E Representations in Point Group C20h

ε=exp(2πi/20)
C_20h		E	C₂₀	C₁₀	(C₂₀)³	C₅	C₄	(C₁₀)³	(C₂₀)⁷	(C₅)²	(C₂₀)⁹	C₂	(C₂₀)¹¹	(C₅)³	(C₂₀)¹³	(C₁₀)⁷	(C₄)³	(C₅)⁴	(C₂₀)¹⁷	(C₁₀)⁹	(C₂₀)¹⁹	i	(S₂₀)¹¹	(S₅)³	(S₂₀)¹³	(S₁₀)⁷	(S₄)³	(S₅)⁹	(S₂₀)¹⁷	(S₁₀)⁹	(S₂₀)¹⁹	σ_h	S₂₀	S₁₀	(S₂₀)³	S₅	S₄	(S₁₀)³	(S₂₀)⁷	(S₅)⁷	(S₂₀)⁹
	A_g	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	1	1	1	1	1	1	1	1	1	1
	B_g	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	1	-1	1	-1	1	-1	1	-1	1	-1
E_1g	E_1g,a E_1g,b	1 1	ε^* ε^*	ε^2* ε^2*	ε^3* ε^3*	ε^4* ε^4*	i -i	-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*	-i i	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*	i -i	-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*	-i i	ε^4* ε^4*	ε^3* ε^3*	ε^2* ε^2*	ε^* ε^*
E_2g	E_2g,a E_2g,b	1 1	ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*	-1 -1	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^4* ε^4*	ε^2* ε^2*	1 1	ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*	-1 -1	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^4* ε^4*	ε^2* ε^2*	1 1	ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*	-1 -1	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^4* ε^4*	ε^2* ε^2*	1 1	ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*	-1 -1	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^4* ε^4*	ε^2* ε^2*
E_3g	E_3g,a E_3g,b	1 1	ε^3* ε^3*	-ε^4* -ε^4*	-ε^* -ε^*	-ε^2* -ε^2*	-i i	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	-ε^3* -ε^3*	-1 -1	-ε^3* -ε^3*	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	i -i	-ε^2* -ε^2*	-ε^* -ε^*	-ε^4* -ε^4*	ε^3* ε^3*	1 1	ε^3* ε^3*	-ε^4* -ε^4*	-ε^* -ε^*	-ε^2* -ε^2*	-i i	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	-ε^3* -ε^3*	-1 -1	-ε^3* -ε^3*	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	i -i	-ε^2* -ε^2*	-ε^* -ε^*	-ε^4* -ε^4*	ε^3* ε^3*
E_4g	E_4g,a E_4g,b	1 1	ε^4* ε^4*	-ε^2* -ε^2*	-ε^2* -ε^2*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^2* -ε^2*	-ε^2* -ε^2*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^2* -ε^2*	-ε^2* -ε^2*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^2* -ε^2*	-ε^2* -ε^2*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^2* -ε^2*	-ε^2* -ε^2*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^2* -ε^2*	-ε^2* -ε^2*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^2* -ε^2*	-ε^2* -ε^2*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^2* -ε^2*	-ε^2* -ε^2*	ε^4* ε^4*
E_5g	E_5g,a E_5g,b	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	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i
E_6g	E_6g,a E_6g,b	1 1	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^2* ε^2*	ε^4* ε^4*	-1 -1	ε^4* ε^4*	ε^2* ε^2*	-ε^2* -ε^2*	-ε^4* -ε^4*	1 1	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^2* ε^2*	ε^4* ε^4*	-1 -1	ε^4* ε^4*	ε^2* ε^2*	-ε^2* -ε^2*	-ε^4* -ε^4*	1 1	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^2* ε^2*	ε^4* ε^4*	-1 -1	ε^4* ε^4*	ε^2* ε^2*	-ε^2* -ε^2*	-ε^4* -ε^4*	1 1	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^2* ε^2*	ε^4* ε^4*	-1 -1	ε^4* ε^4*	ε^2* ε^2*	-ε^2* -ε^2*	-ε^4* -ε^4*
E_7g	E_7g,a E_7g,b	1 1	-ε^3* -ε^3*	-ε^4* -ε^4*	ε^* ε^*	-ε^2* -ε^2*	-i i	ε^2* ε^2*	-ε^* -ε^*	ε^4* ε^4*	ε^3* ε^3*	-1 -1	ε^3* ε^3*	ε^4* ε^4*	-ε^* -ε^*	ε^2* ε^2*	i -i	-ε^2* -ε^2*	ε^* ε^*	-ε^4* -ε^4*	-ε^3* -ε^3*	1 1	-ε^3* -ε^3*	-ε^4* -ε^4*	ε^* ε^*	-ε^2* -ε^2*	-i i	ε^2* ε^2*	-ε^* -ε^*	ε^4* ε^4*	ε^3* ε^3*	-1 -1	ε^3* ε^3*	ε^4* ε^4*	-ε^* -ε^*	ε^2* ε^2*	i -i	-ε^2* -ε^2*	ε^* ε^*	-ε^4* -ε^4*	-ε^3* -ε^3*
E_8g	E_8g,a E_8g,b	1 1	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*
E_9g	E_9g,a E_9g,b	1 1	-ε^* -ε^*	ε^2* ε^2*	-ε^3* -ε^3*	ε^4* ε^4*	i -i	-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*	-i i	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*	i -i	-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*	-i i	ε^4* ε^4*	-ε^3* -ε^3*	ε^2* ε^2*	-ε^* -ε^*
	A_u	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	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1
	B_u	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	-1	1	-1	1	-1	1	-1	1	-1	1
E_1u	E_1u,a E_1u,b	1 1	ε^* ε^*	ε^2* ε^2*	ε^3* ε^3*	ε^4* ε^4*	i -i	-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*	-i i	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*	-i i	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*	i -i	-ε^4* -ε^4*	-ε^3* -ε^3*	-ε^2* -ε^2*	-ε^* -ε^*
E_2u	E_2u,a E_2u,b	1 1	ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*	-1 -1	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^4* ε^4*	ε^2* ε^2*	1 1	ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*	-1 -1	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^4* ε^4*	ε^2* ε^2*	-1 -1	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^4* ε^4*	ε^2* ε^2*	1 1	ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*	-1 -1	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^4* ε^4*	ε^2* ε^2*	1 1	ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*
E_3u	E_3u,a E_3u,b	1 1	ε^3* ε^3*	-ε^4* -ε^4*	-ε^* -ε^*	-ε^2* -ε^2*	-i i	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	-ε^3* -ε^3*	-1 -1	-ε^3* -ε^3*	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	i -i	-ε^2* -ε^2*	-ε^* -ε^*	-ε^4* -ε^4*	ε^3* ε^3*	-1 -1	-ε^3* -ε^3*	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	i -i	-ε^2* -ε^2*	-ε^* -ε^*	-ε^4* -ε^4*	ε^3* ε^3*	1 1	ε^3* ε^3*	-ε^4* -ε^4*	-ε^* -ε^*	-ε^2* -ε^2*	-i i	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	-ε^3* -ε^3*
E_4u	E_4u,a E_4u,b	1 1	ε^4* ε^4*	-ε^2* -ε^2*	-ε^2* -ε^2*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^2* -ε^2*	-ε^2* -ε^2*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^2* -ε^2*	-ε^2* -ε^2*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^2* -ε^2*	-ε^2* -ε^2*	ε^4* ε^4*	-1 -1	-ε^4* -ε^4*	ε^2* ε^2*	ε^2* ε^2*	-ε^4* -ε^4*	-1 -1	-ε^4* -ε^4*	ε^2* ε^2*	ε^2* ε^2*	-ε^4* -ε^4*	-1 -1	-ε^4* -ε^4*	ε^2* ε^2*	ε^2* ε^2*	-ε^4* -ε^4*	-1 -1	-ε^4* -ε^4*	ε^2* ε^2*	ε^2* ε^2*	-ε^4* -ε^4*
E_5u	E_5u,a E_5u,b	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	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i
E_6u	E_6u,a E_6u,b	1 1	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^2* ε^2*	ε^4* ε^4*	-1 -1	ε^4* ε^4*	ε^2* ε^2*	-ε^2* -ε^2*	-ε^4* -ε^4*	1 1	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^2* ε^2*	ε^4* ε^4*	-1 -1	ε^4* ε^4*	ε^2* ε^2*	-ε^2* -ε^2*	-ε^4* -ε^4*	-1 -1	ε^4* ε^4*	ε^2* ε^2*	-ε^2* -ε^2*	-ε^4* -ε^4*	1 1	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^2* ε^2*	ε^4* ε^4*	-1 -1	ε^4* ε^4*	ε^2* ε^2*	-ε^2* -ε^2*	-ε^4* -ε^4*	1 1	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^2* ε^2*	ε^4* ε^4*
E_7u	E_7u,a E_7u,b	1 1	-ε^3* -ε^3*	-ε^4* -ε^4*	ε^* ε^*	-ε^2* -ε^2*	-i i	ε^2* ε^2*	-ε^* -ε^*	ε^4* ε^4*	ε^3* ε^3*	-1 -1	ε^3* ε^3*	ε^4* ε^4*	-ε^* -ε^*	ε^2* ε^2*	i -i	-ε^2* -ε^2*	ε^* ε^*	-ε^4* -ε^4*	-ε^3* -ε^3*	-1 -1	ε^3* ε^3*	ε^4* ε^4*	-ε^* -ε^*	ε^2* ε^2*	i -i	-ε^2* -ε^2*	ε^* ε^*	-ε^4* -ε^4*	-ε^3* -ε^3*	1 1	-ε^3* -ε^3*	-ε^4* -ε^4*	ε^* ε^*	-ε^2* -ε^2*	-i i	ε^2* ε^2*	-ε^* -ε^*	ε^4* ε^4*	ε^3* ε^3*
E_8u	E_8u,a E_8u,b	1 1	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	1 1	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	-1 -1	ε^2* ε^2*	-ε^4* -ε^4*	-ε^4* -ε^4*	ε^2* ε^2*	-1 -1	ε^2* ε^2*	-ε^4* -ε^4*	-ε^4* -ε^4*	ε^2* ε^2*	-1 -1	ε^2* ε^2*	-ε^4* -ε^4*	-ε^4* -ε^4*	ε^2* ε^2*	-1 -1	ε^2* ε^2*	-ε^4* -ε^4*	-ε^4* -ε^4*	ε^2* ε^2*
E_9u	E_9u,a E_9u,b	1 1	-ε^* -ε^*	ε^2* ε^2*	-ε^3* -ε^3*	ε^4* ε^4*	i -i	-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*	-i i	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*	-i i	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*	i -i	-ε^4* -ε^4*	ε^3* ε^3*	-ε^2* -ε^2*	ε^* ε^*

Note on E representations in
C_20h character table

h = ∑ Χ_i(E) Χ_i(E)

c_i = 1/h ∑_R n_R χ_i(R)^irred χ(R)^red

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)

Note on E representations inC20h character table

h = ∑ Χi(E) Χi(E)

ci = 1/h ∑R nR χi(R)irred χ(R)red

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)

Note on E representations in
C_20h character table

h = ∑ Χ_i(E) Χ_i(E)

c_i = 1/h ∑_R n_R χ_i(R)^irred χ(R)^red

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)