| 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: | $C_2^2$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$38$ |
$0$ |
$745$ |
$0$ |
$17094$ |
$0$ |
$427014$ |
$0$ |
$11289564$ |
| $a_2$ |
$1$ |
$2$ |
$12$ |
$83$ |
$750$ |
$7697$ |
$84989$ |
$981766$ |
$11707026$ |
$143022917$ |
$1781477919$ |
$22546777596$ |
$289206317739$ |
| $a_3$ |
$1$ |
$0$ |
$15$ |
$0$ |
$1878$ |
$0$ |
$386170$ |
$0$ |
$94401258$ |
$0$ |
$25461220092$ |
$0$ |
$7330330362168$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$12$ |
$5$ |
$18$ |
$38$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$15$ |
$83$ |
$46$ |
$160$ |
$92$ |
$340$ |
$745$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$117$ |
$750$ |
$425$ |
$250$ |
$1581$ |
$918$ |
$3466$ |
$2005$ |
$7670$ |
$17094$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1166$ |
$7697$ |
$668$ |
$4399$ |
$2532$ |
$16904$ |
$9704$ |
$5595$ |
$37646$ |
$21590$ |
$84290$ |
$48244$ |
$189420$ |
$427014$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1878$ |
$12444$ |
$84989$ |
$7138$ |
$48375$ |
$27636$ |
$190177$ |
$15810$ |
$108264$ |
$61753$ |
$428092$ |
$243353$ |
$138630$ |
$966747$ |
| $$ |
$548786$ |
$2188760$ |
$1240722$ |
$4966164$ |
$11289564$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&5&0&1&0&2&0&0&6\\0&3&0&2&0&10&0&0&13&0&7&0&14&27&0\\1&0&9&0&5&0&17&20&0&9&0&35&0&0&64\\0&2&0&8&0&16&0&0&17&0&8&0&37&46&0\\1&0&5&0&14&0&21&25&0&21&0&49&0&0&86\\0&10&0&16&0&60&0&0&80&0&44&0&124&184&0\\1&0&17&0&21&0&61&59&0&40&0&122&0&0&242\\5&0&20&0&25&0&59&88&0&55&0&132&0&0&284\\0&13&0&17&0&80&0&0&129&0&60&0&176&277&0\\1&0&9&0&21&0&40&55&0&57&0&99&0&0&220\\0&7&0&8&0&44&0&0&60&0&40&0&91&140&0\\2&0&35&0&49&0&122&132&0&99&0&280&0&0&544\\0&14&0&37&0&124&0&0&176&0&91&0&312&428&0\\0&27&0&46&0&184&0&0&277&0&140&0&428&654&0\\6&0&64&0&86&0&242&284&0&220&0&544&0&0&1192\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&9&8&14&60&61&88&129&57&40&280&312&654&1192&546&577&1265&987&279\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
