| 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: | $D_6$ |
| Order: | $12$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{11} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$14$ |
$0$ |
$175$ |
$0$ |
$2884$ |
$0$ |
$55314$ |
$0$ |
$1170972$ |
| $a_2$ |
$1$ |
$2$ |
$7$ |
$32$ |
$196$ |
$1482$ |
$12835$ |
$121151$ |
$1213196$ |
$12698252$ |
$137670397$ |
$1536637324$ |
$17579215978$ |
| $a_3$ |
$1$ |
$0$ |
$7$ |
$0$ |
$410$ |
$0$ |
$50735$ |
$0$ |
$8524390$ |
$0$ |
$1710928758$ |
$0$ |
$387502522044$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$7$ |
$3$ |
$8$ |
$14$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$7$ |
$32$ |
$17$ |
$48$ |
$29$ |
$89$ |
$175$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$38$ |
$196$ |
$112$ |
$69$ |
$350$ |
$213$ |
$691$ |
$420$ |
$1400$ |
$2884$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$272$ |
$1482$ |
$164$ |
$873$ |
$527$ |
$2913$ |
$1750$ |
$1059$ |
$5975$ |
$3585$ |
$12434$ |
$7434$ |
$26124$ |
$55314$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$410$ |
$2228$ |
$12835$ |
$1338$ |
$7587$ |
$4530$ |
$26496$ |
$2715$ |
$15731$ |
$9373$ |
$55738$ |
$33012$ |
$19617$ |
$118281$ |
| $$ |
$69860$ |
$252658$ |
$148806$ |
$542640$ |
$1170972$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&1&0&2&5&0\\1&0&4&0&1&0&4&6&0&1&0&5&0&0&10\\0&1&0&3&0&4&0&0&4&0&1&0&6&7&0\\0&0&1&0&5&0&5&3&0&5&0&9&0&0&12\\0&3&0&4&0&13&0&0&14&0&7&0&18&25&0\\0&0&4&0&5&0&13&9&0&7&0&18&0&0&31\\2&0&6&0&3&0&9&18&0&6&0&16&0&0&34\\0&3&0&4&0&14&0&0&20&0&8&0&22&33&0\\0&0&1&0&5&0&7&6&0&10&0&13&0&0&24\\0&1&0&1&0&7&0&0&8&0&8&0&12&15&0\\0&0&5&0&9&0&18&16&0&13&0&36&0&0&58\\0&2&0&6&0&18&0&0&22&0&12&0&38&44&0\\0&5&0&7&0&25&0&0&33&0&15&0&44&69&0\\1&0&10&0&12&0&31&34&0&24&0&58&0&0&115\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&5&13&13&18&20&10&8&36&38&69&115&51&57&102&76&23\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
