| Name: | $\mathrm{U}(1)\times\mathrm{SU}(2)$ |
| $\mathbb{R}$-dimension: | $4$ |
| Description: | $\left\{\begin{bmatrix}A&0\\0&B\end{bmatrix}:A\in \mathrm{U}(1)\subseteq\mathrm{SU}(2),\ B\in\mathrm{SU}(2)\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0\\0&J_2\end{bmatrix}, J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
| Hodge circle: | $u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$20$ |
$0$ |
$175$ |
$0$ |
$1764$ |
$0$ |
$19404$ |
$0$ |
$226512$ |
| $a_2$ |
$1$ |
$2$ |
$6$ |
$20$ |
$76$ |
$312$ |
$1364$ |
$6232$ |
$29460$ |
$142952$ |
$708328$ |
$3570096$ |
$18251248$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$6$ |
$10$ |
$20$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$20$ |
$38$ |
$80$ |
$175$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$76$ |
$156$ |
$342$ |
$770$ |
$1764$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$312$ |
$678$ |
$1532$ |
$3528$ |
$8232$ |
$19404$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$1364$ |
$3076$ |
$7112$ |
$16672$ |
$39480$ |
$94248$ |
$226512$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&1&0&1&0&1&0&1\\0&3&0&0&4&0&3&0&4&0\\1&0&3&3&0&3&0&5&0&3\\1&0&3&6&0&4&0&9&0&4\\0&4&0&0&10&0&8&0&12&0\\1&0&3&4&0&6&0&9&0&6\\0&3&0&0&8&0&10&0&11&0\\1&0&5&9&0&9&0&21&0&11\\0&4&0&0&12&0&11&0&21&0\\1&0&3&4&0&6&0&11&0&10\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&3&6&10&6&10&21&21&10\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2$ and $n\in\mathbb{Z}$.
