| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(J(C_1))$, $L_1(C_{2,1})$, $L_1(C_2)$ |
| Minimal supergroups: | $L_1(J(D_2))$${}^{\times 3}$, $L(J(C_4),J(C_2))$, $L_2(J(C_2))$, $L_1(J(C_4))$, $L_1(J(C_6))$, $L_1(J(D_3))$, $L(J(D_2),J(C_2))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$54$ |
$0$ |
$1240$ |
$0$ |
$33670$ |
$0$ |
$978264$ |
$0$ |
$29546748$ |
| $a_2$ |
$1$ |
$2$ |
$14$ |
$119$ |
$1290$ |
$15397$ |
$193079$ |
$2492926$ |
$32841146$ |
$439215509$ |
$5944164669$ |
$81223573576$ |
$1118751719655$ |
| $a_3$ |
$1$ |
$0$ |
$22$ |
$0$ |
$3618$ |
$0$ |
$967240$ |
$0$ |
$292522258$ |
$0$ |
$94093303752$ |
$0$ |
$31472714093832$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$14$ |
$6$ |
$24$ |
$54$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$22$ |
$119$ |
$68$ |
$250$ |
$138$ |
$548$ |
$1240$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$186$ |
$1290$ |
$722$ |
$418$ |
$2864$ |
$1616$ |
$6476$ |
$3620$ |
$14720$ |
$33670$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$2140$ |
$15397$ |
$1188$ |
$8590$ |
$4802$ |
$35026$ |
$19526$ |
$10940$ |
$80178$ |
$44640$ |
$184152$ |
$102270$ |
$423948$ |
$978264$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$3618$ |
$26028$ |
$193079$ |
$14530$ |
$107074$ |
$59500$ |
$444194$ |
$33060$ |
$246066$ |
$136476$ |
$1024876$ |
$566952$ |
$314154$ |
$2369352$ |
| $$ |
$1309140$ |
$5486740$ |
$3027780$ |
$12723984$ |
$29546748$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&6&0&5&0&4&0&0&12\\0&4&0&2&0&14&0&0&24&0&12&0&24&50&0\\1&0&11&0&8&0&31&31&0&20&0&63&0&0&140\\0&2&0&14&0&26&0&0&32&0&12&0&82&90&0\\2&0&8&0&20&0&32&51&0&49&0&87&0&0&196\\0&14&0&26&0&104&0&0&166&0&80&0&264&404&0\\1&0&31&0&32&0&123&117&0&84&0&257&0&0&588\\6&0&31&0&51&0&117&176&0&149&0&299&0&0&716\\0&24&0&32&0&166&0&0&296&0&138&0&418&696&0\\5&0&20&0&49&0&84&149&0&155&0&244&0&0&604\\0&12&0&12&0&80&0&0&138&0&74&0&190&330&0\\4&0&63&0&87&0&257&299&0&244&0&614&0&0&1420\\0&24&0&82&0&264&0&0&418&0&190&0&788&1090&0\\0&50&0&90&0&404&0&0&696&0&330&0&1090&1736&0\\12&0&140&0&196&0&588&716&0&604&0&1420&0&0&3424\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&11&14&20&104&123&176&296&155&74&614&788&1736&3424&1758&1751&4272&3593&1168\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
