| 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: | $C_2\times C_6$ |
| Order: | $12$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{11} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{1} \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$33$ |
$0$ |
$570$ |
$0$ |
$12985$ |
$0$ |
$346878$ |
$0$ |
$10103478$ |
| $a_2$ |
$1$ |
$2$ |
$10$ |
$65$ |
$566$ |
$5917$ |
$69107$ |
$860722$ |
$11138862$ |
$147673073$ |
$1989954875$ |
$27133541272$ |
$373330838415$ |
| $a_3$ |
$1$ |
$0$ |
$13$ |
$0$ |
$1455$ |
$0$ |
$335800$ |
$0$ |
$98334999$ |
$0$ |
$31422135528$ |
$0$ |
$10495319391480$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$5$ |
$16$ |
$33$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$13$ |
$65$ |
$38$ |
$126$ |
$75$ |
$264$ |
$570$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$93$ |
$566$ |
$327$ |
$197$ |
$1199$ |
$702$ |
$2626$ |
$1520$ |
$5810$ |
$12985$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$892$ |
$5917$ |
$516$ |
$3371$ |
$1939$ |
$13134$ |
$7477$ |
$4290$ |
$29529$ |
$16740$ |
$66810$ |
$37695$ |
$151900$ |
$346878$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1455$ |
$9748$ |
$69107$ |
$5553$ |
$38781$ |
$21874$ |
$157055$ |
$12374$ |
$87945$ |
$49406$ |
$358968$ |
$200414$ |
$112267$ |
$823444$ |
| $$ |
$458570$ |
$1894620$ |
$1052604$ |
$4370310$ |
$10103478$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&3&0&1&0&2&0&0&4\\0&3&0&2&0&8&0&0&9&0&6&0&10&18&0\\1&0&7&0&5&0&13&14&0&7&0&25&0&0&48\\0&2&0&6&0&12&0&0&13&0&6&0&29&32&0\\1&0&5&0&11&0&15&18&0&17&0&35&0&0&66\\0&8&0&12&0&44&0&0&60&0&32&0&94&140&0\\1&0&13&0&15&0&47&43&0&31&0&92&0&0&200\\3&0&14&0&18&0&43&65&0&48&0&104&0&0&242\\0&9&0&13&0&60&0&0&103&0&47&0&145&235&0\\1&0&7&0&17&0&31&48&0&53&0&83&0&0&204\\0&6&0&6&0&32&0&0&47&0&29&0&67&113&0\\2&0&25&0&35&0&92&104&0&83&0&216&0&0&476\\0&10&0&29&0&94&0&0&145&0&67&0&266&369&0\\0&18&0&32&0&140&0&0&235&0&113&0&369&585&0\\4&0&48&0&66&0&200&242&0&204&0&476&0&0&1150\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&6&11&44&47&65&103&53&29&216&266&585&1150&588&593&1424&1206&385\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/12$ | $1/6$ | $0$ | $1/6$ | $1/12$ |
|---|
| $a_1=0$ | $1/3$ | $1/4$ | $0$ | $1/6$ | $0$ | $0$ | $1/12$ |
|---|
| $a_3=0$ | $1/3$ | $1/4$ | $0$ | $1/6$ | $0$ | $0$ | $1/12$ |
|---|
| $a_1=a_3=0$ | $1/3$ | $1/4$ | $0$ | $1/6$ | $0$ | $0$ | $1/12$ |
|---|
