| 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: | $C_2^3$ |
| Order: | $8$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &0 & i \\0 & 0 & 0 & 0 & i & 0 \\0 & 0 & 0 & i & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0& 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0& 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
| Maximal subgroups: | $L_2(C_2)$, $L(D_{2,1},C_{2,1})$${}^{\times 2}$, $L_2(C_{2,1})$${}^{\times 2}$, $L_1(D_{2,1})$, $L(D_{2,1},C_2)$ |
| Minimal supergroups: | $L_2(D_{6,1})$, $L_2(D_{4,1})$, $L_2(J(D_2))$${}^{\times 3}$, $L_2(D_{6,2})$, $L_2(D_{4,2})$${}^{\times 2}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$39$ |
$0$ |
$780$ |
$0$ |
$19075$ |
$0$ |
$521388$ |
$0$ |
$15246462$ |
| $a_2$ |
$1$ |
$3$ |
$14$ |
$90$ |
$794$ |
$8518$ |
$101411$ |
$1276873$ |
$16615946$ |
$220884750$ |
$2980519769$ |
$40667942923$ |
$559751591135$ |
| $a_3$ |
$1$ |
$0$ |
$15$ |
$0$ |
$2001$ |
$0$ |
$493860$ |
$0$ |
$146834569$ |
$0$ |
$47079682020$ |
$0$ |
$15738294815364$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$14$ |
$6$ |
$19$ |
$39$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$15$ |
$90$ |
$47$ |
$164$ |
$96$ |
$353$ |
$780$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$121$ |
$794$ |
$440$ |
$257$ |
$1672$ |
$963$ |
$3732$ |
$2140$ |
$8410$ |
$19075$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1226$ |
$8518$ |
$702$ |
$4774$ |
$2717$ |
$19024$ |
$10754$ |
$6110$ |
$43278$ |
$24410$ |
$98912$ |
$55650$ |
$226793$ |
$521388$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2001$ |
$13974$ |
$101411$ |
$7885$ |
$56560$ |
$31726$ |
$231818$ |
$17850$ |
$129408$ |
$72438$ |
$533246$ |
$297158$ |
$166037$ |
$1229722$ |
| $$ |
$684173$ |
$2841552$ |
$1578528$ |
$6577158$ |
$15246462$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&5&0&0&0&2&0&0&6\\0&3&0&3&0&10&0&0&12&0&7&0&16&27&0\\2&0&9&0&5&0&16&24&0&9&0&34&0&0&70\\0&3&0&6&0&16&0&0&21&0&10&0&36&50&0\\0&0&5&0&15&0&24&20&0&23&0&56&0&0&98\\0&10&0&16&0&62&0&0&86&0&48&0&140&212&0\\1&0&16&0&24&0&63&63&0&52&0&138&0&0&294\\5&0&24&0&20&0&63&99&0&56&0&154&0&0&358\\0&12&0&21&0&86&0&0&143&0&67&0&222&347&0\\0&0&9&0&23&0&52&56&0&67&0&126&0&0&302\\0&7&0&10&0&48&0&0&67&0&43&0&111&170&0\\2&0&34&0&56&0&138&154&0&126&0&334&0&0&710\\0&16&0&36&0&140&0&0&222&0&111&0&380&561&0\\0&27&0&50&0&212&0&0&347&0&170&0&561&877&0\\6&0&70&0&98&0&294&358&0&302&0&710&0&0&1712\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&9&6&15&62&63&99&143&67&43&334&380&877&1712&862&893&2082&1802&515\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $3/8$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_3=0$ | $3/8$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=a_3=0$ | $3/8$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
