| 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: | $C_4$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$26$ |
$0$ |
$345$ |
$0$ |
$5782$ |
$0$ |
$112686$ |
$0$ |
$2442132$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$51$ |
$359$ |
$2908$ |
$26103$ |
$252822$ |
$2595083$ |
$27872238$ |
$310353971$ |
$3558078967$ |
$41781089633$ |
| $a_3$ |
$1$ |
$0$ |
$12$ |
$0$ |
$812$ |
$0$ |
$105080$ |
$0$ |
$18674068$ |
$0$ |
$3986830992$ |
$0$ |
$956002577904$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$5$ |
$14$ |
$26$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$12$ |
$51$ |
$31$ |
$91$ |
$56$ |
$174$ |
$345$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$71$ |
$359$ |
$216$ |
$134$ |
$689$ |
$420$ |
$1376$ |
$835$ |
$2800$ |
$5782$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$532$ |
$2908$ |
$322$ |
$1736$ |
$1046$ |
$5861$ |
$3506$ |
$2112$ |
$12072$ |
$7205$ |
$25187$ |
$14980$ |
$53060$ |
$112686$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$812$ |
$4477$ |
$26103$ |
$2678$ |
$15411$ |
$9148$ |
$54346$ |
$5451$ |
$32050$ |
$18971$ |
$114622$ |
$67414$ |
$39794$ |
$243903$ |
| $$ |
$143052$ |
$522702$ |
$305718$ |
$1126896$ |
$2442132$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&2&0&1&0&1&0&0&2\\0&3&0&2&0&6&0&0&6&0&3&0&5&9&0\\1&0&6&0&4&0&9&9&0&4&0&12&0&0&20\\0&2&0&5&0&8&0&0&8&0&3&0&12&14&0\\1&0&4&0&7&0&9&10&0&7&0&16&0&0&24\\0&6&0&8&0&26&0&0&28&0&14&0&36&52&0\\1&0&9&0&9&0&24&21&0&13&0&35&0&0&64\\2&0&9&0&10&0&21&30&0&16&0&36&0&0&72\\0&6&0&8&0&28&0&0&41&0&16&0&45&70&0\\1&0&4&0&7&0&13&16&0&18&0&24&0&0&52\\0&3&0&3&0&14&0&0&16&0&13&0&23&32&0\\1&0&12&0&16&0&35&36&0&24&0&71&0&0&120\\0&5&0&12&0&36&0&0&45&0&23&0&75&94&0\\0&9&0&14&0&52&0&0&70&0&32&0&94&147&0\\2&0&20&0&24&0&64&72&0&52&0&120&0&0&250\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&5&7&26&24&30&41&18&13&71&75&147&250&113&121&239&176&53\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
