| 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: | $C_4$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0\\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$24$ |
$0$ |
$470$ |
$0$ |
$11200$ |
$0$ |
$293202$ |
$0$ |
$8124270$ |
| $a_2$ |
$1$ |
$2$ |
$8$ |
$53$ |
$482$ |
$5117$ |
$58715$ |
$704678$ |
$8716118$ |
$110244185$ |
$1419037673$ |
$18526707608$ |
$244745665961$ |
| $a_3$ |
$1$ |
$0$ |
$10$ |
$0$ |
$1254$ |
$0$ |
$277600$ |
$0$ |
$73331510$ |
$0$ |
$21210900480$ |
$0$ |
$6499814269872$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$4$ |
$12$ |
$24$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$10$ |
$53$ |
$30$ |
$102$ |
$60$ |
$216$ |
$470$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$78$ |
$482$ |
$276$ |
$162$ |
$1026$ |
$594$ |
$2256$ |
$1300$ |
$5010$ |
$11200$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$768$ |
$5117$ |
$444$ |
$2916$ |
$1674$ |
$11340$ |
$6468$ |
$3704$ |
$25416$ |
$14460$ |
$57240$ |
$32480$ |
$129360$ |
$293202$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1254$ |
$8424$ |
$58715$ |
$4806$ |
$33192$ |
$18828$ |
$132486$ |
$10708$ |
$74808$ |
$42336$ |
$300348$ |
$169260$ |
$95590$ |
$682800$ |
| $$ |
$384090$ |
$1555806$ |
$873684$ |
$3552066$ |
$8124270$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&1&2&0&1&0&2&0&0&4\\0&2&0&2&0&6&0&0&8&0&4&0&10&16&0\\1&0&5&0&4&0&11&13&0&8&0&21&0&0&44\\0&2&0&4&0&10&0&0&14&0&6&0&22&30&0\\0&0&4&0&8&0&14&15&0&13&0&31&0&0&60\\0&6&0&10&0&38&0&0&54&0&28&0&84&124&0\\1&0&11&0&14&0&39&41&0&32&0&81&0&0&172\\2&0&13&0&15&0&41&54&0&37&0&93&0&0&204\\0&8&0&14&0&54&0&0&86&0&40&0&130&196&0\\1&0&8&0&13&0&32&37&0&35&0&72&0&0&164\\0&4&0&6&0&28&0&0&40&0&24&0&64&96&0\\2&0&21&0&31&0&81&93&0&72&0&188&0&0&396\\0&10&0&22&0&84&0&0&130&0&64&0&214&310&0\\0&16&0&30&0&124&0&0&196&0&96&0&310&474&0\\4&0&44&0&60&0&172&204&0&164&0&396&0&0&900\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&8&38&39&54&86&35&24&188&214&474&900&424&431&990&807&218\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $3/4$ | $0$ | $0$ | $1/2$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
|---|
