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Irreducible corepresentations of the Projective Magnetic Point Group 4π/21'


Table of characters of the unitary symmetry operations


1
4+
2
4-
d1
d4+
d2
d4-
2EB
2
√2e3iπ/4
0
√2e-3iπ/4
2
√2e3iπ/4
0
√2e-3iπ/4
1EA
2
√2e-iπ/4
0
√2eiπ/4
2
√2e-iπ/4
0
√2eiπ/4
1E11E2
2
0
2i
0
-2
0
-2i
0
2E22E2
2
2e3iπ/4
-2i
2e-3iπ/4
-2
2e-iπ/4
2i
2eiπ/4
2E12E1
2
2e-iπ/4
-2i
2eiπ/4
-2
2e3iπ/4
2i
2e-3iπ/4

Multiplication table of the symmetry operations


1
4+
2
4-
d1
d4+
d2
d4-
1'
4+'
2'
4-'
d1'
d4+'
d2'
d4-'
1
1
4+
2
4-
d1
d4+
d2
d4-
1'
4+'
2'
4-'
d1'
d4+'
d2'
d4-'
4+
4+
2
d4-
1
d4+
d2
4-
d1
4+'
2'
d4-'
1'
d4+'
d2'
4-'
d1'
2
2
d4-
d1
4+
d2
4-
1
d4+
2'
d4-'
d1'
4+'
d2'
4-'
1'
d4+'
4-
4-
1
4+
d2
d4-
d1
d4+
2
4-'
1'
4+'
d2'
d4-'
d1'
d4+'
2'
d1
d1
d4+
d2
d4-
1
4+
2
4-
d1'
d4+'
d2'
d4-'
1'
4+'
2'
4-'
d4+
d4+
d2
4-
d1
4+
2
d4-
1
d4+'
d2'
4-'
d1'
4+'
2'
d4-'
1'
d2
d2
4-
1
d4+
2
d4-
d1
4+
d2'
4-'
1'
d4+'
2'
d4-'
d1'
4+'
d4-
d4-
d1
d4+
2
4-
1
4+
d2
d4-'
d1'
d4+'
2'
4-'
1'
4+'
d2'
1'
1'
4+'
2'
4-'
d1'
d4+'
d2'
d4-'
d1
d4+
d2
d4-
1
4+
2
4-
4+'
4+'
2'
d4-'
1'
d4+'
d2'
4-'
d1'
d4+
d2
4-
d1
4+
2
d4-
1
2'
2'
d4-'
d1'
4+'
d2'
4-'
1'
d4+'
d2
4-
1
d4+
2
d4-
d1
4+
4-'
4-'
1'
4+'
d2'
d4-'
d1'
d4+'
2'
d4-
d1
d4+
2
4-
1
4+
d2
d1'
d1'
d4+'
d2'
d4-'
1'
4+'
2'
4-'
1
4+
2
4-
d1
d4+
d2
d4-
d4+'
d4+'
d2'
4-'
d1'
4+'
2'
d4-'
1'
4+
2
d4-
1
d4+
d2
4-
d1
d2'
d2'
4-'
1'
d4+'
2'
d4-'
d1'
4+'
2
d4-
d1
4+
d2
4-
1
d4+
d4-'
d4-'
d1'
d4+'
2'
4-'
1'
4+'
d2'
4-
1
4+
d2
d4-
d1
d4+
2

Table of projective phases in group multiplication


1
4+
2
4-
d1
d4+
d2
d4-
1'
4+'
2'
4-'
d1'
d4+'
d2'
d4-'
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
-i
-i
-i
-i
-i
-i
-i
-i
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
i
i
i
i
i
i
i
i
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
-i
-i
-i
-i
-i
-i
-i
-i
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
i
i
i
i
i
i
i
i
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
i
i
i
i
i
i
i
i
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
-i
-i
-i
-i
-i
-i
-i
-i
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
i
i
i
i
i
i
i
i
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
-i
-i
-i
-i
-i
-i
-i
-i

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbol2EB1EA1E11E22E22E22E12E1
1
(
 1 0
 0 1
)
(
 1 0
 0 1
)
1
(
 1 0
 0 1
)
(
 1 0
 0 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+
(
  i  0
  0 -1
)
(
 -i  0
  0  1
)
(
 e-3iπ/4  0
  0  eiπ/4
)
(
 e3iπ/4  0
  0 e3iπ/4
)
(
 e-iπ/4  0
  0 e-iπ/4
)
3
(
 -1  0
  0 -1
)
(
 -i  0
  0  i
)
2
(
 -1  0
  0  1
)
(
 -1  0
  0  1
)
(
 i 0
 0 i
)
(
 -i  0
  0 -i
)
(
 -i  0
  0 -i
)
4
(
  0  1
 -1  0
)
(
  eiπ/4  0
  0 e-iπ/4
)
4-
(
 -i  0
  0 -1
)
(
 i 0
 0 1
)
(
 e3iπ/4  0
  0 e-iπ/4
)
(
 e-3iπ/4  0
  0 e-3iπ/4
)
(
 eiπ/4  0
  0 eiπ/4
)
5
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
d1
(
 1 0
 0 1
)
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
(
 -1  0
  0 -1
)
6
(
  0 -1
  1  0
)
(
  e3iπ/4  0
  0 e-3iπ/4
)
d4+
(
  i  0
  0 -1
)
(
 -i  0
  0  1
)
(
  eiπ/4  0
  0 e-3iπ/4
)
(
 e-iπ/4  0
  0 e-iπ/4
)
(
 e3iπ/4  0
  0 e3iπ/4
)
7
(
 -1  0
  0 -1
)
(
  i  0
  0 -i
)
d2
(
 -1  0
  0  1
)
(
 -1  0
  0  1
)
(
 -i  0
  0 -i
)
(
 i 0
 0 i
)
(
 i 0
 0 i
)
8
(
  0  1
 -1  0
)
(
 e-3iπ/4  0
  0  e3iπ/4
)
d4-
(
 -i  0
  0 -1
)
(
 i 0
 0 1
)
(
 e-iπ/4  0
  0 e3iπ/4
)
(
 eiπ/4  0
  0 eiπ/4
)
(
 e-3iπ/4  0
  0 e-3iπ/4
)
9
(
 1 0
 0 1
)
(
 1 0
 0 1
)
1'
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
  0 -1
  1  0
)
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
10
(
  0 -1
  1  0
)
(
 e-iπ/4  0
  0  eiπ/4
)
4+'
(
  0 -1
 -i  0
)
(
 0 1
 i 0
)
(
  0 e3iπ/4
 e3iπ/4  0
)
(
  0 e3iπ/4
 e-iπ/4  0
)
(
  0 e-iπ/4
 e3iπ/4  0
)
11
(
 -1  0
  0 -1
)
(
 -i  0
  0  i
)
2'
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0  i
 -i  0
)
(
  0 -i
  i  0
)
(
  0 -i
  i  0
)
12
(
  0  1
 -1  0
)
(
  eiπ/4  0
  0 e-iπ/4
)
4-'
(
  0 -1
  i  0
)
(
  0  1
 -i  0
)
(
  0 e-3iπ/4
 e-3iπ/4  0
)
(
  0 e-3iπ/4
  eiπ/4  0
)
(
  0  eiπ/4
 e-3iπ/4  0
)
13
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
d1'
(
 0 1
 1 0
)
(
 0 1
 1 0
)
(
  0  1
 -1  0
)
(
  0 -1
  1  0
)
(
  0 -1
  1  0
)
14
(
  0 -1
  1  0
)
(
  e3iπ/4  0
  0 e-3iπ/4
)
d4+'
(
  0 -1
 -i  0
)
(
 0 1
 i 0
)
(
  0 e-iπ/4
 e-iπ/4  0
)
(
  0 e-iπ/4
 e3iπ/4  0
)
(
  0 e3iπ/4
 e-iπ/4  0
)
15
(
 -1  0
  0 -1
)
(
  i  0
  0 -i
)
d2'
(
  0  1
 -1  0
)
(
  0  1
 -1  0
)
(
  0 -i
  i  0
)
(
  0  i
 -i  0
)
(
  0  i
 -i  0
)
16
(
  0  1
 -1  0
)
(
 e-3iπ/4  0
  0  e3iπ/4
)
d4-'
(
  0 -1
  i  0
)
(
  0  1
 -i  0
)
(
  0 eiπ/4
 eiπ/4  0
)
(
  0  eiπ/4
 e-3iπ/4  0
)
(
  0 e-3iπ/4
  eiπ/4  0
)
k-Subgroupsmag
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