| 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: | $C_4$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0\\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$42$ |
$0$ |
$915$ |
$0$ |
$22974$ |
$0$ |
$620046$ |
$0$ |
$17537652$ |
| $a_2$ |
$1$ |
$2$ |
$13$ |
$105$ |
$1043$ |
$11448$ |
$133299$ |
$1613118$ |
$20069035$ |
$255004734$ |
$3294805967$ |
$43154218165$ |
$571650528309$ |
| $a_3$ |
$1$ |
$0$ |
$17$ |
$0$ |
$2772$ |
$0$ |
$640322$ |
$0$ |
$171331468$ |
$0$ |
$49904506989$ |
$0$ |
$15362989171068$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$13$ |
$6$ |
$21$ |
$42$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$17$ |
$105$ |
$57$ |
$204$ |
$115$ |
$427$ |
$915$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$159$ |
$1043$ |
$582$ |
$330$ |
$2190$ |
$1237$ |
$4742$ |
$2674$ |
$10393$ |
$22974$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1683$ |
$11448$ |
$950$ |
$6423$ |
$3620$ |
$24975$ |
$14037$ |
$7902$ |
$55229$ |
$30988$ |
$123029$ |
$68894$ |
$275576$ |
$620046$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2772$ |
$18993$ |
$133299$ |
$10682$ |
$74610$ |
$41842$ |
$296901$ |
$23492$ |
$166039$ |
$92954$ |
$665958$ |
$371802$ |
$207780$ |
$1500851$ |
| $$ |
$836546$ |
$3395350$ |
$1889580$ |
$7705938$ |
$17537652$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&2&5&0&1&0&3&0&0&10\\0&3&0&3&0&12&0&0&18&0&6&0&22&38&0\\1&0&10&0&7&0&23&30&0&20&0&44&0&0&104\\0&3&0&8&0&22&0&0&33&0&12&0&46&72&0\\1&0&7&0&13&0&30&33&0&29&0&61&0&0&138\\0&12&0&22&0&80&0&0&126&0&52&0&180&280&0\\2&0&23&0&30&0&86&97&0&76&0&172&0&0&400\\5&0&30&0&33&0&97&126&0&88&0&200&0&0&480\\0&18&0&33&0&126&0&0&204&0&86&0&296&461&0\\1&0&20&0&29&0&76&88&0&80&0&171&0&0&392\\0&6&0&12&0&52&0&0&86&0&42&0&134&200&0\\3&0&44&0&61&0&172&200&0&171&0&387&0&0&894\\0&22&0&46&0&180&0&0&296&0&134&0&462&696&0\\0&38&0&72&0&280&0&0&461&0&200&0&696&1083&0\\10&0&104&0&138&0&400&480&0&392&0&894&0&0&2112\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&10&8&13&80&86&126&204&80&42&387&462&1083&2112&1011&1055&2424&1988&533\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
