| 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: | $D_4$ |
| Order: | $8$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0\\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0& 0 & 0 & 0 & -1 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$22$ |
$0$ |
$460$ |
$0$ |
$11494$ |
$0$ |
$310044$ |
$0$ |
$8768892$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$59$ |
$539$ |
$5772$ |
$66783$ |
$806934$ |
$10035579$ |
$127505390$ |
$1647411635$ |
$21577133943$ |
$285825335837$ |
| $a_3$ |
$1$ |
$0$ |
$9$ |
$0$ |
$1388$ |
$0$ |
$320178$ |
$0$ |
$85665916$ |
$0$ |
$24952255629$ |
$0$ |
$7681494611956$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$3$ |
$11$ |
$22$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$9$ |
$59$ |
$29$ |
$103$ |
$57$ |
$214$ |
$460$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$80$ |
$539$ |
$292$ |
$166$ |
$1097$ |
$619$ |
$2372$ |
$1335$ |
$5197$ |
$11494$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$843$ |
$5772$ |
$474$ |
$3214$ |
$1810$ |
$12492$ |
$7019$ |
$3954$ |
$27616$ |
$15495$ |
$61516$ |
$34440$ |
$137788$ |
$310044$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1388$ |
$9500$ |
$66783$ |
$5342$ |
$37311$ |
$20922$ |
$148461$ |
$11742$ |
$83021$ |
$46476$ |
$332982$ |
$185901$ |
$103900$ |
$750427$ |
| $$ |
$418276$ |
$1697678$ |
$944766$ |
$3852966$ |
$8768892$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&1&0&0&5\\0&2&0&1&0&6&0&0&9&0&3&0&10&20&0\\1&0&6&0&2&0&11&17&0&8&0&21&0&0&52\\0&1&0&5&0&11&0&0&16&0&5&0&25&35&0\\0&0&2&0&9&0&15&14&0&18&0&32&0&0&69\\0&6&0&11&0&40&0&0&63&0&26&0&90&140&0\\0&0&11&0&15&0&46&45&0&37&0&87&0&0&200\\4&0&17&0&14&0&45&70&0&42&0&97&0&0&240\\0&9&0&16&0&63&0&0&103&0&44&0&146&231&0\\0&0&8&0&18&0&37&42&0&46&0&87&0&0&196\\0&3&0&5&0&26&0&0&44&0&23&0&65&100&0\\1&0&21&0&32&0&87&97&0&87&0&195&0&0&447\\0&10&0&25&0&90&0&0&146&0&65&0&237&345&0\\0&20&0&35&0&140&0&0&231&0&100&0&345&544&0\\5&0&52&0&69&0&200&240&0&196&0&447&0&0&1056\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&9&40&46&70&103&46&23&195&237&544&1056&513&539&1220&1002&277\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
