| Name: | $\mathrm{SU}(2)_3$ |
| $\mathbb{R}$-dimension: | $3$ |
| Description: | $\left\{\begin{bmatrix}\alpha I_3&\beta I_3\\ \gamma I_3& \delta I_3\end{bmatrix}: \begin{bmatrix}\alpha&\beta\\\gamma&\delta\end{bmatrix}\in\mathrm{SU}(2)\right\}$ |
Symplectic form: | $\begin{bmatrix} 0 & I_3\\ -I_3 & 0\end{bmatrix}$ |
| Hodge circle: | $u\mapsto \mathrm{diag}(u,u, u, \bar u, \bar u, \bar u)$ |
| Name: | $C_2^2$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 &0 & 0 & 0 & 0 & -1 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$42$ |
$0$ |
$915$ |
$0$ |
$22974$ |
$0$ |
$620046$ |
$0$ |
$17537652$ |
| $a_2$ |
$1$ |
$3$ |
$15$ |
$111$ |
$1059$ |
$11493$ |
$133425$ |
$1613475$ |
$20070051$ |
$255007641$ |
$3294814317$ |
$43154242233$ |
$571650597885$ |
| $a_3$ |
$1$ |
$0$ |
$17$ |
$0$ |
$2772$ |
$0$ |
$640322$ |
$0$ |
$171331468$ |
$0$ |
$49904506989$ |
$0$ |
$15362989171068$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$15$ |
$6$ |
$21$ |
$42$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$17$ |
$111$ |
$57$ |
$204$ |
$114$ |
$426$ |
$915$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$159$ |
$1059$ |
$582$ |
$330$ |
$2190$ |
$1236$ |
$4740$ |
$2670$ |
$10389$ |
$22974$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1683$ |
$11493$ |
$948$ |
$6423$ |
$3618$ |
$24975$ |
$14034$ |
$7902$ |
$55224$ |
$30984$ |
$123021$ |
$68880$ |
$275562$ |
$620046$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2772$ |
$18993$ |
$133425$ |
$10680$ |
$74610$ |
$41838$ |
$296901$ |
$23484$ |
$166032$ |
$92946$ |
$665946$ |
$371790$ |
$207780$ |
$1500831$ |
| $$ |
$836532$ |
$3395322$ |
$1889532$ |
$7705890$ |
$17537652$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&6&0&1&0&3&0&0&10\\0&3&0&3&0&12&0&0&18&0&6&0&21&39&0\\2&0&10&0&6&0&23&33&0&17&0&42&0&0&104\\0&3&0&8&0&22&0&0&33&0&11&0&48&71&0\\0&0&6&0&15&0&30&30&0&33&0&63&0&0&138\\0&12&0&22&0&80&0&0&126&0&52&0&180&280&0\\1&0&23&0&30&0&88&93&0&76&0&174&0&0&400\\6&0&33&0&30&0&93&132&0&87&0&198&0&0&480\\0&18&0&33&0&126&0&0&204&0&87&0&294&462&0\\1&0&17&0&33&0&76&87&0&84&0&171&0&0&392\\0&6&0&11&0&52&0&0&87&0&44&0&132&200&0\\3&0&42&0&63&0&174&198&0&171&0&387&0&0&894\\0&21&0&48&0&180&0&0&294&0&132&0&468&693&0\\0&39&0&71&0&280&0&0&462&0&200&0&693&1085&0\\10&0&104&0&138&0&400&480&0&392&0&894&0&0&2112\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&10&8&15&80&88&132&204&84&44&387&468&1085&2112&1017&1065&2432&1992&541\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
