Corepresentations PG

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COREPRESENTATIONS PG

Irreducible corepresentations of the Projective Magnetic Point Group 6_π/3mm

Table of characters of the unitary symmetry operations

6⁺

3⁺

3^-

6^-

m₁₁
m₁₀
m₀₁

m₂₁
m₁₂
^dm_1-1

m_1-1
^dm₂₁
^dm₁₂

^d1

^d6⁺

^d3⁺

^d2

^d3^-

^d6^-

^dm₁₁
^dm₁₀
^dm₀₁

²E₁²E₂

2e^2iπ/3

2e^-2iπ/3

2e^2iπ/3

2e^-2iπ/3

¹E₂A

√3e^-iπ/6

e^-iπ/3

e^iπ/3

√3e^iπ/6

√3e^-iπ/6

e^-iπ/3

e^iπ/3

√3e^iπ/6

¹E₁B

√3e^5iπ/6

e^-iπ/3

e^iπ/3

√3e^-5iπ/6

√3e^5iπ/6

e^-iπ/3

e^iπ/3

√3e^-5iπ/6

¹E₃'

e^-iπ/6

e^-iπ/3

e^iπ/3

e^iπ/6

-i

-1

e^5iπ/6

e^2iπ/3

-i

e^-2iπ/3

e^-5iπ/6

-i

¹E₃''

e^-iπ/6

e^-iπ/3

e^iπ/3

e^iπ/6

-i

-1

e^5iπ/6

e^2iπ/3

-i

e^-2iπ/3

e^-5iπ/6

²E₂'

e^5iπ/6

e^-iπ/3

-i

e^iπ/3

e^-5iπ/6

-i

-1

e^-iπ/6

e^2iπ/3

e^-2iπ/3

e^iπ/6

-i

²E₂''

e^5iπ/6

e^-iπ/3

-i

e^iπ/3

e^-5iπ/6

-i

-1

e^-iπ/6

e^2iπ/3

e^-2iπ/3

e^iπ/6

¹E₁¹E₂

e^5iπ/6

e^2iπ/3

e^-2iπ/3

e^-5iπ/6

-2

e^-iπ/6

e^-iπ/3

-2i

e^iπ/3

e^iπ/6

²E₁²E₃

e^-iπ/6

e^2iπ/3

-2i

e^-2iπ/3

e^iπ/6

-2

e^5iπ/6

e^-iπ/3

e^iπ/3

e^-5iπ/6

Multiplication table of the symmetry operations

6⁺

3⁺

3^-

6^-

m₁₁

m₂₁

m₁₀

m_1-1

m₀₁

m₁₂

^d1

^d6⁺

^d3⁺

^d2

^d3^-

^d6^-

^dm₁₁

^dm₂₁

^dm₁₀

^dm_1-1

^dm₀₁

^dm₁₂

6⁺

3⁺

3^-

6^-

m₁₁

m₂₁

m₁₀

m_1-1

m₀₁

m₁₂

^d1

^d6⁺

^d3⁺

^d2

^d3^-

^d6^-

^dm₁₁

^dm₂₁

^dm₁₀

^dm_1-1

^dm₀₁

^dm₁₂

6⁺

3⁺

^d2

3^-

6^-

m₁₂

^dm₁₁

^dm₂₁

m₁₀

m_1-1

^dm₀₁

^d6⁺

^d3⁺

^d3^-

^d6^-

^d1

^dm₁₂

m₁₁

m₂₁

^dm₁₀

^dm_1-1

m₀₁

3⁺

^d2

^d3^-

6^-

6⁺

^dm₀₁

^dm₁₂

m₁₁

^dm₂₁

m₁₀

^dm_1-1

^d3⁺

3^-

^d6^-

^d1

^d6⁺

m₀₁

m₁₂

^dm₁₁

m₂₁

^dm₁₀

m_1-1

3^-

6^-

^d1

^d6⁺

^d3⁺

m_1-1

^dm₀₁

^dm₁₂

^dm₁₁

m₂₁

m₁₀

^d2

^d3^-

^d6^-

6⁺

3⁺

^dm_1-1

m₀₁

m₁₂

m₁₁

^dm₂₁

^dm₁₀

3^-

6^-

^d6⁺

^d3⁺

m₁₀

^dm_1-1

m₀₁

^dm₁₂

^dm₁₁

^dm₂₁

^d3^-

^d6^-

^d1

6⁺

3⁺

^d2

^dm₁₀

m_1-1

^dm₀₁

m₁₂

m₁₁

m₂₁

6^-

6⁺

^d3⁺

3^-

^dm₂₁

^dm₁₀

m_1-1

m₀₁

^dm₁₂

m₁₁

^d6^-

^d1

^d6⁺

3⁺

^d2

^d3^-

m₂₁

m₁₀

^dm_1-1

^dm₀₁

m₁₂

^dm₁₁

m₁₁

^dm₂₁

m₁₀

^dm_1-1

^dm₀₁

m₁₂

^d1

6⁺

^d3⁺

3^-

^d6^-

^dm₁₁

m₂₁

^dm₁₀

m_1-1

m₀₁

^dm₁₂

^d6⁺

3⁺

^d2

^d3^-

6^-

m₂₁

^dm₁₀

^dm_1-1

m₀₁

^dm₁₂

^dm₁₁

6^-

^d1

6⁺

3⁺

^d2

3^-

^dm₂₁

m₁₀

m_1-1

^dm₀₁

m₁₂

m₁₁

^d6^-

^d6⁺

^d3⁺

^d3^-

m₁₀

m_1-1

m₀₁

m₁₂

m₁₁

^dm₂₁

^d3^-

6^-

^d1

^d6⁺

^d3⁺

^d2

^dm₁₀

^dm_1-1

^dm₀₁

^dm₁₂

^dm₁₁

m₂₁

3^-

^d6^-

6⁺

3⁺

m_1-1

m₀₁

^dm₁₂

m₁₁

^dm₂₁

m₁₀

^d2

3^-

^d6^-

^d1

^d6⁺

3⁺

^dm_1-1

^dm₀₁

m₁₂

^dm₁₁

m₂₁

^dm₁₀

^d3^-

6^-

6⁺

^d3⁺

m₀₁

^dm₁₂

^dm₁₁

^dm₂₁

m₁₀

m_1-1

3⁺

^d3^-

^d6^-

^d1

6⁺

^dm₀₁

m₁₂

m₁₁

m₂₁

^dm₁₀

^dm_1-1

^d3⁺

^d2

3^-

6^-

^d6⁺

m₁₂

m₁₁

^dm₂₁

^dm₁₀

^dm_1-1

^dm₀₁

^d6⁺

3⁺

3^-

6^-

^d1

^dm₁₂

^dm₁₁

m₂₁

m₁₀

m_1-1

m₀₁

6⁺

^d3⁺

^d2

^d3^-

^d6^-

^d1

^d6⁺

^d3⁺

^d2

^d3^-

^d6^-

^dm₁₁

^dm₂₁

^dm₁₀

^dm_1-1

^dm₀₁

^dm₁₂

6⁺

3⁺

3^-

6^-

m₁₁

m₂₁

m₁₀

m_1-1

m₀₁

m₁₂

^d6⁺

^d3⁺

^d3^-

^d6^-

^d1

^dm₁₂

m₁₁

m₂₁

^dm₁₀

^dm_1-1

m₀₁

6⁺

3⁺

^d2

3^-

6^-

m₁₂

^dm₁₁

^dm₂₁

m₁₀

m_1-1

^dm₀₁

^d3⁺

3^-

^d6^-

^d1

^d6⁺

m₀₁

m₁₂

^dm₁₁

m₂₁

^dm₁₀

m_1-1

3⁺

^d2

^d3^-

6^-

6⁺

^dm₀₁

^dm₁₂

m₁₁

^dm₂₁

m₁₀

^dm_1-1

^d2

^d3^-

^d6^-

6⁺

3⁺

^dm_1-1

m₀₁

m₁₂

m₁₁

^dm₂₁

^dm₁₀

3^-

6^-

^d1

^d6⁺

^d3⁺

m_1-1

^dm₀₁

^dm₁₂

^dm₁₁

m₂₁

m₁₀

^d3^-

^d6^-

^d1

6⁺

3⁺

^d2

^dm₁₀

m_1-1

^dm₀₁

m₁₂

m₁₁

m₂₁

3^-

6^-

^d6⁺

^d3⁺

m₁₀

^dm_1-1

m₀₁

^dm₁₂

^dm₁₁

^dm₂₁

^d6^-

^d1

^d6⁺

3⁺

^d2

^d3^-

m₂₁

m₁₀

^dm_1-1

^dm₀₁

m₁₂

^dm₁₁

6^-

6⁺

^d3⁺

3^-

^dm₂₁

^dm₁₀

m_1-1

m₀₁

^dm₁₂

m₁₁

^dm₁₁

m₂₁

^dm₁₀

m_1-1

m₀₁

^dm₁₂

^d6⁺

3⁺

^d2

^d3^-

6^-

m₁₁

^dm₂₁

m₁₀

^dm_1-1

^dm₀₁

m₁₂

^d1

6⁺

^d3⁺

3^-

^d6^-

^dm₂₁

m₁₀

m_1-1

^dm₀₁

m₁₂

m₁₁

^d6^-

^d6⁺

^d3⁺

^d3^-

m₂₁

^dm₁₀

^dm_1-1

m₀₁

^dm₁₂

^dm₁₁

6^-

^d1

6⁺

3⁺

^d2

3^-

^dm₁₀

^dm_1-1

^dm₀₁

^dm₁₂

^dm₁₁

m₂₁

3^-

^d6^-

6⁺

3⁺

m₁₀

m_1-1

m₀₁

m₁₂

m₁₁

^dm₂₁

^d3^-

6^-

^d1

^d6⁺

^d3⁺

^d2

^dm_1-1

^dm₀₁

m₁₂

^dm₁₁

m₂₁

^dm₁₀

^d3^-

6^-

6⁺

^d3⁺

m_1-1

m₀₁

^dm₁₂

m₁₁

^dm₂₁

m₁₀

^d2

3^-

^d6^-

^d1

^d6⁺

3⁺

^dm₀₁

m₁₂

m₁₁

m₂₁

^dm₁₀

^dm_1-1

^d3⁺

^d2

3^-

6^-

^d6⁺

m₀₁

^dm₁₂

^dm₁₁

^dm₂₁

m₁₀

m_1-1

3⁺

^d3^-

^d6^-

^d1

6⁺

^dm₁₂

^dm₁₁

m₂₁

m₁₀

m_1-1

m₀₁

6⁺

^d3⁺

^d2

^d3^-

^d6^-

m₁₂

m₁₁

^dm₂₁

^dm₁₀

^dm_1-1

^dm₀₁

^d6⁺

3⁺

3^-

6^-

^d1

Table of projective phases in group multiplication

6⁺

3⁺

3^-

6^-

m₁₁

m₂₁

m₁₀

m_1-1

m₀₁

m₁₂

^d1

^d6⁺

^d3⁺

^d2

^d3^-

^d6^-

^dm₁₁

^dm₂₁

^dm₁₀

^dm_1-1

^dm₀₁

^dm₁₂

6⁺

e^5iπ/6

e^-iπ/6

e^5iπ/6

e^-iπ/6

3⁺

e^2iπ/3

e^-iπ/3

e^2iπ/3

e^-iπ/3

-i

3^-

e^iπ/3

e^-2iπ/3

e^iπ/3

e^-2iπ/3

6^-

e^iπ/6

e^-5iπ/6

e^iπ/6

e^-5iπ/6

m₁₁

e^-iπ/6

e^-iπ/3

-i

e^-2iπ/3

e^-5iπ/6

e^iπ/6

e^iπ/3

e^2iπ/3

e^5iπ/6

e^-iπ/6

e^-iπ/3

-i

e^-2iπ/3

e^-5iπ/6

e^iπ/6

e^iπ/3

e^2iπ/3

e^5iπ/6

m₂₁

e^-iπ/6

e^-iπ/3

-i

e^-2iπ/3

e^iπ/6

e^-iπ/6

e^iπ/6

e^iπ/3

e^2iπ/3

e^-iπ/6

e^-iπ/3

-i

e^-2iπ/3

e^iπ/6

e^-iπ/6

e^iπ/6

e^iπ/3

e^2iπ/3

m₁₀

e^-iπ/6

e^-iπ/3

-i

e^iπ/3

e^iπ/6

e^-iπ/3

e^-iπ/6

e^iπ/6

e^iπ/3

e^-iπ/6

e^-iπ/3

-i

e^iπ/3

e^iπ/6

e^-iπ/3

e^-iπ/6

e^iπ/6

e^iπ/3

m_1-1

e^-iπ/6

e^-iπ/3

e^iπ/3

e^iπ/6

-i

e^-iπ/3

e^-iπ/6

e^iπ/6

e^iπ/3

e^-iπ/6

e^-iπ/3

e^iπ/3

e^iπ/6

-i

e^-iπ/3

e^-iπ/6

e^iπ/6

e^iπ/3

m₀₁

e^-iπ/6

e^2iπ/3

e^iπ/3

e^iπ/6

e^-2iπ/3

-i

e^-iπ/3

e^-iπ/6

e^iπ/6

e^-iπ/6

e^2iπ/3

e^iπ/3

e^iπ/6

e^-2iπ/3

-i

e^-iπ/3

e^-iπ/6

e^iπ/6

m₁₂

e^5iπ/6

e^2iπ/3

e^iπ/3

e^iπ/6

e^-5iπ/6

e^-2iπ/3

-i

e^-iπ/3

e^-iπ/6

e^5iπ/6

e^2iπ/3

e^iπ/3

e^iπ/6

e^-5iπ/6

e^-2iπ/3

-i

e^-iπ/3

e^-iπ/6

^d1

^d6⁺

e^5iπ/6

e^-iπ/6

e^5iπ/6

e^-iπ/6

^d3⁺

e^2iπ/3

e^-iπ/3

e^2iπ/3

e^-iπ/3

^d2

-i

^d3^-

e^iπ/3

e^-2iπ/3

e^iπ/3

e^-2iπ/3

^d6^-

e^iπ/6

e^-5iπ/6

e^iπ/6

e^-5iπ/6

^dm₁₁

e^-iπ/6

e^-iπ/3

-i

e^-2iπ/3

e^-5iπ/6

e^iπ/6

e^iπ/3

e^2iπ/3

e^5iπ/6

e^-iπ/6

e^-iπ/3

-i

e^-2iπ/3

e^-5iπ/6

e^iπ/6

e^iπ/3

e^2iπ/3

e^5iπ/6

^dm₂₁

e^-iπ/6

e^-iπ/3

-i

e^-2iπ/3

e^iπ/6

e^-iπ/6

e^iπ/6

e^iπ/3

e^2iπ/3

e^-iπ/6

e^-iπ/3

-i

e^-2iπ/3

e^iπ/6

e^-iπ/6

e^iπ/6

e^iπ/3

e^2iπ/3

^dm₁₀

e^-iπ/6

e^-iπ/3

-i

e^iπ/3

e^iπ/6

e^-iπ/3

e^-iπ/6

e^iπ/6

e^iπ/3

e^-iπ/6

e^-iπ/3

-i

e^iπ/3

e^iπ/6

e^-iπ/3

e^-iπ/6

e^iπ/6

e^iπ/3

^dm_1-1

e^-iπ/6

e^-iπ/3

e^iπ/3

e^iπ/6

-i

e^-iπ/3

e^-iπ/6

e^iπ/6

e^iπ/3

e^-iπ/6

e^-iπ/3

e^iπ/3

e^iπ/6

-i

e^-iπ/3

e^-iπ/6

e^iπ/6

e^iπ/3

^dm₀₁

e^-iπ/6

e^2iπ/3

e^iπ/3

e^iπ/6

e^-2iπ/3

-i

e^-iπ/3

e^-iπ/6

e^iπ/6

e^-iπ/6

e^2iπ/3

e^iπ/3

e^iπ/6

e^-2iπ/3

-i

e^-iπ/3

e^-iπ/6

e^iπ/6

^dm₁₂

e^5iπ/6

e^2iπ/3

e^iπ/3

e^iπ/6

e^-5iπ/6

e^-2iπ/3

-i

e^-iπ/3

e^-iπ/6

e^5iπ/6

e^2iπ/3

e^iπ/3

e^iπ/6

e^-5iπ/6

e^-2iπ/3

-i

e^-iπ/3

e^-iπ/6

Matrices of the representations of the group

The antiunitary operations are written in red color

Matrix presentation

Seitz symbol

²E₁²E₂

¹E₂A

¹E₁B

¹E₃'

¹E₃''

²E₂'

²E₂''

¹E₁¹E₂

²E₁²E₃

 1  0
 0  1

 1  0
 0  1

 1  0
 0  1

 1  0
 0  1

 1  0
 0  1

 1  0
 0  1

 1  0
 0  1

  1  -1
  1   0

  e^iπ/6   0
  0  e^-iπ/6

6⁺

  e^iπ/3   0
  0  e^-2iπ/3

  1   0
  0  e^-iπ/3

  -1   0
  0  e^2iπ/3

e^-iπ/6

e^5iπ/6

  i   0
  0  e^-5iπ/6

  -i   0
  0  e^iπ/6

  0  -1
  1  -1

  e^iπ/3   0
  0  e^-iπ/3

3⁺

 e^2iπ/3   0
  0  e^2iπ/3

  1   0
  0  e^-2iπ/3

  1   0
  0  e^-2iπ/3

e^-iπ/3

  -1   0
  0  e^iπ/3

  -1   0
  0  e^iπ/3

 -1   0
  0  -1

 -i   0
  0   i

 -1   0
  0   1

  1   0
  0  -1

 -1   0
  0   1

-i

 i  0
 0  i

 -i   0
  0  -i

 -1   1
 -1   0

 e^-iπ/3   0
  0   e^iπ/3

3^-

 e^-2iπ/3   0
  0  e^-2iπ/3

  1   0
  0  e^2iπ/3

  1   0
  0  e^2iπ/3

e^iπ/3

  -1   0
  0  e^-iπ/3

  -1   0
  0  e^-iπ/3

  0   1
 -1   1

 e^-iπ/6   0
  0   e^iπ/6

6^-

 e^-iπ/3   0
  0  e^2iπ/3

  1   0
  0  e^iπ/3

  -1   0
  0  e^-2iπ/3

e^iπ/6

e^-5iπ/6

  -i   0
  0  e^5iπ/6

  i   0
  0  e^-iπ/6

  0  -1
 -1   0

  0  e^-2iπ/3
  e^-iπ/3   0

m₁₁

 0  1
 1  0

 0  1
 1  0

 0  1
 1  0

-i

  0  -1
  1   0

  0  -1
  1   0

 -1   0
 -1   1

  0   e^iπ/6
 e^5iπ/6   0

m₂₁

  0  -i
  i   0

  0  e^-iπ/6
  e^iπ/6   0

  0   e^5iπ/6
 e^-5iπ/6   0

-i

  0  e^-2iπ/3
  e^-iπ/3   0

  0   e^iπ/3
 e^2iπ/3   0

 -1   1
  0   1

  0  -1
  1   0

m₁₀

  0  -1
 -1   0

  0  e^-iπ/3
  e^iπ/3   0

  0  e^-iπ/3
  e^iπ/3   0

-i

  0   e^-iπ/3
 e^-2iπ/3   0

  0   e^-iπ/3
 e^-2iπ/3   0

 0  1
 1  0

  0  e^5iπ/6
  e^iπ/6   0

m_1-1

  0   i
 -i   0

  0  -i
  i   0

  0   i
 -i   0

-i

  0  -1
  1   0

  0   1
 -1   0

  1   0
  1  -1

  0  e^2iπ/3
  e^iπ/3   0

m₀₁

 0  1
 1  0

  0  e^-2iπ/3
  e^2iπ/3   0

  0  e^-2iπ/3
  e^2iπ/3   0

-i

  0   e^iπ/3
 e^2iπ/3   0

  0   e^iπ/3
 e^2iπ/3   0

  1  -1
  0  -1

  0  -i
 -i   0

m₁₂

  0  -i
  i   0

  0  e^-5iπ/6
  e^5iπ/6   0

  0   e^iπ/6
 e^-iπ/6   0

-i

  0  e^2iπ/3
  e^iπ/3   0

  0   e^-iπ/3
 e^-2iπ/3   0

 1  0
 0  1

 -1   0
  0  -1

^d1

 1  0
 0  1

 1  0
 0  1

 1  0
 0  1

-1

 -1   0
  0  -1

 -1   0
  0  -1

  1  -1
  1   0

 e^-5iπ/6   0
  0   e^5iπ/6

^d6⁺

  e^iπ/3   0
  0  e^-2iπ/3

  1   0
  0  e^-iπ/3

  -1   0
  0  e^2iπ/3

e^5iπ/6

e^-iπ/6

  -i   0
  0  e^iπ/6

  i   0
  0  e^-5iπ/6

  0  -1
  1  -1

 e^-2iπ/3   0
  0   e^2iπ/3

^d3⁺

 e^2iπ/3   0
  0  e^2iπ/3

  1   0
  0  e^-2iπ/3

  1   0
  0  e^-2iπ/3

e^2iπ/3

  1   0
  0  e^-2iπ/3

  1   0
  0  e^-2iπ/3

 -1   0
  0  -1

  i   0
  0  -i

^d2

 -1   0
  0   1

  1   0
  0  -1

 -1   0
  0   1

-i

 -i   0
  0  -i

 i  0
 0  i

 -1   1
 -1   0

  e^2iπ/3   0
  0  e^-2iπ/3

^d3^-

 e^-2iπ/3   0
  0  e^-2iπ/3

  1   0
  0  e^2iπ/3

  1   0
  0  e^2iπ/3

e^-2iπ/3

  1   0
  0  e^2iπ/3

  1   0
  0  e^2iπ/3

  0   1
 -1   1

  e^5iπ/6   0
  0  e^-5iπ/6

^d6^-

 e^-iπ/3   0
  0  e^2iπ/3

  1   0
  0  e^iπ/3

  -1   0
  0  e^-2iπ/3

e^-5iπ/6

e^iπ/6

  i   0
  0  e^-iπ/6

  -i   0
  0  e^5iπ/6

  0  -1
 -1   0

  0   e^iπ/3
 e^2iπ/3   0

^dm₁₁

 0  1
 1  0

 0  1
 1  0

 0  1
 1  0

-i

  0   1
 -1   0

  0   1
 -1   0

 -1   0
 -1   1

  0  e^-5iπ/6
  e^-iπ/6   0

^dm₂₁

  0  -i
  i   0

  0  e^-iπ/6
  e^iπ/6   0

  0   e^5iπ/6
 e^-5iπ/6   0

-i

  0   e^iπ/3
 e^2iπ/3   0

  0  e^-2iπ/3
  e^-iπ/3   0

 -1   1
  0   1

  0   1
 -1   0

^dm₁₀

  0  -1
 -1   0

  0  e^-iπ/3
  e^iπ/3   0

  0  e^-iπ/3
  e^iπ/3   0

-i

  0  e^2iπ/3
  e^iπ/3   0

  0  e^2iπ/3
  e^iπ/3   0

 0  1
 1  0

  0   e^-iπ/6
 e^-5iπ/6   0

^dm_1-1

  0   i
 -i   0

  0  -i
  i   0

  0   i
 -i   0

-i

  0   1
 -1   0

  0  -1
  1   0

  1   0
  1  -1

  0   e^-iπ/3
 e^-2iπ/3   0

^dm₀₁

 0  1
 1  0

  0  e^-2iπ/3
  e^2iπ/3   0

  0  e^-2iπ/3
  e^2iπ/3   0

-i

  0  e^-2iπ/3
  e^-iπ/3   0

  0  e^-2iπ/3
  e^-iπ/3   0

  1  -1
  0  -1

 0  i
 i  0

^dm₁₂

  0  -i
  i   0

  0  e^-5iπ/6
  e^5iπ/6   0

  0   e^iπ/6
 e^-iπ/6   0

-i

  0   e^-iπ/3
 e^-2iπ/3   0

  0  e^2iπ/3
  e^iπ/3   0

k-Subgroupsmag

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Irreducible corepresentations of the Projective Magnetic Point Group 6π/3mm

Table of characters of the unitary symmetry operations

Multiplication table of the symmetry operations

Table of projective phases in group multiplication

Matrices of the representations of the group

Irreducible corepresentations of the Projective Magnetic Point Group 6_π/3mm