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Irreducible corepresentations of the Projective Magnetic Point Group 41'


Table of characters of the unitary symmetry operations


1
4+
2
4-
d1
d4+
d2
d4-
A
1
1
1
1
1
1
1
1
B
1
-1
1
-1
1
-1
1
-1
2E1E
2
0
-2
0
2
0
-2
0
2E21E2
2
-√2
0
-√2
-2
√2
0
√2
2E11E1
2
√2
0
√2
-2
-√2
0
-√2

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
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
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
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
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

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbolAB2E1E2E21E22E11E1
1
(
 1 0
 0 1
)
(
 1 0
 0 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
(
 -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  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
(
  i  0
  0 -i
)
(
 e-3iπ/4  0
  0  e3iπ/4
)
(
  eiπ/4  0
  0 e-iπ/4
)
5
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
d1
1
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+
1
-1
(
 -i  0
  0  i
)
(
 e-iπ/4  0
  0  eiπ/4
)
(
  e3iπ/4  0
  0 e-3iπ/4
)
7
(
 -1  0
  0 -1
)
(
  i  0
  0 -i
)
d2
1
1
(
 -1  0
  0 -1
)
(
  i  0
  0 -i
)
(
  i  0
  0 -i
)
8
(
  0  1
 -1  0
)
(
 e-3iπ/4  0
  0  e3iπ/4
)
d4-
1
-1
(
  i  0
  0 -i
)
(
  eiπ/4  0
  0 e-iπ/4
)
(
 e-3iπ/4  0
  0  e3iπ/4
)
9
(
 1 0
 0 1
)
(
 1 0
 0 1
)
1'
-1
-1
(
 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+'
-1
1
(
  0 -i
  i  0
)
(
  0  e-iπ/4
 e-3iπ/4  0
)
(
  0 e3iπ/4
  eiπ/4  0
)
11
(
 -1  0
  0 -1
)
(
 -i  0
  0  i
)
2'
-1
-1
(
  0 -1
 -1  0
)
(
 0 i
 i 0
)
(
 0 i
 i 0
)
12
(
  0  1
 -1  0
)
(
  eiπ/4  0
  0 e-iπ/4
)
4-'
-1
1
(
  0  i
 -i  0
)
(
  0  eiπ/4
 e3iπ/4  0
)
(
  0 e-3iπ/4
  e-iπ/4  0
)
13
(
 1 0
 0 1
)
(
 -1  0
  0 -1
)
d1'
-1
-1
(
 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+'
-1
1
(
  0 -i
  i  0
)
(
  0 e3iπ/4
  eiπ/4  0
)
(
  0  e-iπ/4
 e-3iπ/4  0
)
15
(
 -1  0
  0 -1
)
(
  i  0
  0 -i
)
d2'
-1
-1
(
  0 -1
 -1  0
)
(
  0 -i
 -i  0
)
(
  0 -i
 -i  0
)
16
(
  0  1
 -1  0
)
(
 e-3iπ/4  0
  0  e3iπ/4
)
d4-'
-1
1
(
  0  i
 -i  0
)
(
  0 e-3iπ/4
  e-iπ/4  0
)
(
  0  eiπ/4
 e3iπ/4  0
)
k-Subgroupsmag
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