| 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: | $S_4$ |
| Order: | $24$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\1 & -1 & 1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 1 & -1 & 1 \\0 & 0 & 0 & 0 & 0 & 1\\\end{bmatrix}, \begin{bmatrix}0 & 0 & -1 & 0 & 0 & 0 \\1 & 0 & -1 & 0 & 0 & 0\\0 & 1 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 0 & 1 & 0 & -1 \\0 &0 & 0 & 0 & 1 & -1 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$8$ |
$0$ |
$155$ |
$0$ |
$3836$ |
$0$ |
$103362$ |
$0$ |
$2923008$ |
| $a_2$ |
$1$ |
$1$ |
$4$ |
$22$ |
$186$ |
$1941$ |
$22308$ |
$269109$ |
$3345562$ |
$42502843$ |
$549140196$ |
$7192386532$ |
$95275136542$ |
| $a_3$ |
$1$ |
$0$ |
$4$ |
$0$ |
$466$ |
$0$ |
$106744$ |
$0$ |
$28555450$ |
$0$ |
$8317420050$ |
$0$ |
$2560498221908$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$1$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$4$ |
$1$ |
$4$ |
$8$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$4$ |
$22$ |
$10$ |
$35$ |
$19$ |
$72$ |
$155$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$27$ |
$186$ |
$98$ |
$56$ |
$367$ |
$207$ |
$792$ |
$445$ |
$1734$ |
$3836$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$282$ |
$1941$ |
$158$ |
$1073$ |
$604$ |
$4167$ |
$2341$ |
$1320$ |
$9208$ |
$5167$ |
$20509$ |
$11480$ |
$45934$ |
$103362$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$466$ |
$3169$ |
$22308$ |
$1782$ |
$12441$ |
$6976$ |
$49494$ |
$3914$ |
$27677$ |
$15494$ |
$111000$ |
$61971$ |
$34640$ |
$250150$ |
| $$ |
$139432$ |
$565904$ |
$314922$ |
$1284336$ |
$2923008$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&2&0&0&0&0&0&0&1\\0&1&0&0&0&2&0&0&3&0&1&0&3&7&0\\0&0&3&0&0&0&3&6&0&2&0&7&0&0&18\\0&0&0&3&0&3&0&0&5&0&2&0&9&12&0\\0&0&0&0&4&0&5&4&0&7&0&11&0&0&23\\0&2&0&3&0&14&0&0&21&0&8&0&30&46&0\\0&0&3&0&5&0&17&14&0&12&0&29&0&0&66\\2&0&6&0&4&0&14&26&0&13&0&31&0&0&80\\0&3&0&5&0&21&0&0&35&0&15&0&48&77&0\\0&0&2&0&7&0&12&13&0&19&0&30&0&0&64\\0&1&0&2&0&8&0&0&15&0&9&0&21&34&0\\0&0&7&0&11&0&29&31&0&30&0&66&0&0&149\\0&3&0&9&0&30&0&0&48&0&21&0&81&114&0\\0&7&0&12&0&46&0&0&77&0&34&0&114&183&0\\1&0&18&0&23&0&66&80&0&64&0&149&0&0&354\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&3&3&4&14&17&26&35&19&9&66&81&183&354&174&186&410&338&100\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/3$ | $0$ | $1/3$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $1/3$ | $1/3$ | $0$ | $1/3$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
