| Name: | $\mathrm{U}(1)\times\mathrm{USp}(4)$ |
| $\mathbb{R}$-dimension: | $11$ |
| Description: | $\left\{\begin{bmatrix}A&0\\0&B\end{bmatrix}: A\in \mathrm{U}(1)\subseteq\mathrm{SU}(2),\ B\in\mathrm{USp}(4)\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
| Hodge circle: | $\mathrm{diag}(u,\bar u, u,u,\bar u,\bar u)$ |
| Name: | $C_2$ |
| Order: | $2$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$12$ |
$0$ |
$114$ |
$0$ |
$1421$ |
$0$ |
$21195$ |
$0$ |
$358776$ |
| $a_2$ |
$1$ |
$2$ |
$6$ |
$23$ |
$112$ |
$668$ |
$4656$ |
$36312$ |
$307075$ |
$2758452$ |
$25973562$ |
$254074659$ |
$2565728028$ |
| $a_3$ |
$1$ |
$0$ |
$6$ |
$0$ |
$198$ |
$0$ |
$15216$ |
$0$ |
$1786372$ |
$0$ |
$265868064$ |
$0$ |
$46182504984$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$6$ |
$3$ |
$7$ |
$12$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$6$ |
$23$ |
$13$ |
$33$ |
$21$ |
$59$ |
$114$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$26$ |
$112$ |
$67$ |
$43$ |
$188$ |
$120$ |
$356$ |
$228$ |
$702$ |
$1421$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$143$ |
$668$ |
$90$ |
$410$ |
$259$ |
$1243$ |
$783$ |
$498$ |
$2457$ |
$1550$ |
$4971$ |
$3129$ |
$10206$ |
$21195$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$198$ |
$926$ |
$4656$ |
$580$ |
$2868$ |
$1790$ |
$9211$ |
$1124$ |
$5721$ |
$3572$ |
$18769$ |
$11646$ |
$7261$ |
$38771$ |
| $$ |
$24010$ |
$80841$ |
$49950$ |
$169785$ |
$358776$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&1&0&0&0&0&0&0&0\\0&2&0&1&0&2&0&0&1&0&1&0&0&2&0\\1&0&3&0&1&0&2&3&0&0&0&2&0&0&2\\0&1&0&2&0&2&0&0&2&0&0&0&2&2&0\\0&0&1&0&4&0&2&1&0&1&0&5&0&0&2\\0&2&0&2&0&7&0&0&4&0&4&0&6&8&0\\0&0&2&0&2&0&6&3&0&2&0&6&0&0&8\\1&0&3&0&1&0&3&7&0&1&0&5&0&0&8\\0&1&0&2&0&4&0&0&7&0&1&0&6&8&0\\0&0&0&0&1&0&2&1&0&4&0&2&0&0&6\\0&1&0&0&0&4&0&0&1&0&6&0&4&5&0\\0&0&2&0&5&0&6&5&0&2&0&15&0&0&12\\0&0&0&2&0&6&0&0&6&0&4&0&13&10&0\\0&2&0&2&0&8&0&0&8&0&5&0&10&18&0\\0&0&2&0&2&0&8&8&0&6&0&12&0&0&25\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&2&4&7&6&7&7&4&6&15&13&18&25&13&13&18&15&6\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
