| 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: | $C_2^3$ |
| Order: | $8$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1& 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$20$ |
$0$ |
$375$ |
$0$ |
$8554$ |
$0$ |
$213528$ |
$0$ |
$5644848$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$50$ |
$400$ |
$3922$ |
$42712$ |
$491528$ |
$5855430$ |
$71517164$ |
$890755960$ |
$11273439498$ |
$144603310177$ |
| $a_3$ |
$1$ |
$0$ |
$9$ |
$0$ |
$950$ |
$0$ |
$193190$ |
$0$ |
$47201770$ |
$0$ |
$12730623444$ |
$0$ |
$3665165346744$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$3$ |
$10$ |
$20$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$9$ |
$50$ |
$25$ |
$83$ |
$47$ |
$172$ |
$375$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$62$ |
$400$ |
$218$ |
$128$ |
$799$ |
$463$ |
$1739$ |
$1005$ |
$3840$ |
$8554$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$594$ |
$3922$ |
$338$ |
$2216$ |
$1273$ |
$8477$ |
$4863$ |
$2805$ |
$18840$ |
$10805$ |
$42160$ |
$24129$ |
$94724$ |
$213528$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$950$ |
$6254$ |
$42712$ |
$3584$ |
$24236$ |
$13840$ |
$95162$ |
$7915$ |
$54165$ |
$30894$ |
$214096$ |
$121704$ |
$69336$ |
$483416$ |
| $$ |
$274421$ |
$1094422$ |
$620382$ |
$2483124$ |
$5644848$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&0&0&0&3\\0&2&0&1&0&5&0&0&7&0&3&0&6&14&0\\1&0&6&0&1&0&8&12&0&3&0&16&0&0&32\\0&1&0&5&0&8&0&0&8&0&3&0&19&23&0\\0&0&1&0&9&0&11&10&0&13&0&26&0&0&43\\0&5&0&8&0&30&0&0&40&0&22&0&62&92&0\\0&0&8&0&11&0&32&27&0&20&0&62&0&0&121\\4&0&12&0&10&0&27&51&0&25&0&62&0&0&142\\0&7&0&8&0&40&0&0&66&0&30&0&86&139&0\\0&0&3&0&13&0&20&25&0&33&0&51&0&0&110\\0&3&0&3&0&22&0&0&30&0&22&0&46&69&0\\0&0&16&0&26&0&62&62&0&51&0&143&0&0&272\\0&6&0&19&0&62&0&0&86&0&46&0&160&212&0\\0&14&0&23&0&92&0&0&139&0&69&0&212&329&0\\3&0&32&0&43&0&121&142&0&110&0&272&0&0&596\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&9&30&32&51&66&33&22&143&160&329&596&277&303&639&502&146\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
