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Irreducible corepresentations of the Magnetic Point Group 4/mm'm' (N. 15.6.58)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
GM1+
Ag
GM1+
1
1
1
1
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
-1
-1
-1
-1
GM2+
Bg
GM2+
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
GM2-
Bu
GM2-
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
GM3+
2Eg
GM3+
1
-1
i
-i
1
-1
i
-i
1
-1
i
-i
1
-1
i
-i
GM3-
2Eu
GM3-
1
-1
i
-i
-1
1
-i
i
1
-1
i
-i
-1
1
-i
i
GM4+
1Eg
GM4+
1
-1
-i
i
1
-1
-i
i
1
-1
-i
i
1
-1
-i
i
GM4-
1Eu
GM4-
1
-1
-i
i
-1
1
i
-i
1
-1
-i
i
-1
1
i
-i
GM7+
2E2g
GM5
1
-i
-(1-i)2/2
-(1+i)2/2
1
-i
-(1-i)2/2
-(1+i)2/2
-1
i
(1-i)2/2
(1+i)2/2
-1
i
(1-i)2/2
(1+i)2/2
GM5+
2E1g
GM6
1
-i
(1-i)2/2
(1+i)2/2
1
-i
(1-i)2/2
(1+i)2/2
-1
i
-(1-i)2/2
-(1+i)2/2
-1
i
-(1-i)2/2
-(1+i)2/2
GM8+
1E2g
GM7
1
i
-(1+i)2/2
-(1-i)2/2
1
i
-(1+i)2/2
-(1-i)2/2
-1
-i
(1+i)2/2
(1-i)2/2
-1
-i
(1+i)2/2
(1-i)2/2
GM6+
1E1g
GM8
1
i
(1+i)2/2
(1-i)2/2
1
i
(1+i)2/2
(1-i)2/2
-1
-i
-(1+i)2/2
-(1-i)2/2
-1
-i
-(1+i)2/2
-(1-i)2/2
GM7-
2E2u
GM9
1
-i
-(1-i)2/2
-(1+i)2/2
-1
i
(1-i)2/2
(1+i)2/2
-1
i
(1-i)2/2
(1+i)2/2
1
-i
-(1-i)2/2
-(1+i)2/2
GM5-
2E1u
GM10
1
-i
(1-i)2/2
(1+i)2/2
-1
i
-(1-i)2/2
-(1+i)2/2
-1
i
-(1-i)2/2
-(1+i)2/2
1
-i
(1-i)2/2
(1+i)2/2
GM8-
1E2u
GM11
1
i
-(1+i)2/2
-(1-i)2/2
-1
-i
(1+i)2/2
(1-i)2/2
-1
-i
(1+i)2/2
(1-i)2/2
1
i
-(1+i)2/2
-(1-i)2/2
GM6-
1E1u
GM12
1
i
(1+i)2/2
(1-i)2/2
-1
-i
-(1+i)2/2
-(1-i)2/2
-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
C51
C6: m001
C74+001
C84-001
C9d1
C10d2001
C11d4+001
C12d4-001
C13d1
C14dm001
C15d4+001
C16d4-001

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1+GM1-GM2+GM2-GM3+GM3-GM4+GM4-GM5GM6GM7GM8GM9GM10GM11GM12
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
1
1
1
1
2
(
-1 0 0
0 -1 0
0 0 1
)
(
-i 0
0 i
)
2001
1
1
1
1
-1
-1
-1
-1
-i
-i
i
i
-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
-1
-1
i
i
-i
-i
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
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
-1
-1
-i
-i
i
i
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
5
(
-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
-1
-1
-1
-1
6
(
1 0 0
0 1 0
0 0 -1
)
(
-i 0
0 i
)
m001
1
-1
1
-1
-1
1
-1
1
-i
-i
i
i
i
i
-i
-i
7
(
0 1 0
-1 0 0
0 0 -1
)
(
(1-i)2/2 0
0 (1+i)2/2
)
4+001
1
-1
-1
1
i
-i
-i
i
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
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
)
4-001
1
-1
-1
1
-i
i
i
-i
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
9
(
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
-1
-1
-1
-1
10
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2001
1
1
1
1
-1
-1
-1
-1
i
i
-i
-i
i
i
-i
-i
11
(
0 -1 0
1 0 0
0 0 1
)
(
-(1-i)2/2 0
0 -(1+i)2/2
)
d4+001
1
1
-1
-1
i
i
-i
-i
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
12
(
0 1 0
-1 0 0
0 0 1
)
(
-(1+i)2/2 0
0 -(1-i)2/2
)
d4-001
1
1
-1
-1
-i
-i
i
i
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
13
(
-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
1
1
1
1
14
(
1 0 0
0 1 0
0 0 -1
)
(
i 0
0 -i
)
dm001
1
-1
1
-1
-1
1
-1
1
i
i
-i
-i
-i
-i
i
i
15
(
0 1 0
-1 0 0
0 0 -1
)
(
-(1-i)2/2 0
0 -(1+i)2/2
)
d4+001
1
-1
-1
1
i
-i
-i
i
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
16
(
0 -1 0
1 0 0
0 0 -1
)
(
-(1+i)2/2 0
0 -(1-i)2/2
)
d4-001
1
-1
-1
1
-i
i
i
-i
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
17
(
-1 0 0
0 1 0
0 0 -1
)
(
0 -1
1 0
)
2'010
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
18
(
1 0 0
0 -1 0
0 0 -1
)
(
0 -i
-i 0
)
2'100
1
1
1
1
-1
-1
-1
-1
i
i
-i
-i
i
i
-i
-i
19
(
0 1 0
1 0 0
0 0 -1
)
(
0 -(1+i)2/2
(1-i)2/2 0
)
2'110
1
1
-1
-1
-i
-i
i
i
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
20
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(1-i)2/2
(1+i)2/2 0
)
2'1-10
1
1
-1
-1
i
i
-i
-i
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
21
(
1 0 0
0 -1 0
0 0 1
)
(
0 -1
1 0
)
m'010
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
-1
-1
-1
22
(
-1 0 0
0 1 0
0 0 1
)
(
0 -i
-i 0
)
m'100
1
-1
1
-1
-1
1
-1
1
i
i
-i
-i
-i
-i
i
i
23
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -(1+i)2/2
(1-i)2/2 0
)
m'110
1
-1
-1
1
-i
i
i
-i
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
24
(
0 1 0
1 0 0
0 0 1
)
(
0 -(1-i)2/2
(1+i)2/2 0
)
m'1-10
1
-1
-1
1
i
-i
-i
i
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
25
(
-1 0 0
0 1 0
0 0 -1
)
(
0 1
-1 0
)
d2'010
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
26
(
1 0 0
0 -1 0
0 0 -1
)
(
0 i
i 0
)
d2'100
1
1
1
1
-1
-1
-1
-1
-i
-i
i
i
-i
-i
i
i
27
(
0 1 0
1 0 0
0 0 -1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
d2'110
1
1
-1
-1
-i
-i
i
i
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
28
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
d2'1-10
1
1
-1
-1
i
i
-i
-i
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
29
(
1 0 0
0 -1 0
0 0 1
)
(
0 1
-1 0
)
dm'010
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
-1
1
1
1
1
30
(
-1 0 0
0 1 0
0 0 1
)
(
0 i
i 0
)
dm'100
1
-1
1
-1
-1
1
-1
1
-i
-i
i
i
i
i
-i
-i
31
(
0 -1 0
-1 0 0
0 0 1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
dm'110
1
-1
-1
1
-i
i
i
-i
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
32
(
0 1 0
1 0 0
0 0 1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
dm'1-10
1
-1
-1
1
i
-i
-i
i
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
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