| 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: | $C_3$ |
| Order: | $3$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$26$ |
$0$ |
$355$ |
$0$ |
$6258$ |
$0$ |
$128646$ |
$0$ |
$2917332$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$53$ |
$385$ |
$3238$ |
$30163$ |
$301934$ |
$3184749$ |
$34948064$ |
$395586301$ |
$4591357059$ |
$54406139191$ |
| $a_3$ |
$1$ |
$0$ |
$12$ |
$0$ |
$872$ |
$0$ |
$124190$ |
$0$ |
$23426424$ |
$0$ |
$5161147212$ |
$0$ |
$1256936939016$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$5$ |
$14$ |
$26$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$12$ |
$53$ |
$31$ |
$93$ |
$56$ |
$178$ |
$355$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$73$ |
$385$ |
$226$ |
$136$ |
$735$ |
$438$ |
$1470$ |
$875$ |
$3008$ |
$6258$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$570$ |
$3238$ |
$338$ |
$1898$ |
$1122$ |
$6533$ |
$3844$ |
$2272$ |
$13514$ |
$7939$ |
$28359$ |
$16618$ |
$60144$ |
$128646$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$872$ |
$5013$ |
$30163$ |
$2946$ |
$17575$ |
$10282$ |
$63056$ |
$6033$ |
$36728$ |
$21447$ |
$133698$ |
$77710$ |
$45274$ |
$286155$ |
| $$ |
$165942$ |
$616950$ |
$356958$ |
$1338156$ |
$2917332$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&2&0&1&0&1&0&0&2\\0&3&0&2&0&6&0&0&6&0&3&0&5&11&0\\1&0&6&0&4&0&9&11&0&4&0&12&0&0&24\\0&2&0&5&0&8&0&0&10&0&3&0&12&18&0\\1&0&4&0&7&0&9&10&0&7&0&18&0&0&28\\0&6&0&8&0&28&0&0&32&0&14&0&40&62&0\\1&0&9&0&9&0&26&25&0&17&0&39&0&0&78\\2&0&11&0&10&0&25&34&0&18&0&42&0&0&88\\0&6&0&10&0&32&0&0&47&0&16&0&57&86&0\\1&0&4&0&7&0&17&18&0&18&0&30&0&0&64\\0&3&0&3&0&14&0&0&16&0&15&0&27&38&0\\1&0&12&0&18&0&39&42&0&30&0&81&0&0&148\\0&5&0&12&0&40&0&0&57&0&27&0&87&116&0\\0&11&0&18&0&62&0&0&86&0&38&0&116&181&0\\2&0&24&0&28&0&78&88&0&64&0&148&0&0&318\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&5&7&28&26&34&47&18&15&81&87&181&318&137&153&299&218&57\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
