Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group -31' (N. 17.2.63)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
GM1+
Ag
GM1+
1
1
1
1
1
1
1
1
1
1
1
1
GM1-
Au
GM1-
1
1
1
-1
-1
-1
1
1
1
-1
-1
-1
GM2+GM3+
1Eg2Eg
GM2+GM3+
2
-1
-1
2
-1
-1
2
-1
-1
2
-1
-1
GM2-GM3-
1Eu2Eu
GM2-GM3-
2
-1
-1
-2
1
1
2
-1
-1
-2
1
1
GM6+GM6+
EgEg
GM4GM4
2
-2
-2
2
-2
-2
-2
2
2
-2
2
2
GM4+GM5+
1Eg2Eg
GM5GM6
2
1
1
2
1
1
-2
-1
-1
-2
-1
-1
GM6-GM6-
EuEu
GM7GM7
2
-2
-2
-2
2
2
-2
2
2
2
-2
-2
GM4-GM5-
1Eu2Eu
GM8GM9
2
1
1
-2
-1
-1
-2
-1
-1
2
1
1
The notation used in this table is an extension to corepresentations of the following notations used for irreducible representations:
(1): Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press.
(2): 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): 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 unitary symmetry operations in the conjugacy classes

C1: 1
C2: 3+001
C3: 3-001
C4: 1
C5: 3+001
C6: 3-001
C7: d1
C8: d3+001
C9: d3-001
C10: d1
C11: d3+001
C12: d3-001

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1+GM1-GM2+GM3+GM2-GM3-GM4GM4GM5GM6GM7GM7GM8GM9
1
(
 1 0 0
 0 1 0
 0 0 1
)
(
 1 0
 0 1
)
1
1
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
)
2
(
  0 -1  0
  1 -1  0
  0  0  1
)
(
 (1+i√3)/2  0
  0 (1-i√3)/2
)
3+001
1
1
(
  ei2π/3  0
  0 e-i2π/3
)
(
  ei2π/3  0
  0 e-i2π/3
)
(
 -1  0
  0 -1
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
 -1  0
  0 -1
)
(
 e-iπ/3  0
  0  eiπ/3
)
3
(
 -1  1  0
 -1  0  0
  0  0  1
)
(
 (1-i√3)/2  0
  0 (1+i√3)/2
)
3-001
1
1
(
 e-i2π/3  0
  0  ei2π/3
)
(
 e-i2π/3  0
  0  ei2π/3
)
(
 -1  0
  0 -1
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
 -1  0
  0 -1
)
(
  eiπ/3  0
  0 e-iπ/3
)
4
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 1 0
 0 1
)
1
1
-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
)
5
(
  0  1  0
 -1  1  0
  0  0 -1
)
(
 (1+i√3)/2  0
  0 (1-i√3)/2
)
3+001
1
-1
(
  ei2π/3  0
  0 e-i2π/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
 -1  0
  0 -1
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
 1 0
 0 1
)
(
  ei2π/3  0
  0 e-i2π/3
)
6
(
  1 -1  0
  1  0  0
  0  0 -1
)
(
 (1-i√3)/2  0
  0 (1+i√3)/2
)
3-001
1
-1
(
 e-i2π/3  0
  0  ei2π/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
 -1  0
  0 -1
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
 1 0
 0 1
)
(
 e-i2π/3  0
  0  ei2π/3
)
7
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
d1
1
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
)
8
(
  0 -1  0
  1 -1  0
  0  0  1
)
(
 -(1+i√3)/2  0
  0 -(1-i√3)/2
)
d3+001
1
1
(
  ei2π/3  0
  0 e-i2π/3
)
(
  ei2π/3  0
  0 e-i2π/3
)
(
 1 0
 0 1
)
(
  ei2π/3  0
  0 e-i2π/3
)
(
 1 0
 0 1
)
(
  ei2π/3  0
  0 e-i2π/3
)
9
(
 -1  1  0
 -1  0  0
  0  0  1
)
(
 -(1-i√3)/2  0
  0 -(1+i√3)/2
)
d3-001
1
1
(
 e-i2π/3  0
  0  ei2π/3
)
(
 e-i2π/3  0
  0  ei2π/3
)
(
 1 0
 0 1
)
(
 e-i2π/3  0
  0  ei2π/3
)
(
 1 0
 0 1
)
(
 e-i2π/3  0
  0  ei2π/3
)
10
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 -1  0
  0 -1
)
d1
1
-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
)
11
(
  0  1  0
 -1  1  0
  0  0 -1
)
(
 -(1+i√3)/2  0
  0 -(1-i√3)/2
)
d3+001
1
-1
(
  ei2π/3  0
  0 e-i2π/3
)
(
 e-iπ/3  0
  0  eiπ/3
)
(
 1 0
 0 1
)
(
  ei2π/3  0
  0 e-i2π/3
)
(
 -1  0
  0 -1
)
(
 e-iπ/3  0
  0  eiπ/3
)
12
(
  1 -1  0
  1  0  0
  0  0 -1
)
(
 -(1-i√3)/2  0
  0 -(1+i√3)/2
)
d3-001
1
-1
(
 e-i2π/3  0
  0  ei2π/3
)
(
  eiπ/3  0
  0 e-iπ/3
)
(
 1 0
 0 1
)
(
 e-i2π/3  0
  0  ei2π/3
)
(
 -1  0
  0 -1
)
(
  eiπ/3  0
  0 e-iπ/3
)
13
(
 1 0 0
 0 1 0
 0 0 1
)
(
 1 0
 0 1
)
1'
1
-1
(
 0 1
 1 0
)
(
  0 -1
 -1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
14
(
  0 -1  0
  1 -1  0
  0  0  1
)
(
 (1+i√3)/2  0
  0 (1-i√3)/2
)
3'+001
1
-1
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0 e-iπ/3
  eiπ/3  0
)
(
  0  1
 -1  0
)
(
  0 ei2π/3
  eiπ/3  0
)
(
  0 -1
  1  0
)
(
  0  e-iπ/3
 e-i2π/3  0
)
15
(
 -1  1  0
 -1  0  0
  0  0  1
)
(
 (1-i√3)/2  0
  0 (1+i√3)/2
)
3'-001
1
-1
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0  eiπ/3
 e-iπ/3  0
)
(
  0  1
 -1  0
)
(
  0 e-i2π/3
  e-iπ/3  0
)
(
  0 -1
  1  0
)
(
  0  eiπ/3
 ei2π/3  0
)
16
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 1 0
 0 1
)
1'
1
1
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
17
(
  0  1  0
 -1  1  0
  0  0 -1
)
(
 (1+i√3)/2  0
  0 (1-i√3)/2
)
3'+001
1
1
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0  1
 -1  0
)
(
  0 ei2π/3
  eiπ/3  0
)
(
  0  1
 -1  0
)
(
  0 ei2π/3
  eiπ/3  0
)
18
(
  1 -1  0
  1  0  0
  0  0 -1
)
(
 (1-i√3)/2  0
  0 (1+i√3)/2
)
3'-001
1
1
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0  1
 -1  0
)
(
  0 e-i2π/3
  e-iπ/3  0
)
(
  0  1
 -1  0
)
(
  0 e-i2π/3
  e-iπ/3  0
)
19
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
d1'
1
-1
(
 0 1
 1 0
)
(
  0 -1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
20
(
  0 -1  0
  1 -1  0
  0  0  1
)
(
 -(1+i√3)/2  0
  0 -(1-i√3)/2
)
d3'+001
1
-1
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0 e-iπ/3
  eiπ/3  0
)
(
  0 -1
  1  0
)
(
  0  e-iπ/3
 e-i2π/3  0
)
(
  0  1
 -1  0
)
(
  0 ei2π/3
  eiπ/3  0
)
21
(
 -1  1  0
 -1  0  0
  0  0  1
)
(
 -(1-i√3)/2  0
  0 -(1+i√3)/2
)
d3'-001
1
-1
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0  eiπ/3
 e-iπ/3  0
)
(
  0 -1
  1  0
)
(
  0  eiπ/3
 ei2π/3  0
)
(
  0  1
 -1  0
)
(
  0 e-i2π/3
  e-iπ/3  0
)
22
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 -1  0
  0 -1
)
d1'
1
1
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
23
(
  0  1  0
 -1  1  0
  0  0 -1
)
(
 -(1+i√3)/2  0
  0 -(1-i√3)/2
)
d3'+001
1
1
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0  ei2π/3
 e-i2π/3  0
)
(
  0 -1
  1  0
)
(
  0  e-iπ/3
 e-i2π/3  0
)
(
  0 -1
  1  0
)
(
  0  e-iπ/3
 e-i2π/3  0
)
24
(
  1 -1  0
  1  0  0
  0  0 -1
)
(
 -(1-i√3)/2  0
  0 -(1+i√3)/2
)
d3'-001
1
1
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0 e-i2π/3
  ei2π/3  0
)
(
  0 -1
  1  0
)
(
  0  eiπ/3
 ei2π/3  0
)
(
  0 -1
  1  0
)
(
  0  eiπ/3
 ei2π/3  0
)
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