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Irreducible corepresentations of the Magnetic Point Group 4/m1' (N. 11.2.36)


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+GM4+
2Eg1Eg
GM3+GM4+
2
-2
0
0
2
-2
0
0
2
-2
0
0
2
-2
0
0
GM3-GM4-
2Eu1Eu
GM3-GM4-
2
-2
0
0
-2
2
0
0
2
-2
0
0
-2
2
0
0
GM7+GM8+
2E2g1E2g
GM5GM7
2
0
-2
-2
2
0
-2
-2
-2
0
2
2
-2
0
2
2
GM5+GM6+
2E1g1E1g
GM6GM8
2
0
2
2
2
0
2
2
-2
0
-2
-2
-2
0
-2
-2
GM5-GM6-
2E1u1E1u
GM10GM12
2
0
2
2
-2
0
-2
-2
-2
0
-2
-2
2
0
2
2
GM8-GM7-
1E2u2E2u
GM11GM9
2
0
-2
-2
-2
0
2
2
-2
0
2
2
2
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: 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+GM4+GM3-GM4-GM5GM7GM6GM8GM10GM12GM11GM9
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
)
(
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
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
i 0
0 -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 0
0 -i
)
(
i 0
0 -i
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
(
e-iπ/4 0
0 eiπ/4
)
(
e-i3π/4 0
0 ei3π/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 0
0 i
)
(
-i 0
0 i
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
(
eiπ/4 0
0 e-iπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
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
)
(
-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
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
i 0
0 -i
)
(
-i 0
0 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 0
0 -i
)
(
-i 0
0 i
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
(
eiπ/4 0
0 e-iπ/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 0
0 i
)
(
i 0
0 -i
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
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
)
(
-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
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
-i 0
0 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 0
0 -i
)
(
i 0
0 -i
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
(
ei3π/4 0
0 e-i3π/4
)
(
eiπ/4 0
0 e-iπ/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 0
0 i
)
(
-i 0
0 i
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
(
e-i3π/4 0
0 ei3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
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
)
(
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
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
-i 0
0 i
)
(
i 0
0 -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 0
0 -i
)
(
-i 0
0 i
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
(
e-i3π/4 0
0 ei3π/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 0
0 i
)
(
i 0
0 -i
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
17
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1'
1
-1
1
-1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
18
(
-1 0 0
0 -1 0
0 0 1
)
(
-i 0
0 i
)
2'001
1
-1
1
-1
(
0 -1
-1 0
)
(
0 1
1 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 -i
-i 0
)
(
0 i
i 0
)
19
(
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 i
-i 0
)
(
0 -i
i 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
20
(
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 -i
i 0
)
(
0 i
-i 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
21
(
-1 0 0
0 -1 0
0 0 -1
)
(
1 0
0 1
)
1'
1
1
1
1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
22
(
1 0 0
0 1 0
0 0 -1
)
(
-i 0
0 i
)
m'001
1
1
1
1
(
0 -1
-1 0
)
(
0 -1
-1 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 -i
-i 0
)
23
(
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 i
-i 0
)
(
0 i
-i 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
24
(
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 -i
i 0
)
(
0 -i
i 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
25
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1'
1
-1
1
-1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
26
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2'001
1
-1
1
-1
(
0 -1
-1 0
)
(
0 1
1 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 i
i 0
)
(
0 -i
-i 0
)
27
(
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 i
-i 0
)
(
0 -i
i 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
28
(
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 -i
i 0
)
(
0 i
-i 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
29
(
-1 0 0
0 -1 0
0 0 -1
)
(
-1 0
0 -1
)
d1'
1
1
1
1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
30
(
1 0 0
0 1 0
0 0 -1
)
(
i 0
0 -i
)
dm'001
1
1
1
1
(
0 -1
-1 0
)
(
0 -1
-1 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 i
i 0
)
31
(
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 i
-i 0
)
(
0 i
-i 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
32
(
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 -i
i 0
)
(
0 -i
i 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
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