| 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: | $C_4$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$5$ |
$0$ |
$62$ |
$0$ |
$1065$ |
$0$ |
$21854$ |
$0$ |
$503706$ |
$0$ |
$12597948$ |
| $a_2$ |
$1$ |
$3$ |
$17$ |
$123$ |
$1055$ |
$10093$ |
$104331$ |
$1142417$ |
$13075675$ |
$154939089$ |
$1887142331$ |
$23498297353$ |
$297887753801$ |
| $a_3$ |
$1$ |
$0$ |
$24$ |
$0$ |
$2604$ |
$0$ |
$457280$ |
$0$ |
$102500692$ |
$0$ |
$26478326112$ |
$0$ |
$7466732198640$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$5$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$17$ |
$10$ |
$31$ |
$62$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$24$ |
$123$ |
$74$ |
$245$ |
$148$ |
$506$ |
$1065$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$184$ |
$1055$ |
$626$ |
$376$ |
$2203$ |
$1308$ |
$4686$ |
$2770$ |
$10075$ |
$21854$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1646$ |
$10093$ |
$976$ |
$5908$ |
$3476$ |
$21711$ |
$12680$ |
$7436$ |
$47186$ |
$27470$ |
$103301$ |
$59948$ |
$227514$ |
$503706$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2604$ |
$16158$ |
$104331$ |
$9436$ |
$60310$ |
$34980$ |
$228989$ |
$20352$ |
$132028$ |
$76340$ |
$505866$ |
$290874$ |
$167712$ |
$1123263$ |
| $$ |
$644252$ |
$2505118$ |
$1433460$ |
$5608302$ |
$12597948$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&2&0&3&4&0&2&0&5&0&0&8\\0&5&0&5&0&16&0&0&17&0&10&0&22&33&0\\2&0&12&0&12&0&26&28&0&16&0&46&0&0&80\\0&5&0&9&0&24&0&0&29&0&14&0&42&57&0\\2&0&12&0&17&0&31&34&0&22&0&62&0&0&104\\0&16&0&24&0&84&0&0&104&0&56&0&152&220&0\\3&0&26&0&31&0&78&82&0&56&0&147&0&0&284\\4&0&28&0&34&0&82&98&0&64&0&166&0&0&328\\0&17&0&29&0&104&0&0&151&0&70&0&216&317&0\\2&0&16&0&22&0&56&64&0&54&0&114&0&0&248\\0&10&0&14&0&56&0&0&70&0&45&0&109&160&0\\5&0&46&0&62&0&147&166&0&114&0&319&0&0&608\\0&22&0&42&0&152&0&0&216&0&109&0&339&484&0\\0&33&0&57&0&220&0&0&317&0&160&0&484&725&0\\8&0&80&0&104&0&284&328&0&248&0&608&0&0&1304\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&12&9&17&84&78&98&151&54&45&319&339&725&1304&575&595&1305&1006&257\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
