| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$27$ |
$0$ |
$480$ |
$0$ |
$11270$ |
$0$ |
$297738$ |
$0$ |
$8425494$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$56$ |
$492$ |
$5172$ |
$59691$ |
$726945$ |
$9178434$ |
$119000576$ |
$1574504409$ |
$21165153495$ |
$288100598979$ |
| $a_3$ |
$1$ |
$0$ |
$12$ |
$0$ |
$1278$ |
$0$ |
$285300$ |
$0$ |
$79214450$ |
$0$ |
$24435177732$ |
$0$ |
$8010399468456$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$4$ |
$13$ |
$27$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$12$ |
$56$ |
$32$ |
$105$ |
$60$ |
$219$ |
$480$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$80$ |
$492$ |
$280$ |
$168$ |
$1034$ |
$600$ |
$2268$ |
$1300$ |
$5030$ |
$11270$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$778$ |
$5172$ |
$444$ |
$2938$ |
$1682$ |
$11431$ |
$6504$ |
$3730$ |
$25632$ |
$14550$ |
$57820$ |
$32690$ |
$130956$ |
$297738$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1278$ |
$8506$ |
$59691$ |
$4848$ |
$33610$ |
$19008$ |
$134873$ |
$10762$ |
$75858$ |
$42770$ |
$306559$ |
$172020$ |
$96810$ |
$699184$ |
| $$ |
$391524$ |
$1599178$ |
$893592$ |
$3666558$ |
$8425494$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&0&3&0&2&0&1&0&0&4\\0&3&0&1&0&6&0&0&9&0&5&0&7&17&0\\1&0&6&0&3&0&12&12&0&6&0&21&0&0&44\\0&1&0&7&0&10&0&0&11&0&3&0&29&27&0\\1&0&3&0&10&0&11&17&0&18&0&31&0&0&60\\0&6&0&10&0&38&0&0&54&0&28&0&84&125&0\\0&0&12&0&11&0&45&36&0&24&0&82&0&0&174\\3&0&12&0&17&0&36&61&0&45&0&91&0&0&208\\0&9&0&11&0&54&0&0&93&0&43&0&121&203&0\\2&0&6&0&18&0&24&45&0&53&0&70&0&0&170\\0&5&0&3&0&28&0&0&43&0&28&0&56&100&0\\1&0&21&0&31&0&82&91&0&70&0&191&0&0&404\\0&7&0&29&0&84&0&0&121&0&56&0&240&308&0\\0&17&0&27&0&125&0&0&203&0&100&0&308&495&0\\4&0&44&0&60&0&174&208&0&170&0&404&0&0&942\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&7&10&38&45&61&93&53&28&191&240&495&942&487&479&1146&952&329\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/16$ | $0$ | $1/8$ | $0$ | $5/16$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $1/8$ | $1/8$ | $0$ | $0$ | $1/8$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
