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

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
GM1+
A1g
GM1+
1
1
1
1
1
1
1
1
1
1
GM1-
A1u
GM1-
1
1
1
1
1
-1
-1
-1
-1
-1
GM3+
B1g
GM2+
1
1
1
-1
-1
1
1
1
-1
-1
GM3-
B1u
GM2-
1
1
1
-1
-1
-1
-1
-1
1
1
GM2+
A2g
GM3+
1
1
-1
-1
1
1
1
-1
-1
1
GM2-
A2u
GM3-
1
1
-1
-1
1
-1
-1
1
1
-1
GM4+
B2g
GM4+
1
1
-1
1
-1
1
1
-1
1
-1
GM4-
B2u
GM4-
1
1
-1
1
-1
-1
-1
1
-1
1
GM5+
Eg
GM5+
2
-2
0
0
0
2
-2
0
0
0
GM5-
Eu
GM5-
2
-2
0
0
0
-2
2
0
0
0
(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: 2001
C3: 2010, 2100
C4: 21-10, 2110
C5: 4+001, 4-001
C6: -1
C7: m001
C8: m010, m100
C9: m1-10, m110
C10: -4+001, -4-001

Matrices of the representations of the group

N
General position
Seitz Symbol
GM1+(1)
GM1-(1)
GM2+(1)
GM2-(1)
GM3+(1)
GM3-(1)
GM4+(1)
GM4-(1)
GM5+(1)
GM5-(1)
1
(
1 0 0
0 1 0
0 0 1
)
1
1
1
1
1
1
1
1
1
(
1 0
0 1
)
(
1 0
0 1
)
2
(
-1 0 0
0 -1 0
0 0 1
)
2001
1
1
1
1
1
1
1
1
(
-1 0
0 -1
)
(
-1 0
0 -1
)
3
(
0 -1 0
1 0 0
0 0 1
)
4+001
1
1
-1
-1
1
1
-1
-1
(
0 -1
1 0
)
(
0 -1
1 0
)
4
(
0 1 0
-1 0 0
0 0 1
)
4-001
1
1
-1
-1
1
1
-1
-1
(
0 1
-1 0
)
(
0 1
-1 0
)
5
(
-1 0 0
0 1 0
0 0 -1
)
2010
1
1
1
1
-1
-1
-1
-1
(
0 1
1 0
)
(
0 1
1 0
)
6
(
1 0 0
0 -1 0
0 0 -1
)
2100
1
1
1
1
-1
-1
-1
-1
(
0 -1
-1 0
)
(
0 -1
-1 0
)
7
(
0 1 0
1 0 0
0 0 -1
)
2110
1
1
-1
-1
-1
-1
1
1
(
1 0
0 -1
)
(
1 0
0 -1
)
8
(
0 -1 0
-1 0 0
0 0 -1
)
2110
1
1
-1
-1
-1
-1
1
1
(
-1 0
0 1
)
(
-1 0
0 1
)
9
(
-1 0 0
0 -1 0
0 0 -1
)
1
1
-1
1
-1
1
-1
1
-1
(
1 0
0 1
)
(
-1 0
0 -1
)
10
(
1 0 0
0 1 0
0 0 -1
)
m001
1
-1
1
-1
1
-1
1
-1
(
-1 0
0 -1
)
(
1 0
0 1
)
11
(
0 1 0
-1 0 0
0 0 -1
)
4+001
1
-1
-1
1
1
-1
-1
1
(
0 -1
1 0
)
(
0 1
-1 0
)
12
(
0 -1 0
1 0 0
0 0 -1
)
4-001
1
-1
-1
1
1
-1
-1
1
(
0 1
-1 0
)
(
0 -1
1 0
)
13
(
1 0 0
0 -1 0
0 0 1
)
m010
1
-1
1
-1
-1
1
-1
1
(
0 1
1 0
)
(
0 -1
-1 0
)
14
(
-1 0 0
0 1 0
0 0 1
)
m100
1
-1
1
-1
-1
1
-1
1
(
0 -1
-1 0
)
(
0 1
1 0
)
15
(
0 -1 0
-1 0 0
0 0 1
)
m110
1
-1
-1
1
-1
1
1
-1
(
1 0
0 -1
)
(
-1 0
0 1
)
16
(
0 1 0
1 0 0
0 0 1
)
m110
1
-1
-1
1
-1
1
1
-1
(
-1 0
0 1
)
(
1 0
0 -1
)
k-Subgroupsmag
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