| S8 | E | S8 | C4 | S8³ | C2 | S8⁵ | C4³ | S8⁷ | 线性 / 转动linear, rotations | 二次quadratic |
|---|---|---|---|---|---|---|---|---|---|---|
| A | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | Rz | x²+y², z² |
| B | 1 | −1 | 1 | −1 | 1 | −1 | 1 | −1 | z | |
| E1 | 12 | ε√2 | i0 | −ε*−√2 | −1−2 | −ε−√2 | −i0 | ε*√2 | (x, y), (Rx, Ry) | |
| ↳ 共轭conj. | 1 | ε* | −i | −ε | −1 | −ε* | i | ε | ||
| E2 | 12 | i0 | −1−2 | −i0 | 12 | i0 | −1−2 | −i0 | (x²−y², xy) | |
| ↳ 共轭conj. | 1 | −i | −1 | i | 1 | −i | −1 | i | ||
| E3 | 12 | −ε*−√2 | −i0 | ε√2 | −1−2 | ε*√2 | i0 | −ε−√2 | (xz, yz) | |
| ↳ 共轭conj. | 1 | −ε | i | ε* | −1 | ε | −i | −ε* |
\(\varepsilon = \exp(2\pi i/8)\)，\(\varepsilon^{*}\) 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 | 8 |
|---|---|
| 类数Classes | 8 |
| 所属族Family | Sn |
| 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.