| Name: | $\mathrm{SU}(2)\times\mathrm{SU}(2)_2$ |
| $\mathbb{R}$-dimension: | $6$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A,B\in\mathrm{SU}(2)\right\}$ |
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: | $D_4$ |
| Order: | $8$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$14$ |
$0$ |
$175$ |
$0$ |
$2898$ |
$0$ |
$56364$ |
$0$ |
$1221132$ |
| $a_2$ |
$1$ |
$2$ |
$7$ |
$32$ |
$197$ |
$1502$ |
$13185$ |
$126786$ |
$1298603$ |
$13939142$ |
$155185637$ |
$1779064344$ |
$20890616499$ |
| $a_3$ |
$1$ |
$0$ |
$7$ |
$0$ |
$412$ |
$0$ |
$52590$ |
$0$ |
$9337524$ |
$0$ |
$1993420788$ |
$0$ |
$478001349936$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$7$ |
$3$ |
$8$ |
$14$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$7$ |
$32$ |
$17$ |
$48$ |
$29$ |
$89$ |
$175$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$38$ |
$197$ |
$112$ |
$69$ |
$351$ |
$213$ |
$693$ |
$420$ |
$1405$ |
$2898$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$273$ |
$1502$ |
$164$ |
$879$ |
$528$ |
$2948$ |
$1761$ |
$1061$ |
$6049$ |
$3610$ |
$12606$ |
$7497$ |
$26544$ |
$56364$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$412$ |
$2258$ |
$13185$ |
$1348$ |
$7736$ |
$4588$ |
$27221$ |
$2733$ |
$16047$ |
$9498$ |
$57346$ |
$33727$ |
$19911$ |
$121984$ |
| $$ |
$71547$ |
$261386$ |
$152880$ |
$563490$ |
$1221132$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&1&0&2&5&0\\1&0&4&0&1&0&4&6&0&1&0&5&0&0&10\\0&1&0&3&0&4&0&0&4&0&1&0&6&7&0\\0&0&1&0&5&0&5&3&0&5&0&9&0&0&12\\0&3&0&4&0&13&0&0&14&0&7&0&18&26&0\\0&0&4&0&5&0&13&9&0&7&0&18&0&0&32\\2&0&6&0&3&0&9&19&0&6&0&16&0&0&36\\0&3&0&4&0&14&0&0&21&0&8&0&22&35&0\\0&0&1&0&5&0&7&6&0&11&0&13&0&0&26\\0&1&0&1&0&7&0&0&8&0&8&0&12&15&0\\0&0&5&0&9&0&18&16&0&13&0&37&0&0&60\\0&2&0&6&0&18&0&0&22&0&12&0&39&46&0\\0&5&0&7&0&26&0&0&35&0&15&0&46&75&0\\1&0&10&0&12&0&32&36&0&26&0&60&0&0&125\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&5&13&13&19&21&11&8&37&39&75&125&58&67&121&91&28\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
