| Name: | $\mathrm{U}(1)^3$ |
| $\mathbb{R}$-dimension: | $3$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A,B,C\in\mathrm{U}(1)\subseteq\mathrm{SU}(2)\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0&0\\0&J_2&0\\0&0&J_2\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
| Hodge circle: | $u\mapsto\mathrm{diag}(u,\bar u, u, \bar u, u, \bar u)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$6$ |
$0$ |
$90$ |
$0$ |
$1860$ |
$0$ |
$44730$ |
$0$ |
$1172556$ |
$0$ |
$32496156$ |
| $a_2$ |
$1$ |
$3$ |
$21$ |
$183$ |
$1845$ |
$20223$ |
$234129$ |
$2816523$ |
$34857909$ |
$440957055$ |
$5676091641$ |
$74106653283$ |
$978982132401$ |
| $a_3$ |
$1$ |
$0$ |
$32$ |
$0$ |
$4920$ |
$0$ |
$1109120$ |
$0$ |
$293308120$ |
$0$ |
$84843343872$ |
$0$ |
$25999253294784$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$6$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$21$ |
$12$ |
$42$ |
$90$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$32$ |
$183$ |
$108$ |
$390$ |
$228$ |
$846$ |
$1860$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$288$ |
$1845$ |
$1068$ |
$624$ |
$4050$ |
$2340$ |
$8970$ |
$5160$ |
$19980$ |
$44730$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$3000$ |
$20223$ |
$1728$ |
$11556$ |
$6624$ |
$45198$ |
$25764$ |
$14736$ |
$101502$ |
$57720$ |
$228780$ |
$129780$ |
$517230$ |
$1172556$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$4920$ |
$33480$ |
$234129$ |
$19080$ |
$132444$ |
$75096$ |
$529458$ |
$42672$ |
$298908$ |
$169104$ |
$1200906$ |
$676680$ |
$382080$ |
$2730660$ |
| $$ |
$1535940$ |
$6222594$ |
$3494232$ |
$14207508$ |
$32496156$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&4&6&0&4&0&9&0&0&16\\0&6&0&6&0&24&0&0&30&0&18&0&42&66&0\\2&0&16&0&18&0&44&48&0&32&0&90&0&0&176\\0&6&0&14&0&40&0&0&54&0&26&0&90&122&0\\3&0&18&0&27&0&54&66&0&48&0&123&0&0&240\\0&24&0&40&0&152&0&0&216&0&112&0&336&496&0\\4&0&44&0&54&0&154&168&0&124&0&324&0&0&688\\6&0&48&0&66&0&168&204&0&156&0&378&0&0&816\\0&30&0&54&0&216&0&0&342&0&162&0&522&786&0\\4&0&32&0&48&0&124&156&0&132&0&288&0&0&656\\0&18&0&26&0&112&0&0&162&0&92&0&252&386&0\\9&0&90&0&123&0&324&378&0&288&0&741&0&0&1584\\0&42&0&90&0&336&0&0&522&0&252&0&852&1242&0\\0&66&0&122&0&496&0&0&786&0&386&0&1242&1886&0\\16&0&176&0&240&0&688&816&0&656&0&1584&0&0&3600\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&16&14&27&152&154&204&342&132&92&741&852&1886&3600&1686&1695&3956&3210&862\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
