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Irreducible representations of the Point Group m3 (No. 29)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
GM1+
Ag
GM1+
1
1
1
1
1
1
1
1
GM1-
Au
GM1-
1
1
1
1
-1
-1
-1
-1
GM2+
1Eg
GM2+
1
1
-(1+i3)/2
-(1-i3)/2
1
1
-(1+i3)/2
-(1-i3)/2
GM2-
1Eu
GM2-
1
1
-(1+i3)/2
-(1-i3)/2
-1
-1
(1+i3)/2
(1-i3)/2
GM3+
2Eg
GM3+
1
1
-(1-i3)/2
-(1+i3)/2
1
1
-(1-i3)/2
-(1+i3)/2
GM3-
2Eu
GM3-
1
1
-(1-i3)/2
-(1+i3)/2
-1
-1
(1-i3)/2
(1+i3)/2
GM4+
Tg
GM4+
3
-1
0
0
3
-1
0
0
GM4-
Tu
GM4-
3
-1
0
0
-3
1
0
0
(1): Notation of the irreps according to Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press.
(2): Notation of the irreps according to Bradley CJ and Cracknell AP, (1972) The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press, based on Mulliken RS (1933) Phys. Rev. 43, 279-302.
(3): Notation of the irreps according to A. P. Cracknell, B. L. Davies, S. C. Miller and W. F. Love (1979) Kronecher Product Tables, 1, General Introduction and Tables of Irreducible Representations of Space groups. New York: IFI/Plenum, for the GM point.

Lists of symmetry operations in the conjugacy classes

C1: 1
C2: 2001, 2010, 2100
C3: 3--11-1, 3-1-1-1, 3--1-11, 3-111
C4: 3+1-1-1, 3+-1-11, 3+-11-1, 3+111
C5: -1
C6: m001, m010, m100
C7: -3--11-1, -3-1-1-1, -3--1-11, -3-111
C8: -3+1-1-1, -3+-1-11, -3+-11-1, -3+111

List of pairs of conjugated irreducible representations

(*GM2+,*GM3+)
(*GM2-,*GM3-)
Matrices of the representations of the group

The number in parentheses 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+(0)
GM2-(0)
GM3+(0)
GM3-(0)
GM4+(1)
GM4-(1)
1
(
1 0 0
0 1 0
0 0 1
)
1
1
1
1
1
1
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0 0
0 1 0
0 0 1
)
2
(
-1 0 0
0 -1 0
0 0 1
)
2001
1
1
1
1
1
1
(
1 0 0
0 -1 0
0 0 -1
)
(
1 0 0
0 -1 0
0 0 -1
)
3
(
-1 0 0
0 1 0
0 0 -1
)
2010
1
1
1
1
1
1
(
-1 0 0
0 -1 0
0 0 1
)
(
-1 0 0
0 -1 0
0 0 1
)
4
(
1 0 0
0 -1 0
0 0 -1
)
2100
1
1
1
1
1
1
(
-1 0 0
0 1 0
0 0 -1
)
(
-1 0 0
0 1 0
0 0 -1
)
5
(
0 0 1
1 0 0
0 1 0
)
3+111
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
(
0 0 1
1 0 0
0 1 0
)
(
0 0 1
1 0 0
0 1 0
)
6
(
0 0 1
-1 0 0
0 -1 0
)
3+111
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
(
0 0 -1
1 0 0
0 -1 0
)
(
0 0 -1
1 0 0
0 -1 0
)
7
(
0 0 -1
-1 0 0
0 1 0
)
3+111
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
(
0 0 1
-1 0 0
0 -1 0
)
(
0 0 1
-1 0 0
0 -1 0
)
8
(
0 0 -1
1 0 0
0 -1 0
)
3+111
1
1
ei2π/3
ei2π/3
e-i2π/3
e-i2π/3
(
0 0 -1
-1 0 0
0 1 0
)
(
0 0 -1
-1 0 0
0 1 0
)
9
(
0 1 0
0 0 1
1 0 0
)
3-111
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
(
0 1 0
0 0 1
1 0 0
)
(
0 1 0
0 0 1
1 0 0
)
10
(
0 -1 0
0 0 1
-1 0 0
)
3-111
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
(
0 -1 0
0 0 -1
1 0 0
)
(
0 -1 0
0 0 -1
1 0 0
)
11
(
0 1 0
0 0 -1
-1 0 0
)
3-111
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
(
0 -1 0
0 0 1
-1 0 0
)
(
0 -1 0
0 0 1
-1 0 0
)
12
(
0 -1 0
0 0 -1
1 0 0
)
3-111
1
1
e-i2π/3
e-i2π/3
ei2π/3
ei2π/3
(
0 1 0
0 0 -1
-1 0 0
)
(
0 1 0
0 0 -1
-1 0 0
)
13
(
-1 0 0
0 -1 0
0 0 -1
)
1
1
-1
1
-1
1
-1
(
1 0 0
0 1 0
0 0 1
)
(
-1 0 0
0 -1 0
0 0 -1
)
14
(
1 0 0
0 1 0
0 0 -1
)
m001
1
-1
1
-1
1
-1
(
1 0 0
0 -1 0
0 0 -1
)
(
-1 0 0
0 1 0
0 0 1
)
15
(
1 0 0
0 -1 0
0 0 1
)
m010
1
-1
1
-1
1
-1
(
-1 0 0
0 -1 0
0 0 1
)
(
1 0 0
0 1 0
0 0 -1
)
16
(
-1 0 0
0 1 0
0 0 1
)
m100
1
-1
1
-1
1
-1
(
-1 0 0
0 1 0
0 0 -1
)
(
1 0 0
0 -1 0
0 0 1
)
17
(
0 0 -1
-1 0 0
0 -1 0
)
3+111
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
(
0 0 1
1 0 0
0 1 0
)
(
0 0 -1
-1 0 0
0 -1 0
)
18
(
0 0 -1
1 0 0
0 1 0
)
3+111
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
(
0 0 -1
1 0 0
0 -1 0
)
(
0 0 1
-1 0 0
0 1 0
)
19
(
0 0 1
1 0 0
0 -1 0
)
3+111
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
(
0 0 1
-1 0 0
0 -1 0
)
(
0 0 -1
1 0 0
0 1 0
)
20
(
0 0 1
-1 0 0
0 1 0
)
3+111
1
-1
ei2π/3
e-iπ/3
e-i2π/3
eiπ/3
(
0 0 -1
-1 0 0
0 1 0
)
(
0 0 1
1 0 0
0 -1 0
)
21
(
0 -1 0
0 0 -1
-1 0 0
)
3-111
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
(
0 1 0
0 0 1
1 0 0
)
(
0 -1 0
0 0 -1
-1 0 0
)
22
(
0 1 0
0 0 -1
1 0 0
)
3-111
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
(
0 -1 0
0 0 -1
1 0 0
)
(
0 1 0
0 0 1
-1 0 0
)
23
(
0 -1 0
0 0 1
1 0 0
)
3-111
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
(
0 -1 0
0 0 1
-1 0 0
)
(
0 1 0
0 0 -1
1 0 0
)
24
(
0 1 0
0 0 1
-1 0 0
)
3-111
1
-1
e-i2π/3
eiπ/3
ei2π/3
e-iπ/3
(
0 1 0
0 0 -1
-1 0 0
)
(
0 -1 0
0 0 1
1 0 0
)
k-Subgroupsmag
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