| 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: | $C_3$ |
| Order: | $3$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(C_1)$ |
| Minimal supergroups: | $L(C_{6,1},C_3)$, $L_2(C_3)$, $L_1(J(C_3))$, $L(D_3,C_3)$, $L(D_{3,2},C_3)$, $L_1(T)$, $L_1(D_3)$, $L_1(C_{6,1})$, $L_1(C_6)$, $L(C_6,C_3)$, $L_1(D_{3,2})$, $L(J(C_3),C_3)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$6$ |
$0$ |
$90$ |
$0$ |
$1900$ |
$0$ |
$48090$ |
$0$ |
$1346436$ |
$0$ |
$39963924$ |
| $a_2$ |
$1$ |
$3$ |
$21$ |
$185$ |
$1917$ |
$21933$ |
$267271$ |
$3393015$ |
$44278413$ |
$589133693$ |
$7950973731$ |
$108483641115$ |
$1493033374063$ |
| $a_3$ |
$1$ |
$0$ |
$32$ |
$0$ |
$5208$ |
$0$ |
$1319720$ |
$0$ |
$392366296$ |
$0$ |
$125646652152$ |
$0$ |
$41979436937592$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$6$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$21$ |
$12$ |
$42$ |
$90$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$32$ |
$185$ |
$108$ |
$394$ |
$228$ |
$858$ |
$1900$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$292$ |
$1917$ |
$1096$ |
$632$ |
$4230$ |
$2412$ |
$9442$ |
$5360$ |
$21240$ |
$48090$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$3144$ |
$21933$ |
$1788$ |
$12360$ |
$6988$ |
$49482$ |
$27808$ |
$15680$ |
$112326$ |
$62960$ |
$256164$ |
$143220$ |
$586362$ |
$1346436$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$5208$ |
$36760$ |
$267271$ |
$20652$ |
$149040$ |
$83296$ |
$611402$ |
$46656$ |
$340224$ |
$189704$ |
$1403670$ |
$779584$ |
$433808$ |
$3231828$ |
| $$ |
$1791804$ |
$7458822$ |
$4128768$ |
$17249652$ |
$39963924$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&4&6&0&4&0&9&0&0&16\\0&6&0&6&0&24&0&0&30&0&18&0&42&70&0\\2&0&16&0&18&0&44&50&0&34&0&92&0&0&192\\0&6&0&14&0&40&0&0&58&0&26&0&94&134&0\\3&0&18&0&27&0&54&68&0&50&0&129&0&0&264\\0&24&0&40&0&156&0&0&232&0&116&0&364&556&0\\4&0&44&0&54&0&162&182&0&142&0&350&0&0&800\\6&0&50&0&68&0&182&224&0&178&0&420&0&0&968\\0&30&0&58&0&232&0&0&390&0&174&0&606&930&0\\4&0&34&0&50&0&142&178&0&156&0&336&0&0&816\\0&18&0&26&0&116&0&0&174&0&100&0&280&442&0\\9&0&92&0&129&0&350&420&0&336&0&837&0&0&1904\\0&42&0&94&0&364&0&0&606&0&280&0&996&1494&0\\0&70&0&134&0&556&0&0&930&0&442&0&1494&2322&0\\16&0&192&0&264&0&800&968&0&816&0&1904&0&0&4600\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&16&14&27&156&162&224&390&156&100&837&996&2322&4600&2266&2301&5520&4688&1314\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
