| 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: | $\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$ |
| Name: | $C_2$ |
| Order: | $2$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & -i \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$46$ |
$0$ |
$840$ |
$0$ |
$18662$ |
$0$ |
$458136$ |
$0$ |
$11976492$ |
| $a_2$ |
$1$ |
$2$ |
$13$ |
$99$ |
$907$ |
$9188$ |
$99515$ |
$1129418$ |
$13270603$ |
$160175358$ |
$1975265743$ |
$24790641351$ |
$315730866109$ |
| $a_3$ |
$1$ |
$0$ |
$20$ |
$0$ |
$2356$ |
$0$ |
$456720$ |
$0$ |
$107764356$ |
$0$ |
$28401536976$ |
$0$ |
$8045064418512$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$13$ |
$6$ |
$22$ |
$46$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$20$ |
$99$ |
$58$ |
$194$ |
$110$ |
$396$ |
$840$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$150$ |
$907$ |
$524$ |
$312$ |
$1882$ |
$1094$ |
$3998$ |
$2300$ |
$8588$ |
$18662$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1448$ |
$9188$ |
$830$ |
$5284$ |
$3046$ |
$19736$ |
$11332$ |
$6544$ |
$42890$ |
$24572$ |
$93928$ |
$53606$ |
$206892$ |
$458136$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2356$ |
$15006$ |
$99515$ |
$8634$ |
$56790$ |
$32488$ |
$218306$ |
$18588$ |
$124324$ |
$70916$ |
$482154$ |
$273928$ |
$155964$ |
$1070272$ |
| $$ |
$606746$ |
$2385710$ |
$1349460$ |
$5336820$ |
$11976492$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&5&0&4&0&3&0&0&8\\0&4&0&2&0&12&0&0&18&0&8&0&14&32&0\\1&0&10&0&7&0&24&23&0&13&0&38&0&0&80\\0&2&0&12&0&20&0&0&22&0&6&0&50&50&0\\2&0&7&0&15&0&21&34&0&31&0&49&0&0&104\\0&12&0&20&0&70&0&0&100&0&44&0&140&208&0\\1&0&24&0&21&0&80&69&0&43&0&136&0&0&288\\5&0&23&0&34&0&69&102&0&81&0&150&0&0&336\\0&18&0&22&0&100&0&0&160&0&72&0&198&328&0\\4&0&13&0&31&0&43&81&0&82&0&116&0&0&264\\0&8&0&6&0&44&0&0&72&0&40&0&86&152&0\\3&0&38&0&49&0&136&150&0&116&0&287&0&0&616\\0&14&0&50&0&140&0&0&198&0&86&0&368&464&0\\0&32&0&50&0&208&0&0&328&0&152&0&464&732&0\\8&0&80&0&104&0&288&336&0&264&0&616&0&0&1376\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&10&12&15&70&80&102&160&82&40&287&368&732&1376&680&649&1520&1188&396\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
