Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group 4m'm' (N. 13.4.47)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
GM1
A
GM1
1
1
1
1
1
1
1
1
GM2
B
GM2
1
1
-1
-1
1
1
-1
-1
GM3
2E
GM3
1
-1
i
-i
1
-1
i
-i
GM4
1E
GM4
1
-1
-i
i
1
-1
-i
i
GM7
2E2
GM5
1
-i
-(1-i)√2/2
-(1+i)√2/2
-1
i
(1-i)√2/2
(1+i)√2/2
GM5
2E1
GM6
1
-i
(1-i)√2/2
(1+i)√2/2
-1
i
-(1-i)√2/2
-(1+i)√2/2
GM8
1E2
GM7
1
i
-(1+i)√2/2
-(1-i)√2/2
-1
-i
(1+i)√2/2
(1-i)√2/2
GM6
1E1
GM8
1
i
(1+i)√2/2
(1-i)√2/2
-1
-i
-(1+i)√2/2
-(1-i)√2/2
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
C3: 4+001
C4: 4-001
C5: d1
C6: d2001
C7: d4+001
C8: d4-001

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1GM2GM3GM4GM5GM6GM7GM8
1
(
 1 0 0
 0 1 0
 0 0 1
)
(
 1 0
 0 1
)
1
1
1
1
1
1
1
1
1
2
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
 -i  0
  0  i
)
2001
1
1
-1
-1
-i
-i
i
i
3
(
  0 -1  0
  1  0  0
  0  0  1
)
(
 (1-i)√2/2  0
  0 (1+i)√2/2
)
4+001
1
-1
i
-i
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
4
(
  0  1  0
 -1  0  0
  0  0  1
)
(
 (1+i)√2/2  0
  0 (1-i)√2/2
)
4-001
1
-1
-i
i
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
5
(
 1 0 0
 0 1 0
 0 0 1
)
(
 -1  0
  0 -1
)
d1
1
1
1
1
-1
-1
-1
-1
6
(
 -1  0  0
  0 -1  0
  0  0  1
)
(
  i  0
  0 -i
)
d2001
1
1
-1
-1
i
i
-i
-i
7
(
  0 -1  0
  1  0  0
  0  0  1
)
(
 -(1-i)√2/2  0
  0 -(1+i)√2/2
)
d4+001
1
-1
i
-i
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
8
(
  0  1  0
 -1  0  0
  0  0  1
)
(
 -(1+i)√2/2  0
  0 -(1-i)√2/2
)
d4-001
1
-1
-i
i
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
9
(
  1  0  0
  0 -1  0
  0  0  1
)
(
  0 -1
  1  0
)
m'010
1
1
1
1
1
1
1
1
10
(
 -1  0  0
  0  1  0
  0  0  1
)
(
  0 -i
 -i  0
)
m'100
1
1
-1
-1
i
i
-i
-i
11
(
  0 -1  0
 -1  0  0
  0  0  1
)
(
  0 -(1+i)√2/2
  (1-i)√2/2  0
)
m'110
1
-1
-i
i
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
12
(
 0 1 0
 1 0 0
 0 0 1
)
(
  0 -(1-i)√2/2
  (1+i)√2/2  0
)
m'1-10
1
-1
i
-i
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
13
(
  1  0  0
  0 -1  0
  0  0  1
)
(
  0  1
 -1  0
)
dm'010
1
1
1
1
-1
-1
-1
-1
14
(
 -1  0  0
  0  1  0
  0  0  1
)
(
 0 i
 i 0
)
dm'100
1
1
-1
-1
-i
-i
i
i
15
(
  0 -1  0
 -1  0  0
  0  0  1
)
(
  0  (1+i)√2/2
 -(1-i)√2/2  0
)
dm'110
1
-1
-i
i
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
16
(
 0 1 0
 1 0 0
 0 0 1
)
(
  0  (1-i)√2/2
 -(1+i)√2/2  0
)
dm'1-10
1
-1
i
-i
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
k-Subgroupsmag
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