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Irreducible corepresentations of the Magnetic Point Group 4'/mm'm (N. 15.4.56)


Table of characters of the unitary symmetry operations


(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
GM1+
Ag
GM1+
1
1
1
1
1
1
1
1
1
1
GM1-
Au
GM1-
1
1
1
1
-1
-1
-1
-1
1
-1
GM3+
B1g
GM2+
1
1
-1
-1
1
1
-1
-1
1
1
GM3-
B1u
GM2-
1
1
-1
-1
-1
-1
1
1
1
-1
GM4+GM2+
B3gB2g
GM3+GM4+
2
-2
0
0
2
-2
0
0
2
2
GM4-GM2-
B3uB2u
GM3-GM4-
2
-2
0
0
-2
2
0
0
2
-2
GM5+
Eg
GM5
2
0
0
0
2
0
0
0
-2
-2
GM5-
Eu
GM6
2
0
0
0
-2
0
0
0
-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: 2001d2001
C3: 2110d2110
C4: 2110d2110
C51
C6: m001dm001
C7: m110dm110
C8: m110dm110
C9d1
C10d1

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1+GM1-GM2+GM2-GM3+GM4+GM3-GM4-GM5GM6
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
1
1
(
1 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
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
0 -1
1 0
)
(
0 -1
1 0
)
3
(
0 1 0
1 0 0
0 0 -1
)
(
0 -(1+i)2/2
(1-i)2/2 0
)
2110
1
1
-1
-1
(
1 0
0 -1
)
(
1 0
0 -1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
4
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(1-i)2/2
(1+i)2/2 0
)
21-10
1
1
-1
-1
(
-1 0
0 1
)
(
-1 0
0 1
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
5
(
-1 0 0
0 -1 0
0 0 -1
)
(
1 0
0 1
)
1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
6
(
1 0 0
0 1 0
0 0 -1
)
(
-i 0
0 i
)
m001
1
-1
1
-1
(
-1 0
0 -1
)
(
1 0
0 1
)
(
0 -1
1 0
)
(
0 1
-1 0
)
7
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -(1+i)2/2
(1-i)2/2 0
)
m110
1
-1
-1
1
(
1 0
0 -1
)
(
-1 0
0 1
)
(
-i 0
0 i
)
(
i 0
0 -i
)
8
(
0 1 0
1 0 0
0 0 1
)
(
0 -(1-i)2/2
(1+i)2/2 0
)
m1-10
1
-1
-1
1
(
-1 0
0 1
)
(
1 0
0 -1
)
(
0 -i
-i 0
)
(
0 i
i 0
)
9
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
10
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2001
1
1
1
1
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
0 1
-1 0
)
(
0 1
-1 0
)
11
(
0 1 0
1 0 0
0 0 -1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
d2110
1
1
-1
-1
(
1 0
0 -1
)
(
1 0
0 -1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
12
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
d21-10
1
1
-1
-1
(
-1 0
0 1
)
(
-1 0
0 1
)
(
0 i
i 0
)
(
0 i
i 0
)
13
(
-1 0 0
0 -1 0
0 0 -1
)
(
-1 0
0 -1
)
d1
1
-1
1
-1
(
1 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
)
dm001
1
-1
1
-1
(
-1 0
0 -1
)
(
1 0
0 1
)
(
0 1
-1 0
)
(
0 -1
1 0
)
15
(
0 -1 0
-1 0 0
0 0 1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
dm110
1
-1
-1
1
(
1 0
0 -1
)
(
-1 0
0 1
)
(
i 0
0 -i
)
(
-i 0
0 i
)
16
(
0 1 0
1 0 0
0 0 1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
dm1-10
1
-1
-1
1
(
-1 0
0 1
)
(
1 0
0 -1
)
(
0 i
i 0
)
(
0 -i
-i 0
)
17
(
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
(
0 -1
1 0
)
(
0 -1
1 0
)
(
-1/2 -1/2
1/2 -1/2
)
(
-1/2 -1/2
1/2 -1/2
)
18
(
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
(
0 1
-1 0
)
(
0 1
-1 0
)
(
1/2 -1/2
1/2 1/2
)
(
1/2 -1/2
1/2 1/2
)
19
(
-1 0 0
0 1 0
0 0 -1
)
(
0 -1
1 0
)
2'010
1
1
1
1
(
0 1
1 0
)
(
0 1
1 0
)
(
-i/2 i/2
i/2 i/2
)
(
-i/2 i/2
i/2 i/2
)
20
(
1 0 0
0 -1 0
0 0 -1
)
(
0 -i
-i 0
)
2'100
1
1
1
1
(
0 -1
-1 0
)
(
0 -1
-1 0
)
(
i/2 i/2
i/2 -i/2
)
(
i/2 i/2
i/2 -i/2
)
21
(
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
(
0 -1
1 0
)
(
0 1
-1 0
)
(
-1/2 -1/2
1/2 -1/2
)
(
1/2 1/2
-1/2 1/2
)
22
(
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
(
0 1
-1 0
)
(
0 -1
1 0
)
(
1/2 -1/2
1/2 1/2
)
(
-1/2 1/2
-1/2 -1/2
)
23
(
1 0 0
0 -1 0
0 0 1
)
(
0 -1
1 0
)
m'010
1
-1
1
-1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
-i/2 i/2
i/2 i/2
)
(
i/2 -i/2
-i/2 -i/2
)
24
(
-1 0 0
0 1 0
0 0 1
)
(
0 -i
-i 0
)
m'100
1
-1
1
-1
(
0 -1
-1 0
)
(
0 1
1 0
)
(
i/2 i/2
i/2 -i/2
)
(
-i/2 -i/2
-i/2 i/2
)
25
(
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
(
0 -1
1 0
)
(
0 -1
1 0
)
(
1/2 1/2
-1/2 1/2
)
(
1/2 1/2
-1/2 1/2
)
26
(
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
(
0 1
-1 0
)
(
0 1
-1 0
)
(
-1/2 1/2
-1/2 -1/2
)
(
-1/2 1/2
-1/2 -1/2
)
27
(
-1 0 0
0 1 0
0 0 -1
)
(
0 1
-1 0
)
d2'010
1
1
1
1
(
0 1
1 0
)
(
0 1
1 0
)
(
i/2 -i/2
-i/2 -i/2
)
(
i/2 -i/2
-i/2 -i/2
)
28
(
1 0 0
0 -1 0
0 0 -1
)
(
0 i
i 0
)
d2'100
1
1
1
1
(
0 -1
-1 0
)
(
0 -1
-1 0
)
(
-i/2 -i/2
-i/2 i/2
)
(
-i/2 -i/2
-i/2 i/2
)
29
(
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
(
0 -1
1 0
)
(
0 1
-1 0
)
(
1/2 1/2
-1/2 1/2
)
(
-1/2 -1/2
1/2 -1/2
)
30
(
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
(
0 1
-1 0
)
(
0 -1
1 0
)
(
-1/2 1/2
-1/2 -1/2
)
(
1/2 -1/2
1/2 1/2
)
31
(
1 0 0
0 -1 0
0 0 1
)
(
0 1
-1 0
)
dm'010
1
-1
1
-1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
i/2 -i/2
-i/2 -i/2
)
(
-i/2 i/2
i/2 i/2
)
32
(
-1 0 0
0 1 0
0 0 1
)
(
0 i
i 0
)
dm'100
1
-1
1
-1
(
0 -1
-1 0
)
(
0 1
1 0
)
(
-i/2 -i/2
-i/2 i/2
)
(
i/2 i/2
i/2 -i/2
)
k-Subgroupsmag
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