| C5h | E | C5 | C5² | C5³ | C5⁴ | σh | S5 | S5⁷ | S5³ | S5⁹ | 线性 / 转动linear, rotations | 二次quadratic |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A′ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | Rz | x²+y², z² |
| E′1 | 12 | ε2cos72° | ε²2cos144° | ε²*2cos144° | ε*2cos72° | 12 | ε2cos72° | ε²2cos144° | ε²*2cos144° | ε*2cos72° | (x, y) | |
| ↳ 共轭conj. | 1 | ε* | ε²* | ε² | ε | 1 | ε* | ε²* | ε² | ε | ||
| E′2 | 12 | ε²2cos144° | ε*2cos72° | ε2cos72° | ε²*2cos144° | 12 | ε²2cos144° | ε*2cos72° | ε2cos72° | ε²*2cos144° | (x²−y², xy) | |
| ↳ 共轭conj. | 1 | ε²* | ε | ε* | ε² | 1 | ε²* | ε | ε* | ε² | ||
| A″ | 1 | 1 | 1 | 1 | 1 | −1 | −1 | −1 | −1 | −1 | z | |
| E″1 | 12 | ε2cos72° | ε²2cos144° | ε²*2cos144° | ε*2cos72° | −1−2 | −ε−2cos72° | −ε²−2cos144° | −ε²*−2cos144° | −ε*−2cos72° | (Rx, Ry) | (xz, yz) |
| ↳ 共轭conj. | 1 | ε* | ε²* | ε² | ε | −1 | −ε* | −ε²* | −ε² | −ε | ||
| E″2 | 12 | ε²2cos144° | ε*2cos72° | ε2cos72° | ε²*2cos144° | −1−2 | −ε²−2cos144° | −ε*−2cos72° | −ε−2cos72° | −ε²*−2cos144° | ||
| ↳ 共轭conj. | 1 | ε²* | ε | ε* | ε² | −1 | −ε²* | −ε | −ε* | −ε² |
\(\varepsilon = \exp(2\pi i/5)\)，\(\varepsilon^{*}\) 为其复共轭。合并后每对复共轭不可约表示相加成一个二维实表示（物理上简并）。; \(\varepsilon^{*}\) is its complex conjugate. Merging adds each conjugate pair into one two-dimensional real representation (physically degenerate).
| 阶 hOrder h | 10 |
|---|---|
| 类数Classes | 10 |
| 所属族Family | Cnh |
| 代表分子Examples | 少见rare |
约化可约表示Reduce a representation
输入可约表示在各类上的特征标，按约化公式 \(n_i = \frac{1}{h}\sum_R g_R\,\chi(R)\,\chi_i(R)^{*}\) 求出各不可约表示出现的次数。
Enter the characters of a reducible representation; the reduction formula gives how many times each irreducible representation occurs.