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Irreducible representations of the Double Point Group 4/m (No. 11)

Table of characters

(1)
(2)
(3)
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
GM1+
Ag
GM1+
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
GM1-
Au
GM1-
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
GM2+
Bg
GM2+
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
GM2-
Bu
GM2-
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
GM3+
2Eg
GM3+
1
-1
i
-i
1
-1
i
-i
1
-1
i
-i
1
-1
i
-i
GM3-
2Eu
GM3-
1
-1
i
-i
1
-1
i
-i
-1
1
-i
i
-1
1
-i
i
GM4+
1Eg
GM4+
1
-1
-i
i
1
-1
-i
i
1
-1
-i
i
1
-1
-i
i
GM4-
1Eu
GM4-
1
-1
-i
i
1
-1
-i
i
-1
1
i
-i
-1
1
i
-i
GM7+
2E2g
GM5
1
-i
(-1+i)/2
(-1-i)/2
-1
i
(1-i)/2
(1+i)/2
1
-i
(-1+i)/2
(-1-i)/2
-1
i
(1-i)/2
(1+i)/2
GM5+
2E1g
GM6
1
-i
(1-i)/2
(1+i)/2
-1
i
(-1+i)/2
(-1-i)/2
1
-i
(1-i)/2
(1+i)/2
-1
i
(-1+i)/2
(-1-i)/2
GM8+
1E2g
GM7
1
i
(-1-i)/2
(-1+i)/2
-1
-i
(1+i)/2
(1-i)/2
1
i
(-1-i)/2
(-1+i)/2
-1
-i
(1+i)/2
(1-i)/2
GM6+
1E1g
GM8
1
i
(1+i)/2
(1-i)/2
-1
-i
(-1-i)/2
(-1+i)/2
1
i
(1+i)/2
(1-i)/2
-1
-i
(-1-i)/2
(-1+i)/2
GM7-
2E2u
GM9
1
-i
(-1+i)/2
(-1-i)/2
-1
i
(1-i)/2
(1+i)/2
-1
i
(1-i)/2
(1+i)/2
1
-i
(-1+i)/2
(-1-i)/2
GM5-
2E1u
GM10
1
-i
(1-i)/2
(1+i)/2
-1
i
(-1+i)/2
(-1-i)/2
-1
i
(-1+i)/2
(-1-i)/2
1
-i
(1-i)/2
(1+i)/2
GM8-
1E2u
GM11
1
i
(-1-i)/2
(-1+i)/2
-1
-i
(1+i)/2
(1-i)/2
-1
-i
(1+i)/2
(1-i)/2
1
i
(-1-i)/2
(-1+i)/2
GM6-
1E1u
GM12
1
i
(1+i)/2
(1-i)/2
-1
-i
(-1-i)/2
(-1+i)/2
-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
C9: -1
C10: m001
C11: -4+001
C12: -4-001
C13d-1
C14dm001
C15d-4+001
C16d-4-001

List of pairs of conjugated irreducible representations

(*GM3+,*GM4+)
(*GM3-,*GM4-)
(*GM5,*GM7)
(*GM6,*GM8)
(*GM9,*GM11)
(*GM10,*GM12)
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+(0)
GM3-(0)
GM4+(0)
GM4-(0)
GM5(0)
GM6(0)
GM7(0)
GM8(0)
GM9(0)
GM10(0)
GM11(0)
GM12(0)
1
(
1 0 0
0 1 0
0 0 1
)
(
1 0
0 1
)
1
1
1
1
1
1
1
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
-1
-1
-1
-1
-i
-i
i
i
-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
-1
-1
i
i
-i
-i
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
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
-1
-1
-i
-i
i
i
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
5
(
-1 0 0
0 -1 0
0 0 -1
)
(
1 0
0 1
)
1
1
-1
1
-1
1
-1
1
-1
1
1
1
1
-1
-1
-1
-1
6
(
1 0 0
0 1 0
0 0 -1
)
(
-i 0
0 i
)
m001
1
-1
1
-1
-1
1
-1
1
-i
-i
i
i
i
i
-i
-i
7
(
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
i
-i
-i
i
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
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
)
4-001
1
-1
-1
1
-i
i
i
-i
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
9
(
1 0 0
0 1 0
0 0 1
)
(
-1 0
0 -1
)
d1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
10
(
-1 0 0
0 -1 0
0 0 1
)
(
i 0
0 -i
)
d2001
1
1
1
1
-1
-1
-1
-1
i
i
-i
-i
i
i
-i
-i
11
(
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
i
i
-i
-i
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
12
(
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
-i
-i
i
i
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
13
(
-1 0 0
0 -1 0
0 0 -1
)
(
-1 0
0 -1
)
d1
1
-1
1
-1
1
-1
1
-1
-1
-1
-1
-1
1
1
1
1
14
(
1 0 0
0 1 0
0 0 -1
)
(
i 0
0 -i
)
dm001
1
-1
1
-1
-1
1
-1
1
i
i
-i
-i
-i
-i
i
i
15
(
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
i
-i
-i
i
e-iπ/4
ei3π/4
eiπ/4
e-i3π/4
ei3π/4
e-iπ/4
e-i3π/4
eiπ/4
16
(
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
-i
i
i
-i
eiπ/4
e-i3π/4
e-iπ/4
ei3π/4
e-i3π/4
eiπ/4
ei3π/4
e-iπ/4
k-Subgroupsmag
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