| 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: | $C_6$ |
| Order: | $6$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{11} \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$26$ |
$0$ |
$345$ |
$0$ |
$5754$ |
$0$ |
$110586$ |
$0$ |
$2341812$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$51$ |
$357$ |
$2868$ |
$25403$ |
$241552$ |
$2424269$ |
$25390458$ |
$275323491$ |
$3073224927$ |
$35158288591$ |
| $a_3$ |
$1$ |
$0$ |
$12$ |
$0$ |
$808$ |
$0$ |
$101370$ |
$0$ |
$17047800$ |
$0$ |
$3421846932$ |
$0$ |
$775004922120$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$5$ |
$14$ |
$26$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$12$ |
$51$ |
$31$ |
$91$ |
$56$ |
$174$ |
$345$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$71$ |
$357$ |
$216$ |
$134$ |
$687$ |
$420$ |
$1372$ |
$835$ |
$2790$ |
$5754$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$530$ |
$2868$ |
$322$ |
$1724$ |
$1044$ |
$5791$ |
$3484$ |
$2108$ |
$11924$ |
$7155$ |
$24843$ |
$14854$ |
$52220$ |
$110586$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$808$ |
$4417$ |
$25403$ |
$2658$ |
$15113$ |
$9032$ |
$52896$ |
$5415$ |
$31418$ |
$18721$ |
$111406$ |
$65984$ |
$39206$ |
$236497$ |
| $$ |
$139678$ |
$505246$ |
$297570$ |
$1085196$ |
$2341812$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&2&0&1&0&1&0&0&2\\0&3&0&2&0&6&0&0&6&0&3&0&5&9&0\\1&0&6&0&4&0&9&9&0&4&0&12&0&0&20\\0&2&0&5&0&8&0&0&8&0&3&0&12&14&0\\1&0&4&0&7&0&9&10&0&7&0&16&0&0&24\\0&6&0&8&0&26&0&0&28&0&14&0&36&50&0\\1&0&9&0&9&0&24&21&0&13&0&35&0&0&62\\2&0&9&0&10&0&21&28&0&16&0&36&0&0&68\\0&6&0&8&0&28&0&0&39&0&16&0&45&66&0\\1&0&4&0&7&0&13&16&0&16&0&24&0&0&48\\0&3&0&3&0&14&0&0&16&0&13&0&23&32&0\\1&0&12&0&16&0&35&36&0&24&0&69&0&0&116\\0&5&0&12&0&36&0&0&45&0&23&0&73&90&0\\0&9&0&14&0&50&0&0&66&0&32&0&90&135&0\\2&0&20&0&24&0&62&68&0&48&0&116&0&0&230\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&5&7&26&24&28&39&16&13&69&73&135&230&99&101&201&146&43\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
