| 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: | $C_2^2$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0& 0 & 0 & -1 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$33$ |
$0$ |
$570$ |
$0$ |
$12425$ |
$0$ |
$309078$ |
$0$ |
$8337714$ |
| $a_2$ |
$1$ |
$3$ |
$13$ |
$75$ |
$585$ |
$5643$ |
$61609$ |
$721479$ |
$8817265$ |
$110868195$ |
$1422953253$ |
$18551582019$ |
$244905177933$ |
| $a_3$ |
$1$ |
$0$ |
$14$ |
$0$ |
$1398$ |
$0$ |
$284000$ |
$0$ |
$73645110$ |
$0$ |
$21227157504$ |
$0$ |
$6500688536496$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$13$ |
$6$ |
$17$ |
$33$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$14$ |
$75$ |
$40$ |
$129$ |
$78$ |
$267$ |
$570$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$98$ |
$585$ |
$330$ |
$198$ |
$1179$ |
$696$ |
$2555$ |
$1500$ |
$5610$ |
$12425$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$876$ |
$5643$ |
$516$ |
$3222$ |
$1878$ |
$12237$ |
$7066$ |
$4104$ |
$27213$ |
$15660$ |
$60910$ |
$34930$ |
$136955$ |
$309078$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1398$ |
$9036$ |
$61609$ |
$5214$ |
$34986$ |
$20024$ |
$137879$ |
$11508$ |
$78402$ |
$44736$ |
$311355$ |
$176600$ |
$100490$ |
$705590$ |
| $$ |
$399280$ |
$1603483$ |
$905436$ |
$3652614$ |
$8337714$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&4&0&0&0&1&0&0&4\\0&3&0&3&0&8&0&0&9&0&5&0&11&17&0\\2&0&8&0&4&0&12&18&0&6&0&22&0&0&44\\0&3&0&5&0&12&0&0&15&0&7&0&23&31&0\\0&0&4&0&12&0&18&12&0&16&0&38&0&0&60\\0&8&0&12&0&44&0&0&56&0&32&0&88&128&0\\1&0&12&0&18&0&43&40&0&34&0&87&0&0&172\\4&0&18&0&12&0&40&66&0&30&0&92&0&0&204\\0&9&0&15&0&56&0&0&87&0&41&0&131&197&0\\0&0&6&0&16&0&34&30&0&40&0&74&0&0&164\\0&5&0&7&0&32&0&0&41&0&28&0&68&99&0\\1&0&22&0&38&0&87&92&0&74&0&201&0&0&396\\0&11&0&23&0&88&0&0&131&0&68&0&218&313&0\\0&17&0&31&0&128&0&0&197&0&99&0&313&479&0\\4&0&44&0&60&0&172&204&0&164&0&396&0&0&900\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&8&5&12&44&43&66&87&40&28&201&218&479&900&429&454&994&829&223\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $1/4$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_3=0$ | $1/4$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=a_3=0$ | $1/4$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
