| Name: | $\mathrm{SU}(2)\times\mathrm{U}(1)_2$ |
| $\mathbb{R}$-dimension: | $4$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
| Hodge circle: | $u\mapsto\mathrm{diag}(u,\bar u, u, u,\bar u, \bar u)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$9$ |
$0$ |
$146$ |
$0$ |
$2965$ |
$0$ |
$68334$ |
$0$ |
$1707930$ |
$0$ |
$45157860$ |
| $a_2$ |
$1$ |
$5$ |
$35$ |
$299$ |
$2899$ |
$30495$ |
$339085$ |
$3924485$ |
$46820435$ |
$572068847$ |
$7125843673$ |
$90186907585$ |
$1156824665725$ |
| $a_3$ |
$1$ |
$0$ |
$50$ |
$0$ |
$7436$ |
$0$ |
$1543940$ |
$0$ |
$377596884$ |
$0$ |
$101844783768$ |
$0$ |
$29321320245360$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$5$ |
$9$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$35$ |
$20$ |
$69$ |
$146$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$50$ |
$299$ |
$174$ |
$627$ |
$368$ |
$1354$ |
$2965$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$458$ |
$2899$ |
$1680$ |
$980$ |
$6291$ |
$3652$ |
$13838$ |
$8020$ |
$30665$ |
$68334$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$4616$ |
$30495$ |
$2672$ |
$17528$ |
$10108$ |
$67515$ |
$38776$ |
$22330$ |
$150518$ |
$86310$ |
$337095$ |
$192976$ |
$757638$ |
$1707930$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$7436$ |
$49652$ |
$339085$ |
$28476$ |
$193308$ |
$110448$ |
$760415$ |
$63240$ |
$432920$ |
$246962$ |
$1712166$ |
$973312$ |
$554380$ |
$3866823$ |
| $$ |
$2195004$ |
$8754858$ |
$4962888$ |
$19864530$ |
$45157860$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&4&0&4&0&7&10&0&4&0&15&0&0&24\\0&9&0&11&0&40&0&0&45&0&28&0&66&105&0\\4&0&26&0&30&0&68&80&0&46&0&140&0&0&256\\0&11&0&19&0&64&0&0&85&0&42&0&128&187&0\\4&0&30&0&43&0&87&100&0&64&0&196&0&0&344\\0&40&0&64&0&240&0&0&320&0&176&0&496&736&0\\7&0&68&0&87&0&228&256&0&180&0&481&0&0&968\\10&0&80&0&100&0&256&304&0&210&0&556&0&0&1136\\0&45&0&85&0&320&0&0&485&0&230&0&742&1101&0\\4&0&46&0&64&0&180&210&0&170&0&396&0&0&880\\0&28&0&42&0&176&0&0&230&0&147&0&377&560&0\\15&0&140&0&196&0&481&556&0&396&0&1099&0&0&2176\\0&66&0&128&0&496&0&0&742&0&377&0&1187&1732&0\\0&105&0&187&0&736&0&0&1101&0&560&0&1732&2603&0\\24&0&256&0&344&0&968&1136&0&880&0&2176&0&0&4768\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&9&26&19&43&240&228&304&485&170&147&1099&1187&2603&4768&2123&2205&4909&3824&965\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
