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Irreducible corepresentations of the Projective Magnetic Point Group 32π/3m


Table of characters of the unitary symmetry operations


1
3+
3-
m1-1
m21
m12
d1
d3+
d3-
dm1-1
dm21
dm12
2E''
1
e2iπ/3
e-2iπ/3
1
-1
1
e2iπ/3
e-2iπ/3
1
-1
2E'
1
e2iπ/3
e-2iπ/3
-1
1
1
e2iπ/3
e-2iπ/3
-1
1
1EA
2
e-iπ/3
eiπ/3
0
0
2
e-iπ/3
eiπ/3
0
0
1E2
1
e-iπ/3
eiπ/3
i
-i
-1
e2iπ/3
e-2iπ/3
-i
i
1E1
1
e-iπ/3
eiπ/3
-i
i
-1
e2iπ/3
e-2iπ/3
i
-i
2EE
2
e2iπ/3
e-2iπ/3
0
0
-2
e-iπ/3
eiπ/3
0
0

Multiplication table of the symmetry operations


1
3+
3-
m1-1
m12
m21
d1
d3+
d3-
dm1-1
dm12
dm21
1
1
3+
3-
m1-1
m12
m21
d1
d3+
d3-
dm1-1
dm12
dm21
3+
3+
d3-
1
dm21
dm1-1
dm12
d3+
3-
d1
m21
m1-1
m12
3-
3-
1
d3+
dm12
dm21
dm1-1
d3-
d1
3+
m12
m21
m1-1
m1-1
m1-1
dm12
dm21
d1
3+
3-
dm1-1
m12
m21
1
d3+
d3-
m12
m12
dm21
dm1-1
3-
d1
3+
dm12
m21
m1-1
d3-
1
d3+
m21
m21
dm1-1
dm12
3+
3-
d1
dm21
m1-1
m12
d3+
d3-
1
d1
d1
d3+
d3-
dm1-1
dm12
dm21
1
3+
3-
m1-1
m12
m21
d3+
d3+
3-
d1
m21
m1-1
m12
3+
d3-
1
dm21
dm1-1
dm12
d3-
d3-
d1
3+
m12
m21
m1-1
3-
1
d3+
dm12
dm21
dm1-1
dm1-1
dm1-1
m12
m21
1
d3+
d3-
m1-1
dm12
dm21
d1
3+
3-
dm12
dm12
m21
m1-1
d3-
1
d3+
m12
dm21
dm1-1
3-
d1
3+
dm21
dm21
m1-1
m12
d3+
d3-
1
m21
dm1-1
dm12
3+
3-
d1

Table of projective phases in group multiplication


1
3+
3-
m1-1
m12
m21
d1
d3+
d3-
dm1-1
dm12
dm21
1
1
1
1
1
1
1
1
1
1
1
1
1
3+
1
1
1
e2iπ/3
e-iπ/3
e-iπ/3
1
1
1
e2iπ/3
e-iπ/3
e-iπ/3
3-
1
1
1
eiπ/3
eiπ/3
e-2iπ/3
1
1
1
eiπ/3
eiπ/3
e-2iπ/3
m1-1
1
e-iπ/3
e-2iπ/3
1
eiπ/3
e2iπ/3
1
e-iπ/3
e-2iπ/3
1
eiπ/3
e2iπ/3
m12
1
e-iπ/3
eiπ/3
e-iπ/3
1
eiπ/3
1
e-iπ/3
eiπ/3
e-iπ/3
1
eiπ/3
m21
1
e2iπ/3
eiπ/3
e-2iπ/3
e-iπ/3
1
1
e2iπ/3
eiπ/3
e-2iπ/3
e-iπ/3
1
d1
1
1
1
1
1
1
1
1
1
1
1
1
d3+
1
1
1
e2iπ/3
e-iπ/3
e-iπ/3
1
1
1
e2iπ/3
e-iπ/3
e-iπ/3
d3-
1
1
1
eiπ/3
eiπ/3
e-2iπ/3
1
1
1
eiπ/3
eiπ/3
e-2iπ/3
dm1-1
1
e-iπ/3
e-2iπ/3
1
eiπ/3
e2iπ/3
1
e-iπ/3
e-2iπ/3
1
eiπ/3
e2iπ/3
dm12
1
e-iπ/3
eiπ/3
e-iπ/3
1
eiπ/3
1
e-iπ/3
eiπ/3
e-iπ/3
1
eiπ/3
dm21
1
e2iπ/3
eiπ/3
e-2iπ/3
e-iπ/3
1
1
e2iπ/3
eiπ/3
e-2iπ/3
e-iπ/3
1

Matrices of the representations of the group

The antiunitary operations are written in red color
NMatrix presentationSeitz symbol2E''2E'1EA1E21E12EE
1
(
1 0
0 1
)
(
1 0
0 1
)
1
1
1
(
1 0
0 1
)
1
1
(
1 0
0 1
)
2
(
0 -1
1 -1
)
(
eiπ/3 0
0 e-iπ/3
)
3+
e2iπ/3
e2iπ/3
(
1 0
0 e-2iπ/3
)
e-iπ/3
e-iπ/3
(
-1 0
0 eiπ/3
)
3
(
-1 1
-1 0
)
(
e-iπ/3 0
0 eiπ/3
)
3-
e-2iπ/3
e-2iπ/3
(
1 0
0 e2iπ/3
)
eiπ/3
eiπ/3
(
-1 0
0 e-iπ/3
)
4
(
0 1
1 0
)
(
0 e5iπ/6
eiπ/6 0
)
m1-1
1
-1
(
0 1
1 0
)
i
-i
(
0 -1
1 0
)
5
(
1 -1
0 -1
)
(
0 -i
-i 0
)
m12
-1
1
(
0 e-iπ/3
eiπ/3 0
)
-i
i
(
0 e2iπ/3
eiπ/3 0
)
6
(
-1 0
-1 1
)
(
0 eiπ/6
e5iπ/6 0
)
m21
1
-1
(
0 e-2iπ/3
e2iπ/3 0
)
i
-i
(
0 eiπ/3
e2iπ/3 0
)
7
(
1 0
0 1
)
(
-1 0
0 -1
)
d1
1
1
(
1 0
0 1
)
-1
-1
(
-1 0
0 -1
)
8
(
0 -1
1 -1
)
(
e-2iπ/3 0
0 e2iπ/3
)
d3+
e2iπ/3
e2iπ/3
(
1 0
0 e-2iπ/3
)
e2iπ/3
e2iπ/3
(
1 0
0 e-2iπ/3
)
9
(
-1 1
-1 0
)
(
e2iπ/3 0
0 e-2iπ/3
)
d3-
e-2iπ/3
e-2iπ/3
(
1 0
0 e2iπ/3
)
e-2iπ/3
e-2iπ/3
(
1 0
0 e2iπ/3
)
10
(
0 1
1 0
)
(
0 e-iπ/6
e-5iπ/6 0
)
dm1-1
1
-1
(
0 1
1 0
)
-i
i
(
0 1
-1 0
)
11
(
1 -1
0 -1
)
(
0 i
i 0
)
dm12
-1
1
(
0 e-iπ/3
eiπ/3 0
)
i
-i
(
0 e-iπ/3
e-2iπ/3 0
)
12
(
-1 0
-1 1
)
(
0 e-5iπ/6
e-iπ/6 0
)
dm21
1
-1
(
0 e-2iπ/3
e2iπ/3 0
)
-i
i
(
0 e-2iπ/3
e-iπ/3 0
)
k-Subgroupsmag
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