| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0& 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(J(C_1))$, $L(C_{2,1},C_1)$, $L(C_2,C_1)$ |
| Minimal supergroups: | $L(J(D_3),J(C_3))$, $L_2(J(C_2))$, $L(J(D_2),J(C_2))$${}^{\times 2}$, $L(J(C_6),J(C_3))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$51$ |
$0$ |
$1230$ |
$0$ |
$33635$ |
$0$ |
$978138$ |
$0$ |
$29546286$ |
| $a_2$ |
$1$ |
$2$ |
$14$ |
$119$ |
$1290$ |
$15397$ |
$193079$ |
$2492926$ |
$32841146$ |
$439215509$ |
$5944164669$ |
$81223573576$ |
$1118751719655$ |
| $a_3$ |
$1$ |
$0$ |
$19$ |
$0$ |
$3585$ |
$0$ |
$966820$ |
$0$ |
$292516553$ |
$0$ |
$94093223364$ |
$0$ |
$31472712934212$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$14$ |
$5$ |
$22$ |
$51$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$19$ |
$119$ |
$64$ |
$244$ |
$135$ |
$542$ |
$1230$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$179$ |
$1290$ |
$711$ |
$409$ |
$2847$ |
$1604$ |
$6458$ |
$3610$ |
$14700$ |
$33635$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$2118$ |
$15397$ |
$1176$ |
$8557$ |
$4781$ |
$34976$ |
$19493$ |
$10910$ |
$80127$ |
$44600$ |
$184092$ |
$102235$ |
$423878$ |
$978138$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$3585$ |
$25964$ |
$193079$ |
$14485$ |
$106977$ |
$59434$ |
$444047$ |
$33020$ |
$245967$ |
$136406$ |
$1024726$ |
$566842$ |
$314049$ |
$2369182$ |
| $$ |
$1309000$ |
$5486530$ |
$3027654$ |
$12723732$ |
$29546286$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&7&0&3&0&4&0&0&12\\0&3&0&2&0&14&0&0&23&0&12&0&26&50&0\\1&0&11&0&7&0&29&34&0&19&0&63&0&0&140\\0&2&0&12&0&26&0&0&33&0&14&0&79&92&0\\1&0&7&0&21&0&35&46&0&49&0&89&0&0&196\\0&14&0&26&0&104&0&0&166&0&80&0&264&404&0\\1&0&29&0&35&0&121&117&0&89&0&258&0&0&588\\7&0&34&0&46&0&117&181&0&140&0&296&0&0&716\\0&23&0&33&0&166&0&0&293&0&137&0&423&695&0\\3&0&19&0&49&0&89&140&0&151&0&249&0&0&604\\0&12&0&14&0&80&0&0&137&0&71&0&193&329&0\\4&0&63&0&89&0&258&296&0&249&0&614&0&0&1420\\0&26&0&79&0&264&0&0&423&0&193&0&776&1095&0\\0&50&0&92&0&404&0&0&695&0&329&0&1095&1731&0\\12&0&140&0&196&0&588&716&0&604&0&1420&0&0&3424\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&11&12&21&104&121&181&293&151&71&614&776&1731&3424&1742&1763&4250&3588&1139\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_3=0$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
