	ε^6* ε^6*	-ε^6* -ε^6*	-ε^4* -ε^4*	-ε^2* -ε^2*	-1 -1	-ε^2* -ε^2*	-ε^4* -ε^4*	-ε^6* -ε^6*	ε^6* ε^6*	ε^4* ε^4*	ε^2* ε^2*	1 1	ε^2* ε^2*	ε^4* ε^4*	ε^6* ε^6*	-ε^6* -ε^6*	-ε^4* -ε^4*	-ε^2* -ε^2*	-1 -1	-ε^2* -ε^2*	-ε^4* -ε^4*	-ε^6* -ε^6*	ε^6* ε^6*	ε^4* ε^4*	ε^2* ε^2*
E₃	E_3a E_3b	1 1	ε^3* ε^3*	ε^6* ε^6*	-ε^5* -ε^5*	-ε^2* -ε^2*	-ε^* -ε^*	-ε^4* -ε^4*	-i i	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	ε^5* ε^5*	-ε^6* -ε^6*	-ε^3* -ε^3*	-1 -1	-ε^3* -ε^3*	-ε^6* -ε^6*	ε^5* ε^5*	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	i -i	-ε^4* -ε^4*	-ε^* -ε^*	-ε^2* -ε^2*	-ε^5* -ε^5*	ε^6* ε^6*	ε^3* ε^3*
E₄	E_4a E_4b	1 1	ε^4* ε^4*	-ε^6* -ε^6*	-ε^2* -ε^2*	-ε^2* -ε^2*	-ε^6* -ε^6*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^6* -ε^6*	-ε^2* -ε^2*	-ε^2* -ε^2*	-ε^6* -ε^6*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^6* -ε^6*	-ε^2* -ε^2*	-ε^2* -ε^2*	-ε^6* -ε^6*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^6* -ε^6*	-ε^2* -ε^2*	-ε^2* -ε^2*	-ε^6* -ε^6*	ε^4* ε^4*
E₅	E_5a E_5b	1 1	ε^5* ε^5*	-ε^4* -ε^4*	-ε^* -ε^*	-ε^6* -ε^6*	ε^3* ε^3*	ε^2* ε^2*	i -i	-ε^2* -ε^2*	-ε^3* -ε^3*	ε^6* ε^6*	ε^* ε^*	ε^4* ε^4*	-ε^5* -ε^5*	-1 -1	-ε^5* -ε^5*	ε^4* ε^4*	ε^* ε^*	ε^6* ε^6*	-ε^3* -ε^3*	-ε^2* -ε^2*	-i i	ε^2* ε^2*	ε^3* ε^3*	-ε^6* -ε^6*	-ε^* -ε^*	-ε^4* -ε^4*	ε^5* ε^5*
E₆	E_6a E_6b	1 1	ε^6* ε^6*	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^4* ε^4*	ε^2* ε^2*	-ε^6* -ε^6*	-1 -1	-ε^6* -ε^6*	ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^6* ε^6*	1 1	ε^6* ε^6*	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^4* ε^4*	ε^2* ε^2*	-ε^6* -ε^6*	-1 -1	-ε^6* -ε^6*	ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^6* ε^6*
E₇	E_7a E_7b	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i
E₈	E_8a E_8b	1 1	-ε^6* -ε^6*	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	-ε^6* -ε^6*	1 1	-ε^6* -ε^6*	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	-ε^6* -ε^6*	1 1	-ε^6* -ε^6*	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	-ε^6* -ε^6*	1 1	-ε^6* -ε^6*	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	-ε^6* -ε^6*
E₉	E_9a E_9b	1 1	-ε^5* -ε^5*	-ε^4* -ε^4*	ε^* ε^*	-ε^6* -ε^6*	-ε^3* -ε^3*	ε^2* ε^2*	i -i	-ε^2* -ε^2*	ε^3* ε^3*	ε^6* ε^6*	-ε^* -ε^*	ε^4* ε^4*	ε^5* ε^5*	-1 -1	ε^5* ε^5*	ε^4* ε^4*	-ε^* -ε^*	ε^6* ε^6*	ε^3* ε^3*	-ε^2* -ε^2*	-i i	ε^2* ε^2*	-ε^3* -ε^3*	-ε^6* -ε^6*	ε^* ε^*	-ε^4* -ε^4*	-ε^5* -ε^5*
E₁₀	E_10a E_10b	1 1	-ε^4* -ε^4*	-ε^6* -ε^6*	ε^2* ε^2*	-ε^2* -ε^2*	ε^6* ε^6*	ε^4* ε^4*	-1 -1	ε^4* ε^4*	ε^6* ε^6*	-ε^2* -ε^2*	ε^2* ε^2*	-ε^6* -ε^6*	-ε^4* -ε^4*	1 1	-ε^4* -ε^4*	-ε^6* -ε^6*	ε^2* ε^2*	-ε^2* -ε^2*	ε^6* ε^6*	ε^4* ε^4*	-1 -1	ε^4* ε^4*	ε^6* ε^6*	-ε^2* -ε^2*	ε^2* ε^2*	-ε^6* -ε^6*	-ε^4* -ε^4*
E₁₁	E_11a E_11b	1 1	-ε^3* -ε^3*	ε^6* ε^6*	ε^5* ε^5*	-ε^2* -ε^2*	ε^* ε^*	-ε^4* -ε^4*	-i i	ε^4* ε^4*	-ε^* -ε^*	ε^2* ε^2*	-ε^5* -ε^5*	-ε^6* -ε^6*	ε^3* ε^3*	-1 -1	ε^3* ε^3*	-ε^6* -ε^6*	-ε^5* -ε^5*	ε^2* ε^2*	-ε^* -ε^*	ε^4* ε^4*	i -i	-ε^4* -ε^4*	ε^* ε^*	-ε^2* -ε^2*	ε^5* ε^5*	ε^6* ε^6*	-ε^3* -ε^3*
E₁₂	E_12a E_12b	1 1	-ε^2* -ε^2*	ε^4* ε^4*	-ε^6* -ε^6*	-ε^6* -ε^6*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	-ε^6* -ε^6*	-ε^6* -ε^6*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	-ε^6* -ε^6*	-ε^6* -ε^6*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	-ε^6* -ε^6*	-ε^6* -ε^6*	ε^4* ε^4*	-ε^2* -ε^2*
E₁₃	E_13a E_13b	1 1	-ε^* -ε^*	ε^2* ε^2*	-ε^3* -ε^3*	ε^4* ε^4*	-ε^5* -ε^5*	ε^6* ε^6*	i -i	-ε^6* -ε^6*	ε^5* ε^5*	-ε^4* -ε^4*	ε^3* ε^3*	-ε^2* -ε^2*	ε^* ε^*	-1 -1	ε^* ε^*	-ε^2* -ε^2*	ε^3* ε^3*	-ε^4* -ε^4*	ε^5* ε^5*	-ε^6* -ε^6*	-i i	ε^6* ε^6*	-ε^5* -ε^5*	ε^4* ε^4*	-ε^3* -ε^3*	ε^2* ε^2*	-ε^* -ε^*

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:

The number of irreducible representations is usually equal the number of classes. For point group S₂₈ 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 = ∑ Χ_i(E) Χ_i(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
c_i = 1/h ∑_R n_R χ_i(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:
c_i(E) = 1/(2h) ∑_R n_R χ_i(R)^irred χ(R)^red = (∑_R n_R χ_i(R)^irred χ(R)^red) / (∑_R n_R χ_i(R)^irred χ_i(R)^irred)

Last update November, 13^th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement