| 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: | $C_2\times C_4$ |
| Order: | $8$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &0 & \zeta_{8}^{1} \\0 & 0 & 0 & 0 & \zeta_{8}^{3} & 0 \\0 & 0 & 0 & \zeta_{8}^{3} & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$39$ |
$0$ |
$780$ |
$0$ |
$19075$ |
$0$ |
$521388$ |
$0$ |
$15246462$ |
| $a_2$ |
$1$ |
$2$ |
$10$ |
$77$ |
$754$ |
$8397$ |
$101047$ |
$1275780$ |
$16612666$ |
$220874909$ |
$2980490245$ |
$40667854350$ |
$559751325415$ |
| $a_3$ |
$1$ |
$0$ |
$13$ |
$0$ |
$1977$ |
$0$ |
$493540$ |
$0$ |
$146830089$ |
$0$ |
$47079617508$ |
$0$ |
$15738293869188$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$5$ |
$18$ |
$39$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$13$ |
$77$ |
$44$ |
$160$ |
$93$ |
$350$ |
$780$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$115$ |
$754$ |
$431$ |
$251$ |
$1659$ |
$954$ |
$3720$ |
$2130$ |
$8400$ |
$19075$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1208$ |
$8397$ |
$690$ |
$4747$ |
$2699$ |
$18984$ |
$10727$ |
$6090$ |
$43239$ |
$24380$ |
$98872$ |
$55615$ |
$226758$ |
$521388$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1977$ |
$13920$ |
$101047$ |
$7849$ |
$56479$ |
$31672$ |
$231697$ |
$17810$ |
$129327$ |
$72378$ |
$533126$ |
$297068$ |
$165967$ |
$1229592$ |
| $$ |
$684068$ |
$2841412$ |
$1578402$ |
$6577032$ |
$15246462$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&3&0&1&0&4&0&0&6\\0&3&0&2&0&10&0&0&11&0&8&0&16&28&0\\1&0&7&0&7&0&17&20&0&11&0&37&0&0&70\\0&2&0&6&0&16&0&0&21&0&10&0&37&50&0\\1&0&7&0&13&0&21&26&0&19&0&53&0&0&98\\0&10&0&16&0&62&0&0&86&0&48&0&140&212&0\\1&0&17&0&21&0&63&67&0&49&0&136&0&0&294\\3&0&20&0&26&0&67&87&0&64&0&160&0&0&358\\0&11&0&21&0&86&0&0&143&0&67&0&223&347&0\\1&0&11&0&19&0&49&64&0&61&0&123&0&0&302\\0&8&0&10&0&48&0&0&67&0&43&0&109&171&0\\4&0&37&0&53&0&136&160&0&123&0&330&0&0&710\\0&16&0&37&0&140&0&0&223&0&109&0&380&561&0\\0&28&0&50&0&212&0&0&347&0&171&0&561&875&0\\6&0&70&0&98&0&294&358&0&302&0&710&0&0&1712\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&6&13&62&63&87&143&61&43&330&380&875&1712&860&867&2082&1790&513\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
|---|
| $a_1=0$ | $3/8$ | $1/4$ | $0$ | $0$ | $1/4$ | $0$ | $0$ |
|---|
| $a_3=0$ | $5/8$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $3/8$ | $1/4$ | $0$ | $0$ | $1/4$ | $0$ | $0$ |
|---|
