Bilbao Crystallographic Server arrow 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 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: 4+001
C4: 4-001
C5d1
C6d2001
C7d4+001
C8d4-001

List of pairs of conjugated irreducible representations

(*GM3,*GM4)
(*GM5,*GM7)
(*GM6,*GM8)
Matrices of the representations of the group

The number in parenthesis 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
Bilbao Crystallographic Server
http://www.cryst.ehu.es
For comments, please mail to
administrador.bcs@ehu.eus