Name: | $\mathrm{U}(1)\times\mathrm{U}(1)$ |
$\mathbb{R}$-dimension: | $2$ |
Description: | $\left\{\begin{bmatrix}A&0\\0&B\end{bmatrix}:A,B\in \mathrm{U}(1)\subseteq\mathrm{SU}(2)\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0\\0&J_2\end{bmatrix}, J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
Hodge circle: | $u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$36$ |
$0$ |
$400$ |
$0$ |
$4900$ |
$0$ |
$63504$ |
$0$ |
$853776$ |
$a_2$ |
$1$ |
$2$ |
$8$ |
$32$ |
$148$ |
$712$ |
$3584$ |
$18496$ |
$97444$ |
$521096$ |
$2820448$ |
$15414016$ |
$84917584$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$16$ |
$36$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$32$ |
$72$ |
$168$ |
$400$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$148$ |
$344$ |
$824$ |
$2000$ |
$4900$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$712$ |
$1712$ |
$4176$ |
$10280$ |
$25480$ |
$63504$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$3584$ |
$8768$ |
$21672$ |
$53920$ |
$134848$ |
$338688$ |
$853776$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&2&0&2&0&3&0&2\\0&4&0&0&8&0&8&0&12&0\\1&0&5&6&0&6&0&15&0&10\\2&0&6&12&0&12&0&26&0&16\\0&8&0&0&24&0&24&0&40&0\\2&0&6&12&0&16&0&30&0&20\\0&8&0&0&24&0&28&0&44&0\\3&0&15&26&0&30&0&73&0&50\\0&12&0&0&40&0&44&0&80&0\\2&0&10&16&0&20&0&50&0&40\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&5&12&24&16&28&73&80&40\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2$ and $n\in\mathbb{Z}$.