| Name: | $\mathrm{U}(1)\times\mathrm{U}(1)_2$ |
| $\mathbb{R}$-dimension: | $6$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
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: | $u\mapsto\mathrm{diag}(u,\bar u, u, u,\bar u, \bar u)$ |
| Name: | $C_2^3$ |
| Order: | $8$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$39$ |
$0$ |
$780$ |
$0$ |
$19075$ |
$0$ |
$521388$ |
$0$ |
$15246462$ |
| $a_2$ |
$1$ |
$2$ |
$11$ |
$80$ |
$763$ |
$8422$ |
$101117$ |
$1275976$ |
$16613219$ |
$220876478$ |
$2980494721$ |
$40667867176$ |
$559751362309$ |
| $a_3$ |
$1$ |
$0$ |
$14$ |
$0$ |
$1986$ |
$0$ |
$493640$ |
$0$ |
$146831314$ |
$0$ |
$47079633384$ |
$0$ |
$15738294082632$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$11$ |
$5$ |
$18$ |
$39$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$14$ |
$80$ |
$45$ |
$161$ |
$93$ |
$350$ |
$780$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$116$ |
$763$ |
$433$ |
$254$ |
$1662$ |
$957$ |
$3723$ |
$2130$ |
$8400$ |
$19075$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1212$ |
$8422$ |
$690$ |
$4753$ |
$2702$ |
$18993$ |
$10733$ |
$6100$ |
$43248$ |
$24390$ |
$98882$ |
$55615$ |
$226758$ |
$521388$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1986$ |
$13930$ |
$101117$ |
$7858$ |
$56495$ |
$31684$ |
$231722$ |
$17810$ |
$129345$ |
$72388$ |
$533153$ |
$297088$ |
$166002$ |
$1229622$ |
| $$ |
$684103$ |
$2841447$ |
$1578402$ |
$6577032$ |
$15246462$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&4&0&1&0&3&0&0&6\\0&3&0&2&0&10&0&0&12&0&8&0&15&28&0\\1&0&8&0&6&0&17&20&0&10&0&37&0&0&70\\0&2&0&7&0&16&0&0&19&0&9&0&40&49&0\\1&0&6&0&14&0&21&26&0&22&0&53&0&0&98\\0&10&0&16&0&62&0&0&86&0&48&0&140&212&0\\1&0&17&0&21&0&65&64&0&45&0&137&0&0&294\\4&0&20&0&26&0&64&92&0&66&0&158&0&0&358\\0&12&0&19&0&86&0&0&145&0&69&0&218&349&0\\1&0&10&0&22&0&45&66&0&70&0&123&0&0&302\\0&8&0&9&0&48&0&0&69&0&44&0&106&172&0\\3&0&37&0&53&0&137&158&0&123&0&331&0&0&710\\0&15&0&40&0&140&0&0&218&0&106&0&390&557&0\\0&28&0&49&0&212&0&0&349&0&172&0&557&877&0\\6&0&70&0&98&0&294&358&0&302&0&710&0&0&1712\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&8&7&14&62&65&92&145&70&44&331&390&877&1712&873&874&2102&1803&539\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $3/8$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $3/8$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $3/8$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
