| Name: | $\mathrm{U}(1)\times\mathrm{U}(1)_2$ |
| $\mathbb{R}$-dimension: | $6$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$5$ |
$0$ |
$75$ |
$0$ |
$1550$ |
$0$ |
$38115$ |
$0$ |
$1042650$ |
$0$ |
$30492462$ |
| $a_2$ |
$1$ |
$2$ |
$18$ |
$149$ |
$1498$ |
$16767$ |
$202023$ |
$2551362$ |
$33224778$ |
$441748247$ |
$5960976013$ |
$81335695872$ |
$1119502613935$ |
| $a_3$ |
$1$ |
$0$ |
$27$ |
$0$ |
$3969$ |
$0$ |
$987300$ |
$0$ |
$293663433$ |
$0$ |
$94159283652$ |
$0$ |
$31476588471108$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$5$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$18$ |
$9$ |
$34$ |
$75$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$27$ |
$149$ |
$88$ |
$318$ |
$183$ |
$694$ |
$1550$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$227$ |
$1498$ |
$859$ |
$505$ |
$3313$ |
$1908$ |
$7434$ |
$4250$ |
$16780$ |
$38115$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$2414$ |
$16767$ |
$1368$ |
$9489$ |
$5389$ |
$37958$ |
$21445$ |
$12190$ |
$86463$ |
$48760$ |
$197724$ |
$111195$ |
$453446$ |
$1042650$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$3969$ |
$27828$ |
$202023$ |
$15701$ |
$112941$ |
$63338$ |
$463367$ |
$35580$ |
$258639$ |
$144726$ |
$1066222$ |
$594106$ |
$331969$ |
$2459134$ |
| $$ |
$1368136$ |
$5682754$ |
$3156678$ |
$13153812$ |
$30492462$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&3&0&3&7&0&3&0&6&0&0&12\\0&5&0&4&0&20&0&0&25&0&16&0&30&56&0\\1&0&15&0&13&0&35&36&0&21&0&77&0&0&140\\0&4&0&14&0&32&0&0&35&0&18&0&83&98&0\\3&0&13&0&25&0&41&58&0&43&0&103&0&0&196\\0&20&0&32&0&124&0&0&172&0&96&0&280&424&0\\3&0&35&0&41&0&129&129&0&83&0&274&0&0&588\\7&0&36&0&58&0&129&181&0&142&0&316&0&0&716\\0&25&0&35&0&172&0&0&295&0&141&0&427&701&0\\3&0&21&0&43&0&83&142&0&151&0&243&0&0&604\\0&16&0&18&0&96&0&0&141&0&87&0&209&345&0\\6&0&77&0&103&0&274&316&0&243&0&662&0&0&1420\\0&30&0&83&0&280&0&0&427&0&209&0&792&1111&0\\0&56&0&98&0&424&0&0&701&0&345&0&1111&1753&0\\12&0&140&0&196&0&588&716&0&604&0&1420&0&0&3424\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&15&14&25&124&129&181&295&151&87&662&792&1753&3424&1764&1751&4266&3648&1161\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $1/4$ | $1/4$ | $1/4$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $1/4$ | $1/4$ | $1/4$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $1/4$ | $1/4$ | $1/4$ | $0$ | $0$ | $0$ | $0$ |
|---|
