E Representations in Point Group C15h

Note on E representations in
C_15h character table

28 irreducible representations of point group C_15h have complex values. 14 two-dimensional real-valued representations E can be constructed as direct sum of the 14 pairs complex plus conjugate complex irreducible representation.
E'₁ = E'_1a ⊕ E'_1b
E'₂ = E'_2a ⊕ E'_2b
E'₃ = E'_3a ⊕ E'_3b
E'₄ = E'_4a ⊕ E'_4b
E'₅ = E'_5a ⊕ E'_5b
E'₆ = E'_6a ⊕ E'_6b
E'₇ = E'_7a ⊕ E'_7b
E''₁ = E''_1a ⊕ E''_1b
E''₂ = E''_2a ⊕ E''_2b
E''₃ = E''_3a ⊕ E''_3b
E''₄ = E''_4a ⊕ E''_4b
E''₅ = E''_5a ⊕ E''_5b
E''₆ = E''_6a ⊕ E''_6b
E''₇ = E''_7a ⊕ E''_7b

ε=exp(2πi/15)
C_15h		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	σ_h	S₁₅	(S₁₅)¹⁷	S₅	(S₁₅)¹⁹	S₃	(S₅)⁷	(S₁₅)⁷	(S₁₅)²³	(S₅)³	(S₃)⁵	(S₁₅)¹¹	(S₅)⁹	(S₁₅)¹³	(S₁₅)²⁹
	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	1	1
E'₁	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*	ε^7* ε^7*	ε^7* ε^7*	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*	ε^7* ε^7*	ε^7* ε^7*	ε^6* ε^6*	ε^5* ε^5*	ε^4* ε^4*	ε^3* ε^3*	ε^2* ε^2*	ε^* ε^*
E'₂	E'_2a E'_2b	1 1	ε^2* ε^2*	ε^4* ε^4*	ε^6* ε^6*	ε^7* ε^7*	ε^5* ε^5*	ε^3* ε^3*	ε^* ε^*	ε^* ε^*	ε^3* ε^3*	ε^5* ε^5*	ε^7* ε^7*	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^7* ε^7*	ε^5* ε^5*	ε^3* ε^3*	ε^* ε^*	ε^* ε^*	ε^3* ε^3*	ε^5* ε^5*	ε^7* ε^7*	ε^6* ε^6*	ε^4* ε^4*	ε^2* ε^2*
E'₃	E'_3a E'_3b	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^6* ε^6*	ε^3* ε^3*	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^6* ε^6*	ε^3* ε^3*	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^6* ε^6*	ε^3* ε^3*	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^6* ε^6*	ε^3* ε^3*	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^6* ε^6*	ε^3* ε^3*	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^6* ε^6*	ε^3* ε^3*
E'₄	E'_4a E'_4b	1 1	ε^4* ε^4*	ε^7* ε^7*	ε^3* ε^3*	ε^* ε^*	ε^5* ε^5*	ε^6* ε^6*	ε^2* ε^2*	ε^2* ε^2*	ε^6* ε^6*	ε^5* ε^5*	ε^* ε^*	ε^3* ε^3*	ε^7* ε^7*	ε^4* ε^4*	1 1	ε^4* ε^4*	ε^7* ε^7*	ε^3* ε^3*	ε^* ε^*	ε^5* ε^5*	ε^6* ε^6*	ε^2* ε^2*	ε^2* ε^2*	ε^6* ε^6*	ε^5* ε^5*	ε^* ε^*	ε^3* ε^3*	ε^7* ε^7*	ε^4* ε^4*
E'₅	E'_5a E'_5b	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*
E'₆	E'_6a E'_6b	1 1	ε^6* ε^6*	ε^3* ε^3*	ε^3* ε^3*	ε^6* ε^6*	1 1	ε^6* ε^6*	ε^3* ε^3*	ε^3* ε^3*	ε^6* ε^6*	1 1	ε^6* ε^6*	ε^3* ε^3*	ε^3* ε^3*	ε^6* ε^6*	1 1	ε^6* ε^6*	ε^3* ε^3*	ε^3* ε^3*	ε^6* ε^6*	1 1	ε^6* ε^6*	ε^3* ε^3*	ε^3* ε^3*	ε^6* ε^6*	1 1	ε^6* ε^6*	ε^3* ε^3*	ε^3* ε^3*	ε^6* ε^6*
E'₇	E'_7a E'_7b	1 1	ε^7* ε^7*	ε^* ε^*	ε^6* ε^6*	ε^2* ε^2*	ε^5* ε^5*	ε^3* ε^3*	ε^4* ε^4*	ε^4* ε^4*	ε^3* ε^3*	ε^5* ε^5*	ε^2* ε^2*	ε^6* ε^6*	ε^* ε^*	ε^7* ε^7*	1 1	ε^7* ε^7*	ε^* ε^*	ε^6* ε^6*	ε^2* ε^2*	ε^5* ε^5*	ε^3* ε^3*	ε^4* ε^4*	ε^4* ε^4*	ε^3* ε^3*	ε^5* ε^5*	ε^2* ε^2*	ε^6* ε^6*	ε^* ε^*	ε^7* ε^7*
	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	-1	-1
E''₁	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*	ε^7* ε^7*	ε^7* ε^7*	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*	-ε^7* -ε^7*	-ε^7* -ε^7*	-ε^6* -ε^6*	-ε^5* -ε^5*	-ε^4* -ε^4*	-ε^3* -ε^3*	-ε^2* -ε^2*	-ε^* -ε^*
E''₂	E''_2a E''_2b	1 1	ε^2* ε^2*	ε^4* ε^4*	ε^6* ε^6*	ε^7* ε^7*	ε^5* ε^5*	ε^3* ε^3*	ε^* ε^*	ε^* ε^*	ε^3* ε^3*	ε^5* ε^5*	ε^7* ε^7*	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^7* -ε^7*	-ε^5* -ε^5*	-ε^3* -ε^3*	-ε^* -ε^*	-ε^* -ε^*	-ε^3* -ε^3*	-ε^5* -ε^5*	-ε^7* -ε^7*	-ε^6* -ε^6*	-ε^4* -ε^4*	-ε^2* -ε^2*
E''₃	E''_3a E''_3b	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^6* ε^6*	ε^3* ε^3*	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^6* ε^6*	ε^3* ε^3*	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^6* ε^6*	ε^3* ε^3*	-1 -1	-ε^3* -ε^3*	-ε^6* -ε^6*	-ε^6* -ε^6*	-ε^3* -ε^3*	-1 -1	-ε^3* -ε^3*	-ε^6* -ε^6*	-ε^6* -ε^6*	-ε^3* -ε^3*	-1 -1	-ε^3* -ε^3*	-ε^6* -ε^6*	-ε^6* -ε^6*	-ε^3* -ε^3*
E''₄	E''_4a E''_4b	1 1	ε^4* ε^4*	ε^7* ε^7*	ε^3* ε^3*	ε^* ε^*	ε^5* ε^5*	ε^6* ε^6*	ε^2* ε^2*	ε^2* ε^2*	ε^6* ε^6*	ε^5* ε^5*	ε^* ε^*	ε^3* ε^3*	ε^7* ε^7*	ε^4* ε^4*	-1 -1	-ε^4* -ε^4*	-ε^7* -ε^7*	-ε^3* -ε^3*	-ε^* -ε^*	-ε^5* -ε^5*	-ε^6* -ε^6*	-ε^2* -ε^2*	-ε^2* -ε^2*	-ε^6* -ε^6*	-ε^5* -ε^5*	-ε^* -ε^*	-ε^3* -ε^3*	-ε^7* -ε^7*	-ε^4* -ε^4*
E''₅	E''_5a E''_5b	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^5* ε^5*	-1 -1	-ε^5* -ε^5*	-ε^5* -ε^5*	-1 -1	-ε^5* -ε^5*	-ε^5* -ε^5*	-1 -1	-ε^5* -ε^5*	-ε^5* -ε^5*	-1 -1	-ε^5* -ε^5*	-ε^5* -ε^5*	-1 -1	-ε^5* -ε^5*	-ε^5* -ε^5*
E''₆	E''_6a E''_6b	1 1	ε^6* ε^6*	ε^3* ε^3*	ε^3* ε^3*	ε^6* ε^6*	1 1	ε^6* ε^6*	ε^3* ε^3*	ε^3* ε^3*	ε^6* ε^6*	1 1	ε^6* ε^6*	ε^3* ε^3*	ε^3* ε^3*	ε^6* ε^6*	-1 -1	-ε^6* -ε^6*	-ε^3* -ε^3*	-ε^3* -ε^3*	-ε^6* -ε^6*	-1 -1	-ε^6* -ε^6*	-ε^3* -ε^3*	-ε^3* -ε^3*	-ε^6* -ε^6*	-1 -1	-ε^6* -ε^6*	-ε^3* -ε^3*	-ε^3* -ε^3*	-ε^6* -ε^6*
E''₇	E''_7a E''_7b	1 1	ε^7* ε^7*	ε^* ε^*	ε^6* ε^6*	ε^2* ε^2*	ε^5* ε^5*	ε^3* ε^3*	ε^4* ε^4*	ε^4* ε^4*	ε^3* ε^3*	ε^5* ε^5*	ε^2* ε^2*	ε^6* ε^6*	ε^* ε^*	ε^7* ε^7*	-1 -1	-ε^7* -ε^7*	-ε^* -ε^*	-ε^6* -ε^6*	-ε^2* -ε^2*	-ε^5* -ε^5*	-ε^3* -ε^3*	-ε^4* -ε^4*	-ε^4* -ε^4*	-ε^3* -ε^3*	-ε^5* -ε^5*	-ε^2* -ε^2*	-ε^6* -ε^6*	-ε^* -ε^*	-ε^7* -ε^7*

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 C_15h 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