| 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: | $u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$ |
| Name: | $C_2^2$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}, \begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$12$ |
$0$ |
$110$ |
$0$ |
$1260$ |
$0$ |
$16002$ |
$0$ |
$213906$ |
| $a_2$ |
$1$ |
$2$ |
$5$ |
$14$ |
$49$ |
$202$ |
$944$ |
$4720$ |
$24553$ |
$130658$ |
$705880$ |
$3855040$ |
$21232468$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$5$ |
$6$ |
$12$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$14$ |
$22$ |
$48$ |
$110$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$49$ |
$94$ |
$218$ |
$520$ |
$1260$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$202$ |
$444$ |
$1068$ |
$2610$ |
$6440$ |
$16002$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$944$ |
$2224$ |
$5466$ |
$13560$ |
$33852$ |
$84924$ |
$213906$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&1&0&0&0&1\\0&2&0&0&2&0&2&0&2&0\\1&0&2&1&0&2&0&3&0&3\\0&0&1&5&0&1&0&7&0&2\\0&2&0&0&6&0&6&0&10&0\\1&0&2&1&0&6&0&7&0&7\\0&2&0&0&6&0&8&0&10&0\\0&0&3&7&0&7&0&19&0&12\\0&2&0&0&10&0&10&0&22&0\\1&0&3&2&0&7&0&12&0&12\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&2&5&6&6&8&19&22&12\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
|---|
| $-$ | $1$ | $3/4$ | $0$ | $0$ | $0$ | $0$ | $3/4$ |
|---|
| $a_1=0$ | $1/4$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
