| Name: | $\mathrm{U}(1)^2\times\mathrm{SU}(2)$ |
| $\mathbb{R}$-dimension: | $5$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A,B\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),C\in\mathrm{SU}(2)\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0&0\\0&J_2&0\\0&0&J_2\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
| Hodge circle: | $u\mapsto\mathrm{diag}(u,\bar u, u, \bar u, u, \bar u)$ |
| Name: | $C_2$ |
| Order: | $2$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$41$ |
$0$ |
$620$ |
$0$ |
$11739$ |
$0$ |
$255885$ |
$0$ |
$6115626$ |
| $a_2$ |
$1$ |
$3$ |
$14$ |
$84$ |
$627$ |
$5493$ |
$53714$ |
$565512$ |
$6264757$ |
$71996631$ |
$850720122$ |
$10275024576$ |
$126341748135$ |
| $a_3$ |
$1$ |
$0$ |
$18$ |
$0$ |
$1460$ |
$0$ |
$228960$ |
$0$ |
$47862780$ |
$0$ |
$11549742624$ |
$0$ |
$3041666585760$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$14$ |
$8$ |
$22$ |
$41$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$18$ |
$84$ |
$50$ |
$153$ |
$94$ |
$303$ |
$620$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$118$ |
$627$ |
$374$ |
$228$ |
$1251$ |
$754$ |
$2598$ |
$1560$ |
$5490$ |
$11739$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$944$ |
$5493$ |
$568$ |
$3246$ |
$1936$ |
$11553$ |
$6832$ |
$4062$ |
$24756$ |
$14590$ |
$53572$ |
$31458$ |
$116753$ |
$255885$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1460$ |
$8628$ |
$53714$ |
$5100$ |
$31346$ |
$18384$ |
$116417$ |
$10828$ |
$67812$ |
$39642$ |
$254651$ |
$147900$ |
$86184$ |
$560290$ |
| $$ |
$324506$ |
$1238391$ |
$715386$ |
$2747535$ |
$6115626$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&1&0&2&3&0&1&0&2&0&0&4\\0&4&0&4&0&10&0&0&10&0&5&0&11&17&0\\2&0&9&0&7&0&15&17&0&8&0&23&0&0&40\\0&4&0&6&0&14&0&0&16&0&7&0&21&29&0\\1&0&7&0&11&0&18&17&0&12&0&34&0&0&52\\0&10&0&14&0&47&0&0&54&0&30&0&78&110&0\\2&0&15&0&18&0&42&42&0&29&0&76&0&0&140\\3&0&17&0&17&0&42&53&0&30&0&82&0&0&160\\0&10&0&16&0&54&0&0&76&0&35&0&107&155&0\\1&0&8&0&12&0&29&30&0&28&0&57&0&0&120\\0&5&0&7&0&30&0&0&35&0&26&0&58&78&0\\2&0&23&0&34&0&76&82&0&57&0&164&0&0&296\\0&11&0&21&0&78&0&0&107&0&58&0&170&234&0\\0&17&0&29&0&110&0&0&155&0&78&0&234&349&0\\4&0&40&0&52&0&140&160&0&120&0&296&0&0&620\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&6&11&47&42&53&76&28&26&164&170&349&620&273&278&592&456&115\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
