| Name: | $\mathrm{U}(1)^2\times\mathrm{SU}(2)$ |
| $\mathbb{R}$-dimension: | $5$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A,B\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),C\in\mathrm{SU}(2)\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0&0\\0&J_2&0\\0&0&J_2\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
| Hodge circle: | $u\mapsto\mathrm{diag}(u,\bar u, u, \bar u, u, \bar u)$ |
| Name: | $C_4$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\-1 & 0 & 0 & 0& 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1\\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$17$ |
$0$ |
$270$ |
$0$ |
$5439$ |
$0$ |
$123123$ |
$0$ |
$3001284$ |
| $a_2$ |
$1$ |
$2$ |
$7$ |
$38$ |
$284$ |
$2567$ |
$25820$ |
$276845$ |
$3098618$ |
$35803781$ |
$424226048$ |
$5130820433$ |
$63130916975$ |
| $a_3$ |
$1$ |
$0$ |
$7$ |
$0$ |
$658$ |
$0$ |
$111760$ |
$0$ |
$23819390$ |
$0$ |
$5769914640$ |
$0$ |
$1520601479760$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$7$ |
$3$ |
$9$ |
$17$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$7$ |
$38$ |
$20$ |
$64$ |
$38$ |
$129$ |
$270$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$49$ |
$284$ |
$161$ |
$96$ |
$557$ |
$330$ |
$1174$ |
$695$ |
$2515$ |
$5439$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$420$ |
$2567$ |
$248$ |
$1485$ |
$874$ |
$5404$ |
$3164$ |
$1861$ |
$11688$ |
$6830$ |
$25489$ |
$14854$ |
$55895$ |
$123123$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$658$ |
$4038$ |
$25820$ |
$2362$ |
$14927$ |
$8688$ |
$56155$ |
$5074$ |
$32524$ |
$18891$ |
$123459$ |
$71351$ |
$41342$ |
$272746$ |
| $$ |
$157276$ |
$604807$ |
$348012$ |
$1345407$ |
$3001284$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&1&0&0&2\\0&2&0&1&0&4&0&0&4&0&2&0&5&9&0\\1&0&4&0&2&0&6&9&0&3&0&10&0&0&20\\0&1&0&3&0&6&0&0&8&0&3&0&10&14&0\\0&0&2&0&6&0&8&6&0&7&0&17&0&0&26\\0&4&0&6&0&21&0&0&26&0&14&0&38&54&0\\0&0&6&0&8&0&21&19&0&15&0&37&0&0&70\\2&0&9&0&6&0&19&29&0&13&0&38&0&0&80\\0&4&0&8&0&26&0&0&38&0&17&0&53&77&0\\0&0&3&0&7&0&15&13&0&15&0&30&0&0&60\\0&2&0&3&0&14&0&0&17&0&13&0&28&38&0\\1&0&10&0&17&0&37&38&0&30&0&81&0&0&148\\0&5&0&10&0&38&0&0&53&0&28&0&85&116&0\\0&9&0&14&0&54&0&0&77&0&38&0&116&174&0\\2&0&20&0&26&0&70&80&0&60&0&148&0&0&310\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&6&21&21&29&38&15&13&81&85&174&310&136&146&296&229&57\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $3/4$ | $0$ | $0$ | $1/2$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
