| Name: | $\mathrm{U}(1)\times\mathrm{U}(1)_2$ |
| $\mathbb{R}$-dimension: | $6$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
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: | $u\mapsto\mathrm{diag}(u,\bar u, u, u,\bar u, \bar u)$ |
| Name: | $C_2\times D_4$ |
| Order: | $16$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$33$ |
$0$ |
$570$ |
$0$ |
$12495$ |
$0$ |
$314748$ |
$0$ |
$8668968$ |
| $a_2$ |
$1$ |
$2$ |
$10$ |
$65$ |
$555$ |
$5582$ |
$62296$ |
$743465$ |
$9283931$ |
$119681030$ |
$1578937950$ |
$21194304155$ |
$288293747517$ |
| $a_3$ |
$1$ |
$0$ |
$13$ |
$0$ |
$1386$ |
$0$ |
$291220$ |
$0$ |
$79530290$ |
$0$ |
$24452628228$ |
$0$ |
$8011397211048$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$5$ |
$16$ |
$33$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$13$ |
$65$ |
$38$ |
$126$ |
$75$ |
$264$ |
$570$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$93$ |
$555$ |
$323$ |
$195$ |
$1171$ |
$690$ |
$2552$ |
$1490$ |
$5620$ |
$12495$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$863$ |
$5582$ |
$504$ |
$3212$ |
$1865$ |
$12295$ |
$7075$ |
$4100$ |
$27429$ |
$15730$ |
$61580$ |
$35175$ |
$138922$ |
$314748$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1386$ |
$9055$ |
$62296$ |
$5208$ |
$35339$ |
$20147$ |
$140330$ |
$11522$ |
$79452$ |
$45140$ |
$317999$ |
$179550$ |
$101745$ |
$723412$ |
| $$ |
$407456$ |
$1651020$ |
$927738$ |
$3778530$ |
$8668968$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&3&0&1&0&2&0&0&4\\0&3&0&2&0&8&0&0&9&0&6&0&10&18&0\\1&0&7&0&5&0&13&14&0&7&0&25&0&0&44\\0&2&0&6&0&12&0&0&13&0&6&0&27&30&0\\1&0&5&0&11&0&15&18&0&15&0&35&0&0&60\\0&8&0&12&0&44&0&0&56&0&32&0&88&130&0\\1&0&13&0&15&0&45&41&0&27&0&86&0&0&174\\3&0&14&0&18&0&41&60&0&40&0&97&0&0&208\\0&9&0&13&0&56&0&0&90&0&43&0&128&203&0\\1&0&7&0&15&0&27&40&0&45&0&71&0&0&170\\0&6&0&6&0&32&0&0&43&0&29&0&63&103&0\\2&0&25&0&35&0&86&97&0&71&0&201&0&0&404\\0&10&0&27&0&88&0&0&128&0&63&0&231&317&0\\0&18&0&30&0&130&0&0&203&0&103&0&317&497&0\\4&0&44&0&60&0&174&208&0&170&0&404&0&0&942\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&6&11&44&45&60&90&45&29&201&231&497&942&476&471&1117&947&292\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $5/16$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $5/16$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $5/16$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
