Bilbao Crystallographic Server Representations

Irreducible representations of the Double Point Group 4 (No. 9)

Table of characters

 (1) (2) (3) C1 C2 C3 C4 C5 C6 C7 C8 GM1 A GM1 1 1 1 1 1 1 1 1 GM2 B GM2 1 1 -1 -1 1 1 -1 -1 GM3 2E GM3 1 -1 i -i 1 -1 i -i GM4 1E GM4 1 -1 -i i 1 -1 -i i GM7 2E2 GM5 1 -i (-1+i)/√2 (-1-i)/√2 -1 i (1-i)/√2 (1+i)/√2 GM5 2E1 GM6 1 -i (1-i)/√2 (1+i)/√2 -1 i (-1+i)/√2 (-1-i)/√2 GM8 1E2 GM7 1 i (-1-i)/√2 (-1+i)/√2 -1 -i (1+i)/√2 (1-i)/√2 GM6 1E1 GM8 1 i (1+i)/√2 (1-i)/√2 -1 -i (-1-i)/√2 (-1+i)/√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 C3: 4+001 C4: 4-001 C5: d1 C6: d2001 C7: d4+001 C8: d4-001

List of pairs of conjugated irreducible representations

(*GM3,*GM4)
(*GM5,*GM7)
(*GM6,*GM8)
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)
GM2(1)
GM3(0)
GM4(0)
GM5(0)
GM6(0)
GM7(0)
GM8(0)
1
 ` 1 0 0 0 1 0 0 0 1`
 ` 1 0 0 1`
1
 1
 1
 1
 1
 1
 1
 1
 1
2
 ` -1 0 0 0 -1 0 0 0 1`
 ` -i 0 0 i`
2001
 1
 1
 -1
 -1
 -i
 -i
 i
 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
 i
 -i
 ei3π/4
 e-iπ/4
 e-i3π/4
 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
 -i
 i
 e-i3π/4
 eiπ/4
 ei3π/4
 e-iπ/4
5
 ` 1 0 0 0 1 0 0 0 1`
 ` -1 0 0 -1`
d1
 1
 1
 1
 1
 -1
 -1
 -1
 -1
6
 ` -1 0 0 0 -1 0 0 0 1`
 ` i 0 0 -i`
d2001
 1
 1
 -1
 -1
 i
 i
 -i
 -i
7
 ` 0 -1 0 1 0 0 0 0 1`
 ` -(1-i)√2/2 0 0 -(1+i)√2/2`
d4+001
 1
 -1
 i
 -i
 e-iπ/4
 ei3π/4
 eiπ/4
 e-i3π/4
8
 ` 0 1 0 -1 0 0 0 0 1`
 ` -(1+i)√2/2 0 0 -(1-i)√2/2`
d4-001
 1
 -1
 -i
 i
 eiπ/4
 e-i3π/4
 e-iπ/4
 ei3π/4
k-Subgroupsmag