Bilbao Crystallographic Server Representations

## 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- 1Au GM1- 1 1 1 1 -1 -1 -1 -1 GM2+ 1Eg GM2+ 1 1 -(1+i√3)/2 -(1-i√3)/2 1 1 -(1+i√3)/2 -(1-i√3)/2 GM2- 1Eu GM2- 1 1 -(1+i√3)/2 -(1-i√3)/2 -1 -1 (1+i√3)/2 (1-i√3)/2 GM3+ 2Eg GM3+ 1 1 -(1-i√3)/2 -(1+i√3)/2 1 1 -(1-i√3)/2 -(1+i√3)/2 GM3- 2Eu GM3- 1 1 -(1-i√3)/2 -(1+i√3)/2 -1 -1 (1-i√3)/2 (1+i√3)/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