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Irreducible representations of the Double Point Group 4/mmm (No. 15)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
GM1+
A1g
GM1+
1
1
1
1
1
1
1
1
1
1
1
1
1
1
GM1-
A1u
GM1-
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
GM3+
B1g
GM2+
1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
GM3-
B1u
GM2-
1
1
-1
1
-1
1
-1
-1
-1
1
-1
1
-1
1
GM2+
A2g
GM3+
1
1
1
-1
-1
1
1
1
1
1
-1
-1
1
1
GM2-
A2u
GM3-
1
1
1
-1
-1
1
1
-1
-1
-1
1
1
-1
-1
GM4+
B2g
GM4+
1
1
-1
-1
1
1
-1
1
1
-1
-1
1
1
-1
GM4-
B2u
GM4-
1
1
-1
-1
1
1
-1
-1
-1
1
1
-1
-1
1
GM5+
Eg
GM5+
2
-2
0
0
0
2
0
2
-2
0
0
0
2
0
GM5-
Eu
GM5-
2
-2
0
0
0
2
0
-2
2
0
0
0
-2
0
GM7+
E2g
GM6
2
0
-2
0
0
-2
2
2
0
-2
0
0
-2
2
GM6+
E1g
GM7
2
0
2
0
0
-2
-2
2
0
2
0
0
-2
-2
GM7-
E2u
GM8
2
0
-2
0
0
-2
2
-2
0
2
0
0
2
-2
GM6-
E1u
GM9
2
0
2
0
0
-2
-2
-2
0
-2
0
0
2
2
(1): Notation of the irreps according to Koster GF, Dimmok JO, Wheeler RG and Statz H, (1963) Properties of the thirty-two point groups, M.I.T. Press, Cambridge, Mass.
(2): Notation of the irreps according to Mulliken RS (1933) Phys. Rev. 43, 279-302.
(3): Notation of the irreps according to C. J. Bradley, A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon Press, Oxford, 1972) for the GM point.

Lists of symmetry operations in the conjugacy classes

C1: 1
C2: 2001d2001
C3: 4+001, 4-001
C4: 2010, 2100d2010d2100
C5: 2110, 21-10d2110d21-10
C6d1
C7d4+001d4-001
C8: -1
C9: m001dm001
C10: -4+001, -4-001
C11: m010, m100dm010dm100
C12: m110, m1-10dm110dm1-10
C13d-1
C14d-4+001d-4-001

Matrices of the representations of the group

The number in parenthesis after the label of the irrep indicates the "reality" of the irrep: (1) for real, (-1) for pseudoreal and (0) for complex representations.

N
Matrix presentation
Seitz Symbol
GM1+(1)
GM1-(1)
GM2+(1)
GM2-(1)
GM3+(1)
GM3-(1)
GM4+(1)
GM4-(1)
GM5+(1)
GM5-(1)
GM6(-1)
GM7(-1)
GM8(-1)
GM9(-1)
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
1
1
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
1 0
0 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
1
1
1
1
1
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
-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
-1
-1
1
1
-1
-1
(
0 -1
1 0
)
(
0 -1
1 0
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
(
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
-1
-1
1
1
-1
-1
(
0 1
-1 0
)
(
0 1
-1 0
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
5
(
-1 0 0
0 1 0
0 0 -1
)
(
0 -1
1 0
)
2010
1
1
1
1
-1
-1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
6
(
1 0 0
0 -1 0
0 0 -1
)
(
0 -i
-i 0
)
2100
1
1
1
1
-1
-1
-1
-1
(
0 -1
-1 0
)
(
0 -1
-1 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
7
(
0 1 0
1 0 0
0 0 -1
)
(
0 -(1+i)2/2
(1-i)2/2 0
)
2110
1
1
-1
-1
-1
-1
1
1
(
1 0
0 -1
)
(
1 0
0 -1
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
8
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 -(1-i)2/2
(1+i)2/2 0
)
2110
1
1
-1
-1
-1
-1
1
1
(
-1 0
0 1
)
(
-1 0
0 1
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
9
(
-1 0 0
0 -1 0
0 0 -1
)
(
1 0
0 1
)
1
1
-1
1
-1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
1 0
0 1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
10
(
1 0 0
0 1 0
0 0 -1
)
(
-i 0
0 i
)
m001
1
-1
1
-1
1
-1
1
-1
(
-1 0
0 -1
)
(
1 0
0 1
)
(
-i 0
0 i
)
(
-i 0
0 i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
11
(
0 1 0
-1 0 0
0 0 -1
)
(
(1-i)2/2 0
0 (1+i)2/2
)
4+001
1
-1
-1
1
1
-1
-1
1
(
0 -1
1 0
)
(
0 1
-1 0
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
12
(
0 -1 0
1 0 0
0 0 -1
)
(
(1+i)2/2 0
0 (1-i)2/2
)
4-001
1
-1
-1
1
1
-1
-1
1
(
0 1
-1 0
)
(
0 -1
1 0
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
13
(
1 0 0
0 -1 0
0 0 1
)
(
0 -1
1 0
)
m010
1
-1
1
-1
-1
1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
14
(
-1 0 0
0 1 0
0 0 1
)
(
0 -i
-i 0
)
m100
1
-1
1
-1
-1
1
-1
1
(
0 -1
-1 0
)
(
0 1
1 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
15
(
0 -1 0
-1 0 0
0 0 1
)
(
0 -(1+i)2/2
(1-i)2/2 0
)
m110
1
-1
-1
1
-1
1
1
-1
(
1 0
0 -1
)
(
-1 0
0 1
)
(
0 -1
1 0
)
(
0 -1
1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
16
(
0 1 0
1 0 0
0 0 1
)
(
0 -(1-i)2/2
(1+i)2/2 0
)
m110
1
-1
-1
1
-1
1
1
-1
(
-1 0
0 1
)
(
1 0
0 -1
)
(
0 i
i 0
)
(
0 i
i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
17
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
1
1
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
18
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2001
1
1
1
1
1
1
1
1
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
i 0
0 -i
)
19
(
0 -1 0
1 0 0
0 0 1
)
(
-(1-i)2/2 0
0 -(1+i)2/2
)
d4+001
1
1
-1
-1
1
1
-1
-1
(
0 -1
1 0
)
(
0 -1
1 0
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
20
(
0 1 0
-1 0 0
0 0 1
)
(
-(1+i)2/2 0
0 -(1-i)2/2
)
d4-001
1
1
-1
-1
1
1
-1
-1
(
0 1
-1 0
)
(
0 1
-1 0
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
21
(
-1 0 0
0 1 0
0 0 -1
)
(
0 1
-1 0
)
d2010
1
1
1
1
-1
-1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
22
(
1 0 0
0 -1 0
0 0 -1
)
(
0 i
i 0
)
d2100
1
1
1
1
-1
-1
-1
-1
(
0 -1
-1 0
)
(
0 -1
-1 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
23
(
0 1 0
1 0 0
0 0 -1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
d2110
1
1
-1
-1
-1
-1
1
1
(
1 0
0 -1
)
(
1 0
0 -1
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 1
-1 0
)
24
(
0 -1 0
-1 0 0
0 0 -1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
d2110
1
1
-1
-1
-1
-1
1
1
(
-1 0
0 1
)
(
-1 0
0 1
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
25
(
-1 0 0
0 -1 0
0 0 -1
)
(
-1 0
0 -1
)
d1
1
-1
1
-1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
-1 0
0 -1
)
(
1 0
0 1
)
(
1 0
0 1
)
26
(
1 0 0
0 1 0
0 0 -1
)
(
i 0
0 -i
)
dm001
1
-1
1
-1
1
-1
1
-1
(
-1 0
0 -1
)
(
1 0
0 1
)
(
i 0
0 -i
)
(
i 0
0 -i
)
(
-i 0
0 i
)
(
-i 0
0 i
)
27
(
0 1 0
-1 0 0
0 0 -1
)
(
-(1-i)2/2 0
0 -(1+i)2/2
)
d4+001
1
-1
-1
1
1
-1
-1
1
(
0 -1
1 0
)
(
0 1
-1 0
)
(
e-iπ/4 0
0 eiπ/4
)
(
ei3π/4 0
0 e-i3π/4
)
(
ei3π/4 0
0 e-i3π/4
)
(
e-iπ/4 0
0 eiπ/4
)
28
(
0 -1 0
1 0 0
0 0 -1
)
(
-(1+i)2/2 0
0 -(1-i)2/2
)
d4-001
1
-1
-1
1
1
-1
-1
1
(
0 1
-1 0
)
(
0 -1
1 0
)
(
eiπ/4 0
0 e-iπ/4
)
(
e-i3π/4 0
0 ei3π/4
)
(
e-i3π/4 0
0 ei3π/4
)
(
eiπ/4 0
0 e-iπ/4
)
29
(
1 0 0
0 -1 0
0 0 1
)
(
0 1
-1 0
)
dm010
1
-1
1
-1
-1
1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
(
0 ei3π/4
eiπ/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 e-iπ/4
e-i3π/4 0
)
(
0 ei3π/4
eiπ/4 0
)
30
(
-1 0 0
0 1 0
0 0 1
)
(
0 i
i 0
)
dm100
1
-1
1
-1
-1
1
-1
1
(
0 -1
-1 0
)
(
0 1
1 0
)
(
0 e-i3π/4
e-iπ/4 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 eiπ/4
ei3π/4 0
)
(
0 e-i3π/4
e-iπ/4 0
)
31
(
0 -1 0
-1 0 0
0 0 1
)
(
0 (1+i)2/2
-(1-i)2/2 0
)
dm110
1
-1
-1
1
-1
1
1
-1
(
1 0
0 -1
)
(
-1 0
0 1
)
(
0 1
-1 0
)
(
0 1
-1 0
)
(
0 -1
1 0
)
(
0 -1
1 0
)
32
(
0 1 0
1 0 0
0 0 1
)
(
0 (1-i)2/2
-(1+i)2/2 0
)
dm110
1
-1
-1
1
-1
1
1
-1
(
-1 0
0 1
)
(
1 0
0 -1
)
(
0 -i
-i 0
)
(
0 -i
-i 0
)
(
0 i
i 0
)
(
0 i
i 0
)
k-Subgroupsmag
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