Name: | $\mathrm{U}(1)\times\mathrm{SU}(2)^2$ |
$\mathbb{R}$-dimension: | $7$ |
Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B,C\in\mathrm{SU}(2)\right\}$ |
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: | $C_2$ |
Order: | $2$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$25$ |
$0$ |
$320$ |
$0$ |
$5243$ |
$0$ |
$100380$ |
$0$ |
$2132856$ |
$a_2$ |
$1$ |
$3$ |
$12$ |
$60$ |
$373$ |
$2783$ |
$23759$ |
$222701$ |
$2227801$ |
$23362683$ |
$253963512$ |
$2840657196$ |
$32527565937$ |
$a_3$ |
$1$ |
$0$ |
$13$ |
$0$ |
$772$ |
$0$ |
$92970$ |
$0$ |
$15674204$ |
$0$ |
$3165039864$ |
$0$ |
$716306128032$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$3$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$12$ |
$6$ |
$15$ |
$25$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$13$ |
$60$ |
$33$ |
$92$ |
$56$ |
$167$ |
$320$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$75$ |
$373$ |
$216$ |
$131$ |
$662$ |
$401$ |
$1284$ |
$780$ |
$2570$ |
$5243$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$521$ |
$2783$ |
$314$ |
$1643$ |
$989$ |
$5411$ |
$3244$ |
$1956$ |
$10986$ |
$6580$ |
$22716$ |
$13566$ |
$47537$ |
$100380$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$772$ |
$4167$ |
$23759$ |
$2494$ |
$14038$ |
$8365$ |
$48709$ |
$5008$ |
$28875$ |
$17180$ |
$101983$ |
$60318$ |
$35785$ |
$215860$ |
$$ |
$127309$ |
$460488$ |
$270816$ |
$988470$ |
$2132856$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&3&0&0&0&0&0&0&2\\0&3&0&3&0&6&0&0&6&0&1&0&4&9&0\\2&0&7&0&3&0&8&12&0&3&0&8&0&0&18\\0&3&0&4&0&8&0&0&9&0&2&0&8&14&0\\0&0&3&0&7&0&10&6&0&7&0&16&0&0&22\\0&6&0&8&0&24&0&0&26&0&12&0&32&46&0\\1&0&8&0&10&0&21&19&0&14&0&32&0&0&56\\3&0&12&0&6&0&19&30&0&10&0&29&0&0&62\\0&6&0&9&0&26&0&0&34&0&13&0&41&61&0\\0&0&3&0&7&0&14&10&0&13&0&24&0&0&44\\0&1&0&2&0&12&0&0&13&0&13&0&25&26&0\\0&0&8&0&16&0&32&29&0&24&0&65&0&0&106\\0&4&0&8&0&32&0&0&41&0&25&0&65&81&0\\0&9&0&14&0&46&0&0&61&0&26&0&81&125&0\\2&0&18&0&22&0&56&62&0&44&0&106&0&0&210\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&4&7&24&21&30&34&13&13&65&65&125&210&90&104&182&135&34\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.