| 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: | $u\mapsto\mathrm{diag}(u,\bar u, u, \bar u, u, \bar u)$ |
| Name: | $C_2$ |
| Order: | $2$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$45$ |
$0$ |
$930$ |
$0$ |
$22365$ |
$0$ |
$586278$ |
$0$ |
$16248078$ |
| $a_2$ |
$1$ |
$3$ |
$15$ |
$105$ |
$963$ |
$10233$ |
$117429$ |
$1409355$ |
$17432235$ |
$220488369$ |
$2838075345$ |
$37053415215$ |
$489491331921$ |
| $a_3$ |
$1$ |
$0$ |
$16$ |
$0$ |
$2460$ |
$0$ |
$554560$ |
$0$ |
$146654060$ |
$0$ |
$42421671936$ |
$0$ |
$12999626647392$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$15$ |
$6$ |
$21$ |
$45$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$16$ |
$105$ |
$54$ |
$195$ |
$114$ |
$423$ |
$930$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$144$ |
$963$ |
$534$ |
$312$ |
$2025$ |
$1170$ |
$4485$ |
$2580$ |
$9990$ |
$22365$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1500$ |
$10233$ |
$864$ |
$5778$ |
$3312$ |
$22599$ |
$12882$ |
$7368$ |
$50751$ |
$28860$ |
$114390$ |
$64890$ |
$258615$ |
$586278$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2460$ |
$16740$ |
$117429$ |
$9540$ |
$66222$ |
$37548$ |
$264729$ |
$21336$ |
$149454$ |
$84552$ |
$600453$ |
$338340$ |
$191040$ |
$1365330$ |
| $$ |
$767970$ |
$3111297$ |
$1747116$ |
$7103754$ |
$16248078$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&6&0&0&0&3&0&0&8\\0&3&0&3&0&12&0&0&15&0&9&0&21&33&0\\2&0&10&0&6&0&20&30&0&12&0&42&0&0&88\\0&3&0&7&0&20&0&0&27&0&13&0&45&61&0\\0&0&6&0&18&0&30&24&0&30&0&66&0&0&120\\0&12&0&20&0&76&0&0&108&0&56&0&168&248&0\\1&0&20&0&30&0&79&78&0&66&0&165&0&0&344\\6&0&30&0&24&0&78&120&0&66&0&180&0&0&408\\0&15&0&27&0&108&0&0&171&0&81&0&261&393&0\\0&0&12&0&30&0&66&66&0&74&0&150&0&0&328\\0&9&0&13&0&56&0&0&81&0&46&0&126&193&0\\3&0&42&0&66&0&165&180&0&150&0&375&0&0&792\\0&21&0&45&0&168&0&0&261&0&126&0&426&621&0\\0&33&0&61&0&248&0&0&393&0&193&0&621&943&0\\8&0&88&0&120&0&344&408&0&328&0&792&0&0&1800\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&10&7&18&76&79&120&171&74&46&375&426&943&1800&843&888&1978&1623&431\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
