Multiplication Table of irreducible representations of the group D2(222)
D2(222)
A
B1
B2
B3
A
A
B1
B2
B3
B1
·
A
B3
B2
B2
·
·
A
B1
B3
·
·
·
A
[ Note: the table is symmetric ]
Symmetrized Products of Irreps
D2(222)
A
B1
B2
B3
[A x A]
1
·
·
·
[B1 x B1]
1
·
·
·
[B2 x B2]
1
·
·
·
[B3 x B3]
1
·
·
·
Antisymmetrized Products of Irreps
D2(222)
A
B1
B2
B3
{A x A}
·
·
·
·
{B1 x B1}
·
·
·
·
{B2 x B2}
·
·
·
·
{B3 x B3}
·
·
·
·
Irreps Decompositions
D2(222)
A
B1
B2
B3
V
·
1
1
1
[V2]
3
1
1
1
[V3]
1
3
3
3
[V4]
6
3
3
3
A
·
1
1
1
[A2]
3
1
1
1
[A3]
1
3
3
3
[A4]
6
3
3
3
[V2]xV
3
5
5
5
[[V2]2]
9
4
4
4
{V2}
·
1
1
1
{A2}
·
1
1
1
{[V2]2}
3
4
4
4
V ≡ the vector representation
A ≡ the axial representation
IR Selection Rules
IR
A
B1
B2
B3
A
·
x
x
x
B1
x
·
x
x
B2
x
x
·
x
B3
x
x
x
·
[ Note: x means allowed ]
Raman Selection Rules
Raman
A
B1
B2
B3
A
x
x
x
x
B1
x
x
x
x
B2
x
x
x
x
B3
x
x
x
x
[ Note: x means allowed ]
Irreps
Dimensions
Irreps of the point group
Subduction of the rotation group D(L) to irreps of the group D2(222)
L
2L+1
A
B1
B2
B3
0
1
1
·
·
·
1
3
·
1
1
1
2
5
2
1
1
1
3
7
1
2
2
2
4
9
3
2
2
2
5
11
2
3
3
3
6
13
4
3
3
3
7
15
3
4
4
4
8
17
5
4
4
4
9
19
4
5
5
5
10
21
6
5
5
5
*
C. J. Bradley and A. P. Cracknell (1972)
The Mathematical Theory of Symmetry in Solids
Clarendon Press - Oxford
*
Simon L. Altmann and Peter Herzig (1994).
Point-Group Theory Tables.
Oxford Science Publications.
Point Group Tables
