| 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: | $C_2$ |
| Order: | $2$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 &0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$9$ |
$0$ |
$147$ |
$0$ |
$3090$ |
$0$ |
$76195$ |
$0$ |
$2085174$ |
$0$ |
$60984462$ |
| $a_2$ |
$1$ |
$5$ |
$35$ |
$299$ |
$2995$ |
$33535$ |
$404045$ |
$5102725$ |
$66449555$ |
$883496495$ |
$11921952025$ |
$162671391745$ |
$2239005227869$ |
| $a_3$ |
$1$ |
$0$ |
$50$ |
$0$ |
$7890$ |
$0$ |
$1973960$ |
$0$ |
$587317906$ |
$0$ |
$188318438280$ |
$0$ |
$62953175049864$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$5$ |
$9$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$35$ |
$20$ |
$69$ |
$147$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$50$ |
$299$ |
$174$ |
$635$ |
$372$ |
$1391$ |
$3090$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$458$ |
$2995$ |
$1720$ |
$998$ |
$6627$ |
$3810$ |
$14865$ |
$8520$ |
$33570$ |
$76195$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$4824$ |
$33535$ |
$2760$ |
$18976$ |
$10790$ |
$75915$ |
$42896$ |
$24340$ |
$172929$ |
$97500$ |
$395438$ |
$222460$ |
$906927$ |
$2085174$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$7890$ |
$55660$ |
$404045$ |
$31378$ |
$225884$ |
$126664$ |
$926735$ |
$71240$ |
$517272$ |
$289492$ |
$2132441$ |
$1188232$ |
$663798$ |
$4918278$ |
| $$ |
$2736202$ |
$11365473$ |
$6313608$ |
$26307750$ |
$60984462$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&4&0&4&0&7&10&0&4&0&15&0&0&24\\0&9&0&11&0&40&0&0&45&0&29&0&69&109&0\\4&0&26&0&30&0&68&80&0&46&0&148&0&0&280\\0&11&0&19&0&64&0&0&85&0&45&0&139&205&0\\4&0&30&0&44&0&90&104&0&70&0&212&0&0&392\\0&40&0&64&0&248&0&0&344&0&192&0&560&848&0\\7&0&68&0&90&0&239&274&0&204&0&545&0&0&1176\\10&0&80&0&104&0&274&338&0&252&0&644&0&0&1432\\0&45&0&85&0&344&0&0&565&0&267&0&899&1387&0\\4&0&46&0&70&0&204&252&0&226&0&492&0&0&1208\\0&29&0&45&0&192&0&0&267&0&164&0&448&679&0\\15&0&148&0&212&0&545&644&0&492&0&1315&0&0&2840\\0&69&0&139&0&560&0&0&899&0&448&0&1494&2255&0\\0&109&0&205&0&848&0&0&1387&0&679&0&2255&3493&0\\24&0&280&0&392&0&1176&1432&0&1208&0&2840&0&0&6848\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&9&26&19&44&248&239&338&565&226&164&1315&1494&3493&6848&3411&3448&8282&7125&1997\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
