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Irreducible corepresentations of the Projective Magnetic Point Group 4mm


Table of characters of the unitary symmetry operations


1
4+
4-
2
d2
m10
m01
dm10
dm01
m1-1
m11
dm1-1
dm11
d1
d4+
d4-
A1
1
1
1
1
1
1
1
A2
1
1
1
-1
-1
1
1
B1
1
-1
1
1
-1
1
-1
B2
1
-1
1
-1
1
1
-1
E
2
0
-2
0
0
2
0
E2
2
-2
0
0
0
-2
2
E1
2
2
0
0
0
-2
-2

Multiplication table of the symmetry operations


1
4+
2
4-
m10
m1-1
m01
m11
d1
d4+
d2
d4-
dm10
dm1-1
dm01
dm11
1
1
4+
2
4-
m10
m1-1
m01
m11
d1
d4+
d2
d4-
dm10
dm1-1
dm01
dm11
4+
4+
2
d4-
1
m11
dm10
m1-1
m01
d4+
d2
4-
d1
dm11
m10
dm1-1
dm01
2
2
d4-
d1
4+
m01
dm11
dm10
m1-1
d2
4-
1
d4+
dm01
m11
m10
dm1-1
4-
4-
1
4+
d2
dm1-1
m01
m11
m10
d4-
d1
d4+
2
m1-1
dm01
dm11
dm10
m10
m10
dm1-1
dm01
m11
d1
4+
2
d4-
dm10
m1-1
m01
dm11
1
d4+
d2
4-
m1-1
m1-1
m01
m11
dm10
4-
d1
d4+
d2
dm1-1
dm01
dm11
m10
d4-
1
4+
2
m01
m01
m11
m10
m1-1
d2
d4-
d1
d4+
dm01
dm11
dm10
dm1-1
2
4-
1
4+
m11
m11
m10
dm1-1
m01
d4+
2
d4-
d1
dm11
dm10
m1-1
dm01
4+
d2
4-
1
d1
d1
d4+
d2
d4-
dm10
dm1-1
dm01
dm11
1
4+
2
4-
m10
m1-1
m01
m11
d4+
d4+
d2
4-
d1
dm11
m10
dm1-1
dm01
4+
2
d4-
1
m11
dm10
m1-1
m01
d2
d2
4-
1
d4+
dm01
m11
m10
dm1-1
2
d4-
d1
4+
m01
dm11
dm10
m1-1
d4-
d4-
d1
d4+
2
m1-1
dm01
dm11
dm10
4-
1
4+
d2
dm1-1
m01
m11
m10
dm10
dm10
m1-1
m01
dm11
1
d4+
d2
4-
m10
dm1-1
dm01
m11
d1
4+
2
d4-
dm1-1
dm1-1
dm01
dm11
m10
d4-
1
4+
2
m1-1
m01
m11
dm10
4-
d1
d4+
d2
dm01
dm01
dm11
dm10
dm1-1
2
4-
1
4+
m01
m11
m10
m1-1
d2
d4-
d1
d4+
dm11
dm11
dm10
m1-1
dm01
4+
d2
4-
1
m11
m10
dm1-1
m01
d4+
2
d4-
d1

Table of projective phases in group multiplication


1
4+
2
4-
m10
m1-1
m01
m11
d1
d4+
d2
d4-
dm10
dm1-1
dm01
dm11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4+
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4-
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
m10
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
m1-1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
m01
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
m11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
d1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
d4+
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
d2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
d4-
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
dm10
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
dm1-1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
dm01
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
dm11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolA1A2B1B2EE2E1
1
(
1 0
0 1
)
(
1 0
0 1
)
1
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 1
)
2
(
0 -1
1 0
)
(
e-iπ/4 0
0 eiπ/4
)
4+
1
1
-1
-1
(
-i 0
0 i
)
(
e3iπ/4 0
0 e-3iπ/4
)
(
e-iπ/4 0
0 eiπ/4
)
3
(
-1 0
0 -1
)
(
-i 0
0 i
)
2
1
1
1
1
(
-1 0
0 -1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
4
(
0 1
-1 0
)
(
eiπ/4 0
0 e-iπ/4
)
4-
1
1
-1
-1
(
i 0
0 -i
)
(
e-3iπ/4 0
0 e3iπ/4
)
(
eiπ/4 0
0 e-iπ/4
)
5
(
-1 0
0 1
)
(
0 -i
-i 0
)
m10
1
-1
1
-1
(
0 1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
6
(
0 1
1 0
)
(
0 e3iπ/4
eiπ/4 0
)
m1-1
1
-1
-1
1
(
0 i
-i 0
)
(
0 e-3iπ/4
e-iπ/4 0
)
(
0 eiπ/4
e3iπ/4 0
)
7
(
1 0
0 -1
)
(
0 -1
1 0
)
m01
1
-1
1
-1
(
0 -1
-1 0
)
(
0 i
i 0
)
(
0 i
i 0
)
8
(
0 -1
-1 0
)
(
0 e-3iπ/4
e-iπ/4 0
)
m11
1
-1
-1
1
(
0 -i
i 0
)
(
0 e-iπ/4
e-3iπ/4 0
)
(
0 e3iπ/4
eiπ/4 0
)
9
(
1 0
0 1
)
(
-1 0
0 -1
)
d1
1
1
1
1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
10
(
0 -1
1 0
)
(
e3iπ/4 0
0 e-3iπ/4
)
d4+
1
1
-1
-1
(
-i 0
0 i
)
(
e-iπ/4 0
0 eiπ/4
)
(
e3iπ/4 0
0 e-3iπ/4
)
11
(
-1 0
0 -1
)
(
i 0
0 -i
)
d2
1
1
1
1
(
-1 0
0 -1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
12
(
0 1
-1 0
)
(
e-3iπ/4 0
0 e3iπ/4
)
d4-
1
1
-1
-1
(
i 0
0 -i
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-3iπ/4 0
0 e3iπ/4
)
13
(
-1 0
0 1
)
(
0 i
i 0
)
dm10
1
-1
1
-1
(
0 1
1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
14
(
0 1
1 0
)
(
0 e-iπ/4
e-3iπ/4 0
)
dm1-1
1
-1
-1
1
(
0 i
-i 0
)
(
0 eiπ/4
e3iπ/4 0
)
(
0 e-3iπ/4
e-iπ/4 0
)
15
(
1 0
0 -1
)
(
0 1
-1 0
)
dm01
1
-1
1
-1
(
0 -1
-1 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
16
(
0 -1
-1 0
)
(
0 eiπ/4
e3iπ/4 0
)
dm11
1
-1
-1
1
(
0 -i
i 0
)
(
0 e3iπ/4
eiπ/4 0
)
(
0 e-iπ/4
e-3iπ/4 0
)
k-Subgroupsmag
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