k-Subgroupsmag

Irreducible representations of the Double Point Group 3 (No. 17)

Table of characters

(1)	(2)	(3)	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
GM₁⁺	A_g	GM₁⁺	1	1	1	1	1	1	1	1	1	1	1	1
GM₁^-	A_u	GM₁^-	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1
GM₂⁺	²E_g	GM₂⁺	1	-(1-i√3)/2	-(1+i√3)/2	1	-(1-i√3)/2	-(1+i√3)/2	1	-(1-i√3)/2	-(1+i√3)/2	1	-(1-i√3)/2	-(1+i√3)/2
GM₂^-	²E_u	GM₂^-	1	-(1-i√3)/2	-(1+i√3)/2	1	-(1-i√3)/2	-(1+i√3)/2	-1	(1-i√3)/2	(1+i√3)/2	-1	(1-i√3)/2	(1+i√3)/2
GM₃⁺	¹E_g	GM₃⁺	1	-(1+i√3)/2	-(1-i√3)/2	1	-(1+i√3)/2	-(1-i√3)/2	1	-(1+i√3)/2	-(1-i√3)/2	1	-(1+i√3)/2	-(1-i√3)/2
GM₃^-	¹E_u	GM₃^-	1	-(1+i√3)/2	-(1-i√3)/2	1	-(1+i√3)/2	-(1-i√3)/2	-1	(1+i√3)/2	(1-i√3)/2	-1	(1+i√3)/2	(1-i√3)/2
GM₆⁺	E_g	GM₄	1	-1	-1	-1	1	1	1	-1	-1	-1	1	1
GM₄⁺	¹E_g	GM₅	1	(1-i√3)/2	(1+i√3)/2	-1	-(1-i√3)/2	-(1+i√3)/2	1	(1-i√3)/2	(1+i√3)/2	-1	-(1-i√3)/2	-(1+i√3)/2
GM₅⁺	²E_g	GM₆	1	(1+i√3)/2	(1-i√3)/2	-1	-(1+i√3)/2	-(1-i√3)/2	1	(1+i√3)/2	(1-i√3)/2	-1	-(1+i√3)/2	-(1-i√3)/2
GM₆^-	E_u	GM₇	1	-1	-1	-1	1	1	-1	1	1	1	-1	-1
GM₄^-	¹E_u	GM₈	1	(1-i√3)/2	(1+i√3)/2	-1	-(1-i√3)/2	-(1+i√3)/2	-1	-(1-i√3)/2	-(1+i√3)/2	1	(1-i√3)/2	(1+i√3)/2
GM₅^-	²E_u	GM₉	1	(1+i√3)/2	(1-i√3)/2	-1	-(1+i√3)/2	-(1-i√3)/2	-1	-(1+i√3)/2	-(1-i√3)/2	1	(1+i√3)/2	(1-i√3)/2

(1): Notation of the irreps according to Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press.

(2): Notation of the irreps according to Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press, based on Mulliken RS (1933) Phys. Rev. 43, 279-302.

(3): Notation of the irreps according to A. P. Cracknell, B. L. Davies, S. C. Miller and W. F. Love (1979) Kronecher Product Tables, 1, General Introduction and Tables of Irreducible Representations of Space groups. New York: IFI/Plenum, for the GM point.

Lists of symmetry operations in the conjugacy classes

C₁: 1

C₂: 3⁺₀₀₁

C₃: 3^-₀₀₁

C₄: ^d1

C₅: ^d3⁺₀₀₁

C₆: ^d3^-₀₀₁

C₇: -1

C₈: -3⁺₀₀₁

C₉: -3^-₀₀₁

C₁₀: ^d-1

C₁₁: ^d-3⁺₀₀₁

C₁₂: ^d-3^-₀₀₁

List of pairs of conjugated irreducible representations

(^*GM₂⁺,^*GM₃⁺)
(^*GM₂^-,^*GM₃^-)
(^*GM₅,^*GM₆)
(^*GM₈,^*GM₉)
Matrices of the representations of the group

The number in parentheses after the label of the irrep indicates the "reality" of the irrep: (1) for real, (-1) for pseudoreal and (0) for complex representations.

Matrix presentation

Seitz Symbol

GM₁⁺(1)

GM₁^-(1)

GM₂⁺(0)

GM₂^-(0)

GM₃⁺(0)

GM₃^-(0)

GM₄(1)

GM₅(0)

GM₆(0)

GM₇(1)

GM₈(0)

GM₉(0)

 1  0  0
 0  1  0
 0  0  1

 1  0
 0  1

  0  -1   0
  1  -1   0
  0   0   1

 (1+i√3)/2   0
  0  (1-i√3)/2

3⁺₀₀₁

e^i2π/3

e^-i2π/3

-1

e^-iπ/3

e^iπ/3

-1

e^-iπ/3

e^iπ/3

 -1   1   0
 -1   0   0
  0   0   1

 (1-i√3)/2   0
  0  (1+i√3)/2

3^-₀₀₁

e^-i2π/3

e^i2π/3

-1

e^iπ/3

e^-iπ/3

-1

e^iπ/3

e^-iπ/3

 -1   0   0
  0  -1   0
  0   0  -1

 1  0
 0  1

-1

  0   1   0
 -1   1   0
  0   0  -1

 (1+i√3)/2   0
  0  (1-i√3)/2

3⁺₀₀₁

-1

e^i2π/3

e^-iπ/3

e^-i2π/3

e^iπ/3

-1

e^-iπ/3

e^iπ/3

e^i2π/3

e^-i2π/3

  1  -1   0
  1   0   0
  0   0  -1

 (1-i√3)/2   0
  0  (1+i√3)/2

3^-₀₀₁

-1

e^-i2π/3

e^iπ/3

e^i2π/3

e^-iπ/3

-1

e^iπ/3

e^-iπ/3

e^-i2π/3

e^i2π/3

 1  0  0
 0  1  0
 0  0  1

 -1   0
  0  -1

^d1

-1

  0  -1   0
  1  -1   0
  0   0   1

 -(1+i√3)/2   0
  0  -(1-i√3)/2

^d3⁺₀₀₁

e^i2π/3

e^-i2π/3

e^i2π/3

e^-i2π/3

e^i2π/3

e^-i2π/3

 -1   1   0
 -1   0   0
  0   0   1

 -(1-i√3)/2   0
  0  -(1+i√3)/2

^d3^-₀₀₁

e^-i2π/3

e^i2π/3

e^-i2π/3

e^i2π/3

e^-i2π/3

e^i2π/3

 -1   0   0
  0  -1   0
  0   0  -1

 -1   0
  0  -1

^d1

-1

  0   1   0
 -1   1   0
  0   0  -1

 -(1+i√3)/2   0
  0  -(1-i√3)/2

^d3⁺₀₀₁

-1

e^i2π/3

e^-iπ/3

e^-i2π/3

e^iπ/3

e^i2π/3

e^-i2π/3

-1

e^-iπ/3

e^iπ/3

  1  -1   0
  1   0   0
  0   0  -1

 -(1-i√3)/2   0
  0  -(1+i√3)/2

^d3^-₀₀₁

-1

e^-i2π/3

e^iπ/3

e^i2π/3

e^-iπ/3

e^-i2π/3

e^i2π/3

-1

e^iπ/3

e^-iπ/3

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