Name: | $\mathrm{U}(1)\times\mathrm{SU}(2)_2$ |
$\mathbb{R}$-dimension: | $4$ |
Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B\in\mathrm{SU}(2)\right\}$ |
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: | $\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$ |
Name: | $C_6$ |
Order: | $6$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{11} \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$42$ |
$0$ |
$660$ |
$0$ |
$12810$ |
$0$ |
$281736$ |
$0$ |
$6727908$ |
$a_2$ |
$1$ |
$2$ |
$11$ |
$75$ |
$621$ |
$5768$ |
$57953$ |
$615940$ |
$6830181$ |
$78310746$ |
$922531581$ |
$11115725877$ |
$136522207423$ |
$a_3$ |
$1$ |
$0$ |
$16$ |
$0$ |
$1512$ |
$0$ |
$249000$ |
$0$ |
$52009720$ |
$0$ |
$12486132792$ |
$0$ |
$3292873266936$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$11$ |
$6$ |
$20$ |
$42$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$16$ |
$75$ |
$46$ |
$150$ |
$90$ |
$310$ |
$660$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$110$ |
$621$ |
$372$ |
$228$ |
$1296$ |
$778$ |
$2754$ |
$1640$ |
$5910$ |
$12810$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$960$ |
$5768$ |
$570$ |
$3408$ |
$2022$ |
$12376$ |
$7296$ |
$4328$ |
$26802$ |
$15760$ |
$58420$ |
$34230$ |
$127988$ |
$281736$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1512$ |
$9166$ |
$57953$ |
$5402$ |
$33802$ |
$19784$ |
$126614$ |
$11604$ |
$73680$ |
$43000$ |
$278174$ |
$161480$ |
$94040$ |
$613792$ |
$$ |
$355502$ |
$1359274$ |
$785484$ |
$3019716$ |
$6727908$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&3&0&2&0&3&0&0&4\\0&4&0&2&0&10&0&0&10&0&8&0&10&20&0\\1&0&8&0&7&0&16&15&0&7&0&28&0&0&44\\0&2&0&8&0&14&0&0&14&0&6&0&30&30&0\\2&0&7&0&13&0&15&22&0&15&0&37&0&0&56\\0&10&0&14&0&50&0&0&58&0&34&0&86&122&0\\1&0&16&0&15&0&50&43&0&25&0&84&0&0&156\\3&0&15&0&22&0&43&58&0&41&0&92&0&0&176\\0&10&0&14&0&58&0&0&88&0&40&0&114&172&0\\2&0&7&0&15&0&25&41&0&40&0&60&0&0&132\\0&8&0&6&0&34&0&0&40&0&32&0&54&92&0\\3&0&28&0&37&0&84&92&0&60&0&181&0&0&324\\0&10&0&30&0&86&0&0&114&0&54&0&204&252&0\\0&20&0&30&0&122&0&0&172&0&92&0&252&384&0\\4&0&44&0&56&0&156&176&0&132&0&324&0&0&680\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&8&8&13&50&50&58&88&40&32&181&204&384&680&312&299&664&504&160\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.