| Name: | $\mathrm{U}(1)\times\mathrm{SU}(2)_2$ |
| $\mathbb{R}$-dimension: | $4$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B\in\mathrm{SU}(2)\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
| Hodge circle: | $\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$6$ |
$0$ |
$86$ |
$0$ |
$1660$ |
$0$ |
$37254$ |
$0$ |
$916020$ |
$0$ |
$23952060$ |
| $a_2$ |
$1$ |
$4$ |
$25$ |
$197$ |
$1811$ |
$18370$ |
$199015$ |
$2258800$ |
$26541115$ |
$320350484$ |
$3950530883$ |
$49581281117$ |
$631461728005$ |
| $a_3$ |
$1$ |
$0$ |
$36$ |
$0$ |
$4676$ |
$0$ |
$913040$ |
$0$ |
$215523812$ |
$0$ |
$56803010448$ |
$0$ |
$16090127983248$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$4$ |
$6$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$25$ |
$14$ |
$44$ |
$86$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$36$ |
$197$ |
$114$ |
$386$ |
$226$ |
$792$ |
$1660$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$302$ |
$1811$ |
$1048$ |
$612$ |
$3762$ |
$2182$ |
$7990$ |
$4620$ |
$17176$ |
$37254$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$2892$ |
$18370$ |
$1678$ |
$10564$ |
$6098$ |
$39466$ |
$22664$ |
$13048$ |
$85774$ |
$49124$ |
$187836$ |
$107282$ |
$413784$ |
$916020$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$4676$ |
$30010$ |
$199015$ |
$17250$ |
$113574$ |
$64964$ |
$436600$ |
$37236$ |
$248636$ |
$141852$ |
$964290$ |
$547856$ |
$311788$ |
$2140524$ |
| $$ |
$1213422$ |
$4771350$ |
$2699172$ |
$10673640$ |
$23952060$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&3&0&2&0&5&7&0&4&0&7&0&0&16\\0&6&0&8&0&24&0&0&32&0&12&0&38&60&0\\3&0&18&0&17&0&44&51&0&33&0&74&0&0&160\\0&8&0&14&0&40&0&0&56&0&22&0&74&110&0\\2&0&17&0&23&0&53&58&0&45&0&101&0&0&208\\0&24&0&40&0&140&0&0&200&0&88&0&280&416&0\\5&0&44&0&53&0&140&155&0&117&0&268&0&0&576\\7&0&51&0&58&0&155&186&0&133&0&306&0&0&672\\0&32&0&56&0&200&0&0&302&0&132&0&430&644&0\\4&0&33&0&45&0&117&133&0&110&0&240&0&0&528\\0&12&0&22&0&88&0&0&132&0&68&0&202&292&0\\7&0&74&0&101&0&268&306&0&240&0&571&0&0&1232\\0&38&0&74&0&280&0&0&430&0&202&0&656&960&0\\0&60&0&110&0&416&0&0&644&0&292&0&960&1450&0\\16&0&160&0&208&0&576&672&0&528&0&1232&0&0&2752\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&18&14&23&140&140&186&302&110&68&571&656&1450&2752&1262&1269&2880&2288&590\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
