Reduction formula for point group D6h
Characters of input representation
E |
2C6 (z) |
2C3 |
C2 |
3C'2 |
3C''2 |
i |
2S3 |
2S6 |
h (xy) |
3d |
3v |
6 |
0 |
0 |
0 |
-2 |
0 |
0 |
0 |
0 |
-6 |
0 |
2 |
Decomposition into Irreducible representations
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
Symmetric Powers of Representation
Characters of symmetric powers
Tensor Order |
E |
2C6 (z) |
2C3 |
C2 |
3C'2 |
3C''2 |
i |
2S3 |
2S6 |
h (xy) |
3d |
3v |
1 |
6 |
0 |
0 |
0 |
-2 |
0 |
0 |
0 |
0 |
-6 |
0 |
2 |
2 |
21 |
0 |
0 |
3 |
5 |
3 |
3 |
0 |
0 |
21 |
3 |
5 |
3 |
56 |
0 |
2 |
0 |
-8 |
0 |
0 |
-2 |
0 |
-56 |
0 |
8 |
4 |
126 |
0 |
0 |
6 |
14 |
6 |
6 |
0 |
0 |
126 |
6 |
14 |
5 |
252 |
0 |
0 |
0 |
-20 |
0 |
0 |
0 |
0 |
-252 |
0 |
20 |
6 |
462 |
1 |
3 |
10 |
30 |
10 |
10 |
3 |
1 |
462 |
10 |
30 |
Decomposition into Irreducible representations
Tensor Order |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
1 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
2 |
4 |
0 |
0 |
0 |
0 |
4 |
0 |
0 |
2 |
1 |
3 |
0 |
3 |
0 |
0 |
3 |
7 |
9 |
0 |
3 |
7 |
0 |
0 |
0 |
9 |
4 |
16 |
6 |
0 |
0 |
0 |
22 |
0 |
0 |
12 |
8 |
20 |
0 |
5 |
0 |
0 |
16 |
26 |
42 |
0 |
16 |
26 |
0 |
0 |
0 |
42 |
6 |
50 |
30 |
0 |
0 |
0 |
78 |
0 |
0 |
43 |
33 |
75 |
0 |
Antisymmetric Powers of Representation
Characters of antisymmetric powers
Tensor Order |
E |
2C6 (z) |
2C3 |
C2 |
3C'2 |
3C''2 |
i |
2S3 |
2S6 |
h (xy) |
3d |
3v |
1 | 6 |
0 |
0 |
0 |
-2 |
0 |
0 |
0 |
0 |
-6 |
0 |
2 |
2 | 15 |
0 |
0 |
-3 |
-1 |
-3 |
-3 |
0 |
0 |
15 |
-3 |
-1 |
3 | 20 |
0 |
2 |
0 |
4 |
0 |
0 |
-2 |
0 |
-20 |
0 |
-4 |
4
| 15 |
0 |
0 |
3 |
-1 |
3 |
3 |
0 |
0 |
15 |
3 |
-1 |
5
| 6 |
0 |
0 |
0 |
-2 |
0 |
0 |
0 |
0 |
-6 |
0 |
2 |
6
*
| 1 |
-1 |
1 |
-1 |
1 |
-1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
* Tensor rank equal tensor order
Decomposition into Irreducible representations
Tensor Order |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
1
| 0 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
2
| 0 |
2 |
0 |
0 |
0 |
2 |
0 |
0 |
2 |
1 |
3 |
0 |
3
| 0 |
0 |
3 |
1 |
3 |
0 |
3 |
1 |
0 |
0 |
0 |
3 |
4
| 2 |
1 |
0 |
0 |
0 |
3 |
0 |
0 |
0 |
2 |
2 |
0 |
5
| 0 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
6
*
| 0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
* Tensor rank equal tensor order
Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement