| 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: | $D_4$ |
| Order: | $8$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$21$ |
$0$ |
$330$ |
$0$ |
$6419$ |
$0$ |
$142548$ |
$0$ |
$3469554$ |
| $a_2$ |
$1$ |
$2$ |
$8$ |
$44$ |
$329$ |
$2962$ |
$29790$ |
$321400$ |
$3642343$ |
$42827054$ |
$518216132$ |
$6415400884$ |
$80904420195$ |
| $a_3$ |
$1$ |
$0$ |
$8$ |
$0$ |
$762$ |
$0$ |
$129760$ |
$0$ |
$28649026$ |
$0$ |
$7309867296$ |
$0$ |
$2038903662312$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$3$ |
$10$ |
$21$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$8$ |
$44$ |
$23$ |
$75$ |
$45$ |
$155$ |
$330$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$55$ |
$329$ |
$186$ |
$114$ |
$649$ |
$389$ |
$1379$ |
$820$ |
$2960$ |
$6419$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$482$ |
$2962$ |
$285$ |
$1714$ |
$1013$ |
$6244$ |
$3666$ |
$2168$ |
$13520$ |
$7920$ |
$29486$ |
$17213$ |
$64666$ |
$142548$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$762$ |
$4645$ |
$29790$ |
$2725$ |
$17199$ |
$10014$ |
$64772$ |
$5844$ |
$37480$ |
$21758$ |
$142383$ |
$82184$ |
$47612$ |
$314538$ |
| $$ |
$181139$ |
$697745$ |
$400890$ |
$1553388$ |
$3469554$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&1&0&0&2\\0&2&0&1&0&5&0&0&5&0&4&0&5&10&0\\1&0&5&0&2&0&7&10&0&2&0&13&0&0&22\\0&1&0&4&0&7&0&0&7&0&3&0&15&15&0\\0&0&2&0&9&0&9&7&0&10&0&20&0&0&28\\0&5&0&7&0&25&0&0&29&0&17&0&43&62&0\\0&0&7&0&9&0&26&19&0&14&0&43&0&0&80\\3&0&10&0&7&0&19&37&0&17&0&43&0&0&92\\0&5&0&7&0&29&0&0&46&0&20&0&57&90&0\\0&0&2&0&10&0&14&17&0&26&0&32&0&0&72\\0&4&0&3&0&17&0&0&20&0&16&0&27&46&0\\1&0&13&0&20&0&43&43&0&32&0&93&0&0&166\\0&5&0&15&0&43&0&0&57&0&27&0&104&130&0\\0&10&0&15&0&62&0&0&90&0&46&0&130&202&0\\2&0&22&0&28&0&80&92&0&72&0&166&0&0&368\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&9&25&26&37&46&26&16&93&104&202&368&182&181&392&316&104\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
