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Irreducible corepresentations of the Magnetic Point Group -3 (N. 17.1.62)


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+
2Eg
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-
2Eu
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+
1Eg
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-
1Eu
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+
Eg
GM4
1
-1
-1
1
-1
-1
-1
1
1
-1
1
1
GM4+
1Eg
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+
2Eg
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-
Eu
GM7
1
-1
-1
-1
1
1
-1
1
1
1
-1
-1
GM4-
1Eu
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-
2Eu
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
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+GM2-GM3+GM3-GM4GM5GM6GM7GM8GM9
1
(
 1 0 0
 0 1 0
 0 0 1
)
(
 1 0
 0 1
)
1
1
1
1
1
1
1
1
1
1
1
1
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
ei2π/3
e-i2π/3
e-i2π/3
-1
e-iπ/3
eiπ/3
-1
e-iπ/3
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
e-i2π/3
ei2π/3
ei2π/3
-1
eiπ/3
e-iπ/3
-1
eiπ/3
e-iπ/3
4
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 1 0
 0 1
)
1
1
-1
1
-1
1
-1
1
1
1
-1
-1
-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
e-iπ/3
e-i2π/3
eiπ/3
-1
e-iπ/3
eiπ/3
1
ei2π/3
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
eiπ/3
ei2π/3
e-iπ/3
-1
eiπ/3
e-iπ/3
1
e-i2π/3
ei2π/3
7
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
d1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-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
ei2π/3
e-i2π/3
e-i2π/3
1
ei2π/3
e-i2π/3
1
ei2π/3
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
e-i2π/3
ei2π/3
ei2π/3
1
e-i2π/3
ei2π/3
1
e-i2π/3
ei2π/3
10
(
 -1  0  0
  0 -1  0
  0  0 -1
)
(
 -1  0
  0 -1
)
d1
1
-1
1
-1
1
-1
-1
-1
-1
1
1
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
e-iπ/3
e-i2π/3
eiπ/3
1
ei2π/3
e-i2π/3
-1
e-iπ/3
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
eiπ/3
ei2π/3
e-iπ/3
1
e-i2π/3
ei2π/3
-1
eiπ/3
e-iπ/3
k-Subgroupsmag
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