群表示论备忘
Character is a function of class
Dimensions of the irreducible representations:
$$ \begin{align} \sum_r d_r^2 = N(G) \end{align} $$Column orthogonality:
$$ \begin{align} \sum_c n_c (\chi^{(r)}(c))^* \, \chi^{(s)}(c) = N(G) \delta^{rs} \end{align} $$Row orthogonality:
$$ \begin{align} \sum_r \chi^{(r)} (c)^* \, \chi^{(r)}(c') = \frac{N(G)}{n_c} \delta^{c c'} \end{align} $$The character table is square:
$$ \begin{align} N(C) = N(R) \end{align} $$Great orthogonality theorem
$$ \begin{align} \sum_g D^{(r) \dagger}(g)^i_{\, j} D^{(s)} (g)^k_{\, l} = \frac{N(G)}{d_r} \delta^{rs} \delta^i_{\,l}\delta^k_{\,j} \end{align} $$Character table
Cyclic groups $Z_n$
$Z_n$ 是 Abel 群。
Abel 群的等价类的个数与群元的个数相同。
Abel 群所有的不可约表示都是一维的。
不可约表示的个数又与类的个数相同。
所以 $Z_n$ 有
如 $Z_3$ 的群元 $\{I, g, g^2\}$，特征值表为
$$ \begin{array}{c|ccc} \hline Z_3 & I & g & g^2 \\ \hline 1 & 1 & 1 & 1 \\ 1' & 1 & \omega & \omega^2 \\ 1' & 1 & \omega^2 & \omega ,\\ \end{array} $$其中 $\omega=e^{2\pi i /3}$ 。
Even permutation group $A_3$
$$ \begin{array}{ccc|ccc} \hline A_3 & n_c & & 1 & 1' & 1'' \\ \hline & 1 & I & 1 & 1 & 1 \\ Z_3 & 1 & c=(123) & 1 & \omega & \omega^* \\ Z_3 & 1 & a=(132) & 1 & \omega^* & \omega \\ \hline \end{array} $$Even permutation group $A_4$
$$ \begin{array}{ccc|cccc} \hline A_4 & n_c & c & 1 & 1' & 1'' & 3 \\ \hline & 1 & I & 1 & 1 & 1 & 3 \\ Z_2 & 3 & (12)(34) & 1 & 1 & 1 & -1 \\ Z_3 & 4 & (123) & 1 & \omega & \omega^* & 0 \\ Z_3 & 4 & (132) & 1 & \omega^* & \omega & 0 \\ \hline \end{array} $$Permutation group $S_3$
$$ \begin{array}{ccc|ccc} \hline S_3 & n_c & & 1 & \bar{1} & 2 \\ \hline & 1 & I & 1 & 1 & 2 \\ Z_3 & 2 & (123),(132) & 1 & 1 & -1_x \\ Z_2 & 3 & (12),(23),(31) & 1 & -1 & 0_y \\ \hline \end{array} $$Permutation group $S_4$
$$ \begin{array}{ccc|ccccc} \hline S_4 & n_c & & 1 & \bar{1} & 2 & 3 & \bar{3} \\ \hline & 1 & I & 1 & 1 & 2 & 3 & 3 \\ Z_2 & 3 & (12)(34) & 1 & 1 & 2 & -1 & -1 \\ Z_3 & 8 & (123) & 1 & 1 & -1 & 0 & 0 \\ Z_2 & 6 & (12) & 1 & -1 & 0 & 1 & -1 \\ Z_4 & 6 & (1234) & 1 & -1 & 0 & -1 & 1 \\ \hline \end{array} $$Reference
- 2016 - A. Zee - Group Theory in a Nutshell for Physicists (2016, Princeton University Press)