| Name: | $\mathrm{SU}(2)\times\mathrm{U}(1)_2$ |
| $\mathbb{R}$-dimension: | $4$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &0 & \zeta_{12}^{1} \\0 & 0 & 0 & 0 & \zeta_{12}^{5} & 0 \\0 & 0 & 0 & \zeta_{12}^{5} & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0& 0 & 0 & -1 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$17$ |
$0$ |
$280$ |
$0$ |
$5999$ |
$0$ |
$145593$ |
$0$ |
$3801006$ |
| $a_2$ |
$1$ |
$2$ |
$8$ |
$42$ |
$307$ |
$2827$ |
$29686$ |
$335022$ |
$3949931$ |
$47979189$ |
$595837954$ |
$7529147881$ |
$96494854418$ |
| $a_3$ |
$1$ |
$0$ |
$8$ |
$0$ |
$696$ |
$0$ |
$131965$ |
$0$ |
$31660496$ |
$0$ |
$8500033542$ |
$0$ |
$2444374091412$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$3$ |
$9$ |
$17$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$8$ |
$42$ |
$21$ |
$66$ |
$38$ |
$132$ |
$280$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$51$ |
$307$ |
$168$ |
$99$ |
$590$ |
$344$ |
$1254$ |
$730$ |
$2725$ |
$5999$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$444$ |
$2827$ |
$256$ |
$1601$ |
$926$ |
$5980$ |
$3447$ |
$1998$ |
$13120$ |
$7555$ |
$29090$ |
$16709$ |
$64918$ |
$145593$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$696$ |
$4442$ |
$29686$ |
$2556$ |
$16885$ |
$9677$ |
$65431$ |
$5561$ |
$37348$ |
$21373$ |
$146188$ |
$83305$ |
$47587$ |
$328421$ |
| $$ |
$186816$ |
$740712$ |
$420609$ |
$1675758$ |
$3801006$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&0&0&0&2\\0&2&0&1&0&4&0&0&5&0&2&0&4&10&0\\1&0&5&0&1&0&6&10&0&2&0&10&0&0&22\\0&1&0&4&0&6&0&0&7&0&2&0&12&16&0\\0&0&1&0&7&0&8&6&0&9&0&19&0&0&29\\0&4&0&6&0&22&0&0&28&0&15&0&42&63&0\\0&0&6&0&8&0&23&19&0&15&0&42&0&0&82\\3&0&10&0&6&0&19&36&0&15&0&42&0&0&96\\0&5&0&7&0&28&0&0&44&0&19&0&60&94&0\\0&0&2&0&9&0&15&15&0&21&0&35&0&0&74\\0&2&0&2&0&15&0&0&19&0&16&0&32&46&0\\0&0&10&0&19&0&42&42&0&35&0&97&0&0&182\\0&4&0&12&0&42&0&0&60&0&32&0&106&142&0\\0&10&0&16&0&63&0&0&94&0&46&0&142&222&0\\2&0&22&0&29&0&82&96&0&74&0&182&0&0&400\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&7&22&23&36&44&21&16&97&106&222&400&184&205&419&330&89\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $1/6$ | $0$ | $0$ | $1/3$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
