Bilbao Crystallographic Server Representations

## 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 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, d2001 C3: 4+001, 4-001 C4: 2010, 2100, d2010, d2100 C5: 2110, 21-10, d2110, d21-10 C6: d1 C7: d4+001, d4-001 C8: -1 C9: m001, dm001 C10: -4+001, -4-001 C11: m010, m100, dm010, dm100 C12: m110, m1-10, dm110, dm1-10 C13: d-1 C14: d-4+001, d-4-001

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+(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