| Name: | $\mathrm{U}(1)\times\mathrm{SU}(2)_2$ |
| $\mathbb{R}$-dimension: | $4$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B\in\mathrm{SU}(2)\right\}$ |
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: | $\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$ |
| Name: | $C_4$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$42$ |
$0$ |
$660$ |
$0$ |
$12838$ |
$0$ |
$285096$ |
$0$ |
$6939108$ |
| $a_2$ |
$1$ |
$2$ |
$11$ |
$75$ |
$623$ |
$5828$ |
$59313$ |
$642050$ |
$7282563$ |
$85648062$ |
$1036414961$ |
$12830752047$ |
$161808697025$ |
| $a_3$ |
$1$ |
$0$ |
$16$ |
$0$ |
$1524$ |
$0$ |
$259520$ |
$0$ |
$57298052$ |
$0$ |
$14619734592$ |
$0$ |
$4077807324624$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$11$ |
$6$ |
$20$ |
$42$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$16$ |
$75$ |
$46$ |
$150$ |
$90$ |
$310$ |
$660$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$110$ |
$623$ |
$372$ |
$228$ |
$1298$ |
$778$ |
$2758$ |
$1640$ |
$5920$ |
$12838$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$964$ |
$5828$ |
$570$ |
$3428$ |
$2026$ |
$12488$ |
$7332$ |
$4336$ |
$27040$ |
$15840$ |
$58972$ |
$34426$ |
$129332$ |
$285096$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1524$ |
$9290$ |
$59313$ |
$5450$ |
$34398$ |
$20028$ |
$129544$ |
$11688$ |
$74960$ |
$43516$ |
$284766$ |
$164368$ |
$95224$ |
$629076$ |
| $$ |
$362278$ |
$1395490$ |
$801780$ |
$3106776$ |
$6939108$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&3&0&2&0&3&0&0&4\\0&4&0&2&0&10&0&0&10&0&8&0&10&20&0\\1&0&8&0&7&0&16&15&0&7&0&28&0&0&44\\0&2&0&8&0&14&0&0&14&0&6&0&30&30&0\\2&0&7&0&13&0&15&22&0&15&0&37&0&0&56\\0&10&0&14&0&50&0&0&58&0&34&0&86&124&0\\1&0&16&0&15&0&50&43&0&25&0&84&0&0&160\\3&0&15&0&22&0&43&60&0&43&0&92&0&0&184\\0&10&0&14&0&58&0&0&92&0&40&0&114&180&0\\2&0&7&0&15&0&25&43&0&46&0&60&0&0&144\\0&8&0&6&0&34&0&0&40&0&32&0&54&92&0\\3&0&28&0&37&0&84&92&0&60&0&183&0&0&332\\0&10&0&30&0&86&0&0&114&0&54&0&208&260&0\\0&20&0&30&0&124&0&0&180&0&92&0&260&404&0\\4&0&44&0&56&0&160&184&0&144&0&332&0&0&736\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&8&8&13&50&50&60&92&46&32&183&208&404&736&364&339&784&622&208\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
