| 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: | $S_3$ |
| Order: | $6$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & -1/2 & \sqrt{3}/2 & 0 & 0 & 0 \\0 &-\sqrt{3}/2 & -1/2 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -1/2 & \sqrt{3}/2 \\0 & 0 & 0 & 0 & -\sqrt{3}/2 & -1/2 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0& 0 & 0 & 0 & -1 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$28$ |
$0$ |
$610$ |
$0$ |
$15316$ |
$0$ |
$413364$ |
$0$ |
$11691768$ |
| $a_2$ |
$1$ |
$2$ |
$10$ |
$74$ |
$706$ |
$7662$ |
$88950$ |
$1075650$ |
$13380034$ |
$170005094$ |
$2196542878$ |
$28769494822$ |
$381100398590$ |
| $a_3$ |
$1$ |
$0$ |
$12$ |
$0$ |
$1850$ |
$0$ |
$426888$ |
$0$ |
$114221002$ |
$0$ |
$33269671410$ |
$0$ |
$10241992781020$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$4$ |
$14$ |
$28$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$12$ |
$74$ |
$38$ |
$136$ |
$76$ |
$284$ |
$610$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$106$ |
$706$ |
$388$ |
$220$ |
$1460$ |
$824$ |
$3160$ |
$1780$ |
$6926$ |
$15316$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1122$ |
$7662$ |
$632$ |
$4282$ |
$2412$ |
$16650$ |
$9356$ |
$5268$ |
$36816$ |
$20656$ |
$82014$ |
$45920$ |
$183708$ |
$413364$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1850$ |
$12662$ |
$88950$ |
$7120$ |
$49740$ |
$27892$ |
$197934$ |
$15656$ |
$110688$ |
$61964$ |
$443964$ |
$247860$ |
$138520$ |
$1000554$ |
| $$ |
$557688$ |
$2263548$ |
$1259688$ |
$5137260$ |
$11691768$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&1&4&0&1&0&2&0&0&6\\0&2&0&2&0&8&0&0&12&0&4&0&14&26&0\\1&0&7&0&4&0&15&22&0&11&0&28&0&0&70\\0&2&0&6&0&14&0&0&22&0&8&0&32&48&0\\0&0&4&0&10&0&20&20&0&22&0&42&0&0&92\\0&8&0&14&0&54&0&0&84&0&34&0&120&186&0\\1&0&15&0&20&0&59&62&0&51&0&116&0&0&266\\4&0&22&0&20&0&62&88&0&58&0&132&0&0&320\\0&12&0&22&0&84&0&0&136&0&58&0&196&308&0\\1&0&11&0&22&0&51&58&0&57&0&114&0&0&260\\0&4&0&8&0&34&0&0&58&0&30&0&88&134&0\\2&0&28&0&42&0&116&132&0&114&0&258&0&0&596\\0&14&0&32&0&120&0&0&196&0&88&0&312&462&0\\0&26&0&48&0&186&0&0&308&0&134&0&462&724&0\\6&0&70&0&92&0&266&320&0&260&0&596&0&0&1410\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&7&6&10&54&59&88&136&57&30&258&312&724&1410&678&712&1622&1328&364\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/3$ | $0$ | $1/3$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $1/3$ | $1/3$ | $0$ | $1/3$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
