| Name: | $\mathrm{U}(1)\times\mathrm{SU}(2)^2$ |
| $\mathbb{R}$-dimension: | $7$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B,C\in\mathrm{SU}(2)\right\}$ |
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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$23$ |
$0$ |
$295$ |
$0$ |
$4984$ |
$0$ |
$97734$ |
$0$ |
$2105004$ |
| $a_2$ |
$1$ |
$2$ |
$8$ |
$44$ |
$308$ |
$2507$ |
$22530$ |
$216967$ |
$2199888$ |
$23221661$ |
$253228082$ |
$2836717798$ |
$32505982085$ |
| $a_3$ |
$1$ |
$0$ |
$10$ |
$0$ |
$698$ |
$0$ |
$90780$ |
$0$ |
$15599430$ |
$0$ |
$3162206628$ |
$0$ |
$716190214872$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$4$ |
$12$ |
$23$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$10$ |
$44$ |
$26$ |
$77$ |
$46$ |
$147$ |
$295$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$60$ |
$308$ |
$184$ |
$114$ |
$590$ |
$358$ |
$1180$ |
$710$ |
$2405$ |
$4984$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$456$ |
$2507$ |
$274$ |
$1496$ |
$900$ |
$5065$ |
$3030$ |
$1826$ |
$10454$ |
$6240$ |
$21837$ |
$12978$ |
$46018$ |
$97734$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$698$ |
$3870$ |
$22530$ |
$2316$ |
$13340$ |
$7934$ |
$47007$ |
$4728$ |
$27802$ |
$16490$ |
$99240$ |
$58540$ |
$34644$ |
$211171$ |
| $$ |
$124236$ |
$452179$ |
$265272$ |
$973350$ |
$2105004$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&0&2&0&1&0&1&0&0&2\\0&3&0&1&0&5&0&0&5&0&3&0&3&9&0\\1&0&5&0&3&0&8&8&0&3&0&10&0&0&18\\0&1&0&5&0&7&0&0&7&0&1&0&13&11&0\\1&0&3&0&7&0&6&9&0&8&0&14&0&0&22\\0&5&0&7&0&22&0&0&25&0&12&0&32&45&0\\0&0&8&0&6&0&24&17&0&9&0&31&0&0&56\\2&0&8&0&9&0&17&26&0&16&0&32&0&0&62\\0&5&0&7&0&25&0&0&35&0&15&0&39&61&0\\1&0&3&0&8&0&9&16&0&17&0&22&0&0&44\\0&3&0&1&0&12&0&0&15&0&13&0&17&30&0\\1&0&10&0&14&0&31&32&0&22&0&61&0&0&106\\0&3&0&13&0&32&0&0&39&0&17&0&73&78&0\\0&9&0&11&0&45&0&0&61&0&30&0&78&125&0\\2&0&18&0&22&0&56&62&0&44&0&106&0&0&210\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&5&5&7&22&24&26&35&17&13&61&73&125&210&99&92&191&135&45\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
