Ih 점군
비가환군, 10(32) 기약 표현Ih 점군에 대한 지표표
| E | 12C5 | 12(C5)2 | 20C3 | 15C2 | i | 12S10 | 12(S10)3 | 20S6 | 15σ | linear, rotations | quadratic | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ag | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | x2+y2+z2 | |
| T1g | 3 | -2cos(4π/5) | -2cos(2π/5) | 0 | -1 | 3 | -2cos(2π/5) | -2cos(4π/5) | 0 | -1 | (Rx, Ry, Rz) | |
| T2g | 3 | -2cos(2π/5) | -2cos(4π/5) | 0 | -1 | 3 | -2cos(4π/5) | -2cos(2π/5) | 0 | -1 | ||
| Gg | 4 | -1 | -1 | 1 | 0 | 4 | -1 | -1 | 1 | 0 | ||
| Hg | 5 | 0 | 0 | -1 | 1 | 5 | 0 | 0 | -1 | 1 | [2z2-x2-y2, x2-y2, xy, xz, yz] | |
| Au | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | ||
| T1u | 3 | -2cos(4π/5) | -2cos(2π/5) | 0 | -1 | -3 | 2cos(2π/5) | 2cos(4π/5) | 0 | 1 | (x, y, z) | |
| T2u | 3 | -2cos(2π/5) | -2cos(4π/5) | 0 | -1 | -3 | 2cos(4π/5) | 2cos(2π/5) | 0 | 1 | ||
| Gu | 4 | -1 | -1 | 1 | 0 | -4 | 1 | 1 | -1 | 0 | ||
| Hu | 5 | 0 | 0 | -1 | 1 | -5 | 0 | 0 | 1 | -1 |
Ih 점군의 곱셈표
| Ag | T1g | T2g | Gg | Hg | Au | T1u | T2u | Gu | Hu | |
|---|---|---|---|---|---|---|---|---|---|---|
| Ag | Ag | T1g | T2g | Gg | Hg | Au | T1u | T2u | Gu | Hu |
| T1g | T1g | Ag+T1g+Hg | Gg+Hg | T2g+Gg+Hg | T1g+T2g+Gg+Hg | T1u | Au+T1u+Hu | Gu+Hu | T2u+Gu+Hu | T1u+T2u+Gu+Hu |
| T2g | T2g | Gg+Hg | Ag+T2g+Hg | T1g+Gg+Hg | T1g+T2g+Gg+Hg | T2u | Gu+Hu | Au+T2u+Hu | T1u+Gu+Hu | T1u+T2u+Gu+Hu |
| Gg | Gg | T2g+Gg+Hg | T1g+Gg+Hg | Ag+T1g+T2g+Gg+Hg | T1g+T2g+Gg+2Hg | Gu | T2u+Gu+Hu | T1u+Gu+Hu | Au+T1u+T2u+Gu+Hu | T1u+T2u+Gu+2Hu |
| Hg | Hg | T1g+T2g+Gg+Hg | T1g+T2g+Gg+Hg | T1g+T2g+Gg+2Hg | Ag+T1g+T2g+2Gg+2Hg | Hu | T1u+T2u+Gu+Hu | T1u+T2u+Gu+Hu | T1u+T2u+Gu+2Hu | Au+T1u+T2u+2Gu+2Hu |
| Au | Au | T1u | T2u | Gu | Hu | Ag | T1g | T2g | Gg | Hg |
| T1u | T1u | Au+T1u+Hu | Gu+Hu | T2u+Gu+Hu | T1u+T2u+Gu+Hu | T1g | Ag+T1g+Hg | Gg+Hg | T2g+Gg+Hg | T1g+T2g+Gg+Hg |
| T2u | T2u | Gu+Hu | Au+T2u+Hu | T1u+Gu+Hu | T1u+T2u+Gu+Hu | T2g | Gg+Hg | Ag+T2g+Hg | T1g+Gg+Hg | T1g+T2g+Gg+Hg |
| Gu | Gu | T2u+Gu+Hu | T1u+Gu+Hu | Au+T1u+T2u+Gu+Hu | T1u+T2u+Gu+2Hu | Gg | T2g+Gg+Hg | T1g+Gg+Hg | Ag+T1g+T2g+Gg+Hg | T1g+T2g+Gg+2Hg |
| Hu | Hu | T1u+T2u+Gu+Hu | T1u+T2u+Gu+Hu | T1u+T2u+Gu+2Hu | Au+T1u+T2u+2Gu+2Hu | Hg | T1g+T2g+Gg+Hg | T1g+T2g+Gg+Hg | T1g+T2g+Gg+2Hg | Ag+T1g+T2g+2Gg+2Hg |
| C1 | Cs | Ci | ||
| C2 | C3 | C4 | C5 | C6 |
| C2v | C3v | C4v | C5v | C6v |
| C2h | C3h | C4h | C5h | C6h |
| D2 | D3 | D4 | D5 | D6 |
| D2h | D3h | D4h | D5h | D6h |
| D2d | D3d | D4d | D5d | D6d |
| S4 | S6 | S8 | S10 | |
| Td | Oh | Ih | ||
| C∞v | D∞h |
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