| 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: | $D_6$ |
| Order: | $12$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$27$ |
$0$ |
$490$ |
$0$ |
$12075$ |
$0$ |
$336798$ |
$0$ |
$9991674$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$56$ |
$507$ |
$5562$ |
$67046$ |
$848920$ |
$11071563$ |
$147289214$ |
$1987760604$ |
$27120961320$ |
$373258495506$ |
| $a_3$ |
$1$ |
$0$ |
$12$ |
$0$ |
$1344$ |
$0$ |
$330460$ |
$0$ |
$98098784$ |
$0$ |
$31411764972$ |
$0$ |
$10494860709564$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$4$ |
$13$ |
$27$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$12$ |
$56$ |
$32$ |
$106$ |
$60$ |
$222$ |
$490$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$80$ |
$507$ |
$286$ |
$170$ |
$1077$ |
$618$ |
$2383$ |
$1350$ |
$5335$ |
$12075$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$810$ |
$5562$ |
$456$ |
$3126$ |
$1768$ |
$12426$ |
$6988$ |
$3960$ |
$28140$ |
$15790$ |
$64116$ |
$35840$ |
$146678$ |
$336798$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1344$ |
$9256$ |
$67046$ |
$5214$ |
$37362$ |
$20896$ |
$153008$ |
$11694$ |
$85164$ |
$47496$ |
$351084$ |
$195016$ |
$108592$ |
$808152$ |
| $$ |
$448126$ |
$1864968$ |
$1032318$ |
$4312728$ |
$9991674$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&0&3&0&2&0&1&0&0&4\\0&3&0&1&0&6&0&0&9&0&5&0&7&18&0\\1&0&6&0&3&0&12&12&0&6&0&22&0&0&48\\0&1&0&7&0&10&0&0&11&0&3&0&31&30&0\\1&0&3&0&10&0&11&18&0&19&0&32&0&0&66\\0&6&0&10&0&39&0&0&58&0&29&0&91&139&0\\0&0&12&0&11&0&47&39&0&27&0&89&0&0&200\\3&0&12&0&18&0&39&66&0&52&0&101&0&0&242\\0&9&0&11&0&58&0&0&105&0&47&0&139&236&0\\2&0&6&0&19&0&27&52&0&60&0&81&0&0&204\\0&5&0&3&0&29&0&0&47&0&30&0&62&113&0\\1&0&22&0&32&0&89&101&0&81&0&213&0&0&476\\0&7&0&31&0&91&0&0&139&0&62&0&276&363&0\\0&18&0&30&0&139&0&0&236&0&113&0&363&588&0\\4&0&48&0&66&0&200&242&0&204&0&476&0&0&1150\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&7&10&39&47&66&105&60&30&213&276&588&1150&602&595&1444&1214&412\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/12$ | $0$ | $0$ | $1/6$ | $1/4$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
