Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group 23 (N. 28.1.107)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
GM1
A
GM1
1
1
1
1
1
1
1
GM2
1E
GM2
1
1
-(1-i√3)/2
-(1+i√3)/2
1
-(1-i√3)/2
-(1+i√3)/2
GM3
2E
GM3
1
1
-(1+i√3)/2
-(1-i√3)/2
1
-(1+i√3)/2
-(1-i√3)/2
GM4
T
GM4
3
-1
0
0
3
0
0
GM5
E
GM5
2
0
1
1
-2
-1
-1
GM7
2F
GM6
2
0
-(1-i√3)/2
-(1+i√3)/2
-2
(1-i√3)/2
(1+i√3)/2
GM6
1F
GM7
2
0
-(1+i√3)/2
-(1-i√3)/2
-2
(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: 2001, 2010, 2100, d2001, d2010, d2100
C3: 3+111, 3+111, 3+111, 3+111
C4: 3-111, 3-111, 3-111, 3-111
C5: d1
C6: d3+111, d3+111, d3+111, d3+111
C7: d3-111, d3-111, d3-111, d3-111

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1GM2GM3GM4GM5GM6GM7
1
(
 1 0 0
 0 1 0
 0 0 1
)
(
 1 0
 0 1
)
1
1
1
1
(
 1 0 0
 0 1 0
 0 0 1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 1 0
 0 1
)
2
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
 -i  0
  0  i
)
2001
1
1
1
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
 -i  0
  0  i
)
(
 -i  0
  0  i
)
(
 -i  0
  0  i
)
3
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
  0 -1
  1  0
)
2010
1
1
1
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
4
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
  0 -i
 -i  0
)
2100
1
1
1
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
5
(
 0 0 1
 1 0 0
 0 1 0
)
(
  (1-i)/2 -(1+i)/2
  (1-i)/2  (1+i)/2
)
3+111
1
ei2π/3
e-i2π/3
(
 0 0 1
 1 0 0
 0 1 0
)
(
  e-iπ/4√2/2 e-i3π/4√2/2
  e-iπ/4√2/2  eiπ/4√2/2
)
(
  ei5π/12√2/2  e-iπ/12√2/2
  ei5π/12√2/2 ei11π/12√2/2
)
(
 e-i11π/12√2/2  ei7π/12√2/2
 e-i11π/12√2/2  e-i5π/12√2/2
)
6
(
  0  0  1
 -1  0  0
  0 -1  0
)
(
  (1+i)/2 -(1-i)/2
  (1+i)/2  (1-i)/2
)
3+-11-1
1
ei2π/3
e-i2π/3
(
  0  0 -1
  1  0  0
  0 -1  0
)
(
  eiπ/4√2/2 ei3π/4√2/2
  eiπ/4√2/2 e-iπ/4√2/2
)
(
 ei11π/12√2/2 e-i7π/12√2/2
 ei11π/12√2/2  ei5π/12√2/2
)
(
  e-i5π/12√2/2  eiπ/12√2/2
  e-i5π/12√2/2 e-i11π/12√2/2
)
7
(
  0  0 -1
 -1  0  0
  0  1  0
)
(
  (1+i)/2  (1-i)/2
 -(1+i)/2  (1-i)/2
)
3+1-1-1
1
ei2π/3
e-i2π/3
(
  0  0  1
 -1  0  0
  0 -1  0
)
(
  eiπ/4√2/2  e-iπ/4√2/2
 e-i3π/4√2/2  e-iπ/4√2/2
)
(
 ei11π/12√2/2  ei5π/12√2/2
  e-iπ/12√2/2  ei5π/12√2/2
)
(
  e-i5π/12√2/2 e-i11π/12√2/2
  ei7π/12√2/2 e-i11π/12√2/2
)
8
(
  0  0 -1
  1  0  0
  0 -1  0
)
(
  (1-i)/2  (1+i)/2
 -(1-i)/2  (1+i)/2
)
3+-1-11
1
ei2π/3
e-i2π/3
(
  0  0 -1
 -1  0  0
  0  1  0
)
(
 e-iπ/4√2/2  eiπ/4√2/2
 ei3π/4√2/2  eiπ/4√2/2
)
(
  ei5π/12√2/2 ei11π/12√2/2
 e-i7π/12√2/2 ei11π/12√2/2
)
(
 e-i11π/12√2/2  e-i5π/12√2/2
  eiπ/12√2/2  e-i5π/12√2/2
)
9
(
 0 1 0
 0 0 1
 1 0 0
)
(
  (1+i)/2  (1+i)/2
 -(1-i)/2  (1-i)/2
)
3-111
1
e-i2π/3
ei2π/3
(
 0 1 0
 0 0 1
 1 0 0
)
(
  eiπ/4√2/2  eiπ/4√2/2
 ei3π/4√2/2 e-iπ/4√2/2
)
(
  e-i5π/12√2/2  e-i5π/12√2/2
  eiπ/12√2/2 e-i11π/12√2/2
)
(
 ei11π/12√2/2 ei11π/12√2/2
 e-i7π/12√2/2  ei5π/12√2/2
)
10
(
  0 -1  0
  0  0  1
 -1  0  0
)
(
  (1-i)/2 -(1-i)/2
  (1+i)/2  (1+i)/2
)
3-1-1-1
1
e-i2π/3
ei2π/3
(
  0 -1  0
  0  0 -1
  1  0  0
)
(
 e-iπ/4√2/2 ei3π/4√2/2
  eiπ/4√2/2  eiπ/4√2/2
)
(
 e-i11π/12√2/2  eiπ/12√2/2
  e-i5π/12√2/2  e-i5π/12√2/2
)
(
  ei5π/12√2/2 e-i7π/12√2/2
 ei11π/12√2/2 ei11π/12√2/2
)
11
(
  0  1  0
  0  0 -1
 -1  0  0
)
(
  (1+i)/2 -(1+i)/2
  (1-i)/2  (1-i)/2
)
3--1-11
1
e-i2π/3
ei2π/3
(
  0 -1  0
  0  0  1
 -1  0  0
)
(
  eiπ/4√2/2 e-i3π/4√2/2
  e-iπ/4√2/2  e-iπ/4√2/2
)
(
  e-i5π/12√2/2  ei7π/12√2/2
 e-i11π/12√2/2 e-i11π/12√2/2
)
(
 ei11π/12√2/2  e-iπ/12√2/2
  ei5π/12√2/2  ei5π/12√2/2
)
12
(
  0 -1  0
  0  0 -1
  1  0  0
)
(
  (1-i)/2  (1-i)/2
 -(1+i)/2  (1+i)/2
)
3--11-1
1
e-i2π/3
ei2π/3
(
  0  1  0
  0  0 -1
 -1  0  0
)
(
  e-iπ/4√2/2  e-iπ/4√2/2
 e-i3π/4√2/2  eiπ/4√2/2
)
(
 e-i11π/12√2/2 e-i11π/12√2/2
  ei7π/12√2/2  e-i5π/12√2/2
)
(
  ei5π/12√2/2  ei5π/12√2/2
  e-iπ/12√2/2 ei11π/12√2/2
)
13
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
d1
1
1
1
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
14
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  i  0
  0 -i
)
d2001
1
1
1
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
15
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
  0  1
 -1  0
)
d2010
1
1
1
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
16
(
  1  0  0
  0 -1  0
  0  0 -1
)
(
 0 i
 i 0
)
d2100
1
1
1
(
 -1  0  0
  0  1  0
  0  0 -1
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
17
(
 0 0 1
 1 0 0
 0 1 0
)
(
 -(1-i)/2  (1+i)/2
 -(1-i)/2 -(1+i)/2
)
d3+111
1
ei2π/3
e-i2π/3
(
 0 0 1
 1 0 0
 0 1 0
)
(
  ei3π/4√2/2  eiπ/4√2/2
  ei3π/4√2/2 e-i3π/4√2/2
)
(
 e-i7π/12√2/2 ei11π/12√2/2
 e-i7π/12√2/2  e-iπ/12√2/2
)
(
  eiπ/12√2/2 e-i5π/12√2/2
  eiπ/12√2/2  ei7π/12√2/2
)
18
(
  0  0  1
 -1  0  0
  0 -1  0
)
(
 -(1+i)/2  (1-i)/2
 -(1+i)/2 -(1-i)/2
)
d3+-11-1
1
ei2π/3
e-i2π/3
(
  0  0 -1
  1  0  0
  0 -1  0
)
(
 e-i3π/4√2/2  e-iπ/4√2/2
 e-i3π/4√2/2  ei3π/4√2/2
)
(
  e-iπ/12√2/2  ei5π/12√2/2
  e-iπ/12√2/2 e-i7π/12√2/2
)
(
  ei7π/12√2/2 e-i11π/12√2/2
  ei7π/12√2/2  eiπ/12√2/2
)
19
(
  0  0 -1
 -1  0  0
  0  1  0
)
(
 -(1+i)/2 -(1-i)/2
  (1+i)/2 -(1-i)/2
)
d3+1-1-1
1
ei2π/3
e-i2π/3
(
  0  0  1
 -1  0  0
  0 -1  0
)
(
 e-i3π/4√2/2  ei3π/4√2/2
  eiπ/4√2/2  ei3π/4√2/2
)
(
  e-iπ/12√2/2 e-i7π/12√2/2
 ei11π/12√2/2 e-i7π/12√2/2
)
(
  ei7π/12√2/2  eiπ/12√2/2
 e-i5π/12√2/2  eiπ/12√2/2
)
20
(
  0  0 -1
  1  0  0
  0 -1  0
)
(
 -(1-i)/2 -(1+i)/2
  (1-i)/2 -(1+i)/2
)
d3+-1-11
1
ei2π/3
e-i2π/3
(
  0  0 -1
 -1  0  0
  0  1  0
)
(
  ei3π/4√2/2 e-i3π/4√2/2
  e-iπ/4√2/2 e-i3π/4√2/2
)
(
 e-i7π/12√2/2  e-iπ/12√2/2
  ei5π/12√2/2  e-iπ/12√2/2
)
(
  eiπ/12√2/2  ei7π/12√2/2
 e-i11π/12√2/2  ei7π/12√2/2
)
21
(
 0 1 0
 0 0 1
 1 0 0
)
(
 -(1+i)/2 -(1+i)/2
  (1-i)/2 -(1-i)/2
)
d3-111
1
e-i2π/3
ei2π/3
(
 0 1 0
 0 0 1
 1 0 0
)
(
 e-i3π/4√2/2 e-i3π/4√2/2
  e-iπ/4√2/2  ei3π/4√2/2
)
(
  ei7π/12√2/2  ei7π/12√2/2
 e-i11π/12√2/2  eiπ/12√2/2
)
(
  e-iπ/12√2/2  e-iπ/12√2/2
  ei5π/12√2/2 e-i7π/12√2/2
)
22
(
  0 -1  0
  0  0  1
 -1  0  0
)
(
 -(1-i)/2  (1-i)/2
 -(1+i)/2 -(1+i)/2
)
d3-1-1-1
1
e-i2π/3
ei2π/3
(
  0 -1  0
  0  0 -1
  1  0  0
)
(
  ei3π/4√2/2  e-iπ/4√2/2
 e-i3π/4√2/2 e-i3π/4√2/2
)
(
  eiπ/12√2/2 e-i11π/12√2/2
  ei7π/12√2/2  ei7π/12√2/2
)
(
 e-i7π/12√2/2  ei5π/12√2/2
  e-iπ/12√2/2  e-iπ/12√2/2
)
23
(
  0  1  0
  0  0 -1
 -1  0  0
)
(
 -(1+i)/2  (1+i)/2
 -(1-i)/2 -(1-i)/2
)
d3--1-11
1
e-i2π/3
ei2π/3
(
  0 -1  0
  0  0  1
 -1  0  0
)
(
 e-i3π/4√2/2  eiπ/4√2/2
  ei3π/4√2/2  ei3π/4√2/2
)
(
  ei7π/12√2/2 e-i5π/12√2/2
  eiπ/12√2/2  eiπ/12√2/2
)
(
  e-iπ/12√2/2 ei11π/12√2/2
 e-i7π/12√2/2 e-i7π/12√2/2
)
24
(
  0 -1  0
  0  0 -1
  1  0  0
)
(
 -(1-i)/2 -(1-i)/2
  (1+i)/2 -(1+i)/2
)
d3--11-1
1
e-i2π/3
ei2π/3
(
  0  1  0
  0  0 -1
 -1  0  0
)
(
  ei3π/4√2/2  ei3π/4√2/2
  eiπ/4√2/2 e-i3π/4√2/2
)
(
  eiπ/12√2/2  eiπ/12√2/2
 e-i5π/12√2/2  ei7π/12√2/2
)
(
 e-i7π/12√2/2 e-i7π/12√2/2
 ei11π/12√2/2  e-iπ/12√2/2
)
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