| 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: | $u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$9$ |
$0$ |
$100$ |
$0$ |
$1225$ |
$0$ |
$15876$ |
$0$ |
$213444$ |
| $a_2$ |
$1$ |
$1$ |
$3$ |
$10$ |
$41$ |
$186$ |
$912$ |
$4656$ |
$24425$ |
$130402$ |
$705368$ |
$3854016$ |
$21230420$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$1$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$3$ |
$4$ |
$9$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$10$ |
$18$ |
$42$ |
$100$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$41$ |
$86$ |
$206$ |
$500$ |
$1225$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$186$ |
$428$ |
$1044$ |
$2570$ |
$6370$ |
$15876$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$912$ |
$2192$ |
$5418$ |
$13480$ |
$33712$ |
$84672$ |
$213444$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&1&0&1&0&1\\0&1&0&0&2&0&2&0&3&0\\0&0&2&1&0&2&0&3&0&3\\0&0&1&4&0&2&0&7&0&3\\0&2&0&0&6&0&6&0&10&0\\1&0&2&2&0&5&0&7&0&6\\0&2&0&0&6&0&7&0&11&0\\1&0&3&7&0&7&0&19&0&12\\0&3&0&0&10&0&11&0&20&0\\1&0&3&3&0&6&0&12&0&11\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&4&6&5&7&19&20&11\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
|---|
| $-$ | $1$ | $3/4$ | $0$ | $0$ | $1/2$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $3/4$ | $3/4$ | $0$ | $0$ | $1/2$ | $0$ | $1/4$ |
|---|
