| Name: | $\mathrm{U}(1)\times\mathrm{U}(1)$ |
| $\mathbb{R}$-dimension: | $2$ |
| Description: | $\left\{\begin{bmatrix}A&0\\0&B\end{bmatrix}:A,B\in \mathrm{U}(1)\subseteq\mathrm{SU}(2)\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0\\0&J_2\end{bmatrix}, J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
| Hodge circle: | $u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$21$ |
$0$ |
$210$ |
$0$ |
$2485$ |
$0$ |
$31878$ |
$0$ |
$427350$ |
| $a_2$ |
$1$ |
$2$ |
$6$ |
$20$ |
$82$ |
$372$ |
$1824$ |
$9312$ |
$48850$ |
$260804$ |
$1410736$ |
$7708032$ |
$42460840$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$6$ |
$10$ |
$21$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$20$ |
$40$ |
$90$ |
$210$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$82$ |
$180$ |
$424$ |
$1020$ |
$2485$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$372$ |
$872$ |
$2112$ |
$5180$ |
$12810$ |
$31878$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$1824$ |
$4416$ |
$10884$ |
$27040$ |
$67564$ |
$169596$ |
$427350$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&1&0&1&0&1&0&1\\0&3&0&0&4&0&4&0&5&0\\1&0&3&3&0&3&0&7&0&5\\1&0&3&7&0&5&0&13&0&7\\0&4&0&0&12&0&12&0&20&0\\1&0&3&5&0&9&0&15&0&11\\0&4&0&0&12&0&15&0&21&0\\1&0&7&13&0&15&0&37&0&25\\0&5&0&0&20&0&21&0&42&0\\1&0&5&7&0&11&0&25&0&21\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&3&7&12&9&15&37&42&21\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
