Bilbao Crystallographic Server arrow COREPRESENTATIONS PG

Irreducible corepresentations of the Magnetic Point Group 41' (N. 9.2.30)


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
GM3GM4
2E1E
GM3GM4
2
-2
0
0
2
-2
0
0
GM7GM8
2E21E2
GM5GM7
2
0
-2
-2
-2
0
2
2
GM5GM6
2E11E1
GM6GM8
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
C5d1
C6d2001
C7d4+001
C8d4-001

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolGM1GM2GM3GM4GM5GM7GM6GM8
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
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 0
0 -1
)
(
-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
(
i 0
0 -i
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 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 0
0 i
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
5
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
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
)
d2001
1
1
(
-1 0
0 -1
)
(
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
)
d4+001
1
-1
(
i 0
0 -i
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 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 0
0 i
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
9
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1'
1
1
(
0 1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
10
(
-1 0 0
0 -1 0
0 0 1
)
(
-i 0
0 i
)
2'001
1
1
(
0 -1
-1 0
)
(
0 i
i 0
)
(
0 i
i 0
)
11
(
0 -1 0
1 0 0
0 0 1
)
(
(1-i)2/2 0
0 (1+i)2/2
)
4'+001
1
-1
(
0 i
-i 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
12
(
0 1 0
-1 0 0
0 0 1
)
(
(1+i)2/2 0
0 (1-i)2/2
)
4'-001
1
-1
(
0 -i
i 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
13
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1'
1
1
(
0 1
1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
14
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2'001
1
1
(
0 -1
-1 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
15
(
0 -1 0
1 0 0
0 0 1
)
(
-(1-i)2/2 0
0 -(1+i)2/2
)
d4'+001
1
-1
(
0 i
-i 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
16
(
0 1 0
-1 0 0
0 0 1
)
(
-(1+i)2/2 0
0 -(1-i)2/2
)
d4'-001
1
-1
(
0 -i
i 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
k-Subgroupsmag
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