| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$48$ |
$0$ |
$940$ |
$0$ |
$22470$ |
$0$ |
$595224$ |
$0$ |
$16850064$ |
| $a_2$ |
$1$ |
$2$ |
$12$ |
$95$ |
$934$ |
$10197$ |
$118947$ |
$1452600$ |
$18353034$ |
$237989741$ |
$3148974817$ |
$42330205590$ |
$576200895343$ |
| $a_3$ |
$1$ |
$0$ |
$18$ |
$0$ |
$2490$ |
$0$ |
$569760$ |
$0$ |
$158417490$ |
$0$ |
$48870194688$ |
$0$ |
$16020796617672$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$12$ |
$6$ |
$22$ |
$48$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$18$ |
$95$ |
$56$ |
$198$ |
$114$ |
$426$ |
$940$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$146$ |
$934$ |
$538$ |
$318$ |
$2034$ |
$1176$ |
$4500$ |
$2580$ |
$10020$ |
$22470$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1512$ |
$10197$ |
$864$ |
$5810$ |
$3322$ |
$22762$ |
$12942$ |
$7400$ |
$51162$ |
$29020$ |
$115520$ |
$65310$ |
$261772$ |
$595224$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2490$ |
$16884$ |
$118947$ |
$9606$ |
$67026$ |
$37884$ |
$269452$ |
$21444$ |
$151518$ |
$85400$ |
$612818$ |
$343820$ |
$193410$ |
$1398028$ |
| $$ |
$782768$ |
$3197936$ |
$1786932$ |
$7332612$ |
$16850064$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&4&0&3&0&4&0&0&8\\0&4&0&2&0&12&0&0&16&0&10&0&18&34&0\\1&0&9&0&8&0&23&23&0&14&0&45&0&0&88\\0&2&0&10&0&20&0&0&24&0&10&0&52&58&0\\2&0&8&0&16&0&24&35&0&29&0&61&0&0&120\\0&12&0&20&0&76&0&0&108&0&56&0&168&250&0\\1&0&23&0&24&0&83&79&0&54&0&163&0&0&348\\4&0&23&0&35&0&79&110&0&87&0&187&0&0&416\\0&16&0&24&0&108&0&0&180&0&84&0&252&404&0\\3&0&14&0&29&0&54&87&0&87&0&142&0&0&340\\0&10&0&10&0&56&0&0&84&0&50&0&118&198&0\\4&0&45&0&61&0&163&187&0&142&0&376&0&0&808\\0&18&0&52&0&168&0&0&252&0&118&0&456&626&0\\0&34&0&58&0&250&0&0&404&0&198&0&626&980&0\\8&0&88&0&120&0&348&416&0&340&0&808&0&0&1884\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&10&16&76&83&110&180&87&50&376&456&980&1884&942&935&2248&1871&600\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/8$ | $0$ | $1/4$ | $0$ | $1/8$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $1/4$ | $1/4$ | $0$ | $0$ | $1/4$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
