| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &0 & i \\0 & 0 & 0 & 0 & i & 0 \\0 & 0 & 0 & i & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0& 0 & 0 & -1 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(C_{2,1})$${}^{\times 2}$, $L_1(C_2)$ |
| Minimal supergroups: | $L_1(J(D_2))$${}^{\times 3}$, $L(D_{4,2},D_{2,1})$, $L_1(D_{4,2})$${}^{\times 2}$, $L_1(D_{6,2})$, $L(D_{4,1},D_{2,1})$, $L(J(D_2),D_{2,1})$, $L_2(D_{2,1})$, $L_1(D_{4,1})$, $L_1(D_{6,1})$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$54$ |
$0$ |
$1240$ |
$0$ |
$33670$ |
$0$ |
$978264$ |
$0$ |
$29546748$ |
| $a_2$ |
$1$ |
$3$ |
$16$ |
$126$ |
$1310$ |
$15458$ |
$193261$ |
$2493473$ |
$32842786$ |
$439220430$ |
$5944179431$ |
$81223617863$ |
$1118751852515$ |
| $a_3$ |
$1$ |
$0$ |
$22$ |
$0$ |
$3618$ |
$0$ |
$967240$ |
$0$ |
$292522258$ |
$0$ |
$94093303752$ |
$0$ |
$31472714093832$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$16$ |
$8$ |
$26$ |
$54$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$22$ |
$126$ |
$70$ |
$254$ |
$144$ |
$554$ |
$1240$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$194$ |
$1310$ |
$732$ |
$418$ |
$2878$ |
$1622$ |
$6488$ |
$3640$ |
$14740$ |
$33670$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$2156$ |
$15458$ |
$1212$ |
$8616$ |
$4826$ |
$35066$ |
$19556$ |
$10940$ |
$80220$ |
$44660$ |
$184192$ |
$102340$ |
$424018$ |
$978264$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$3618$ |
$26084$ |
$193261$ |
$14554$ |
$107156$ |
$59548$ |
$444316$ |
$33140$ |
$246144$ |
$136556$ |
$1024996$ |
$567052$ |
$314154$ |
$2369492$ |
| $$ |
$1309210$ |
$5486880$ |
$3028032$ |
$12724236$ |
$29546748$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&1&0&2&5&0&3&0&4&0&0&12\\0&4&0&4&0&14&0&0&22&0&10&0&28&48&0\\2&0&11&0&9&0&29&35&0&24&0&60&0&0&140\\0&4&0&10&0&26&0&0&40&0&16&0&68&94&0\\1&0&9&0&18&0&37&44&0&41&0&90&0&0&196\\0&14&0&26&0&104&0&0&166&0&80&0&264&404&0\\2&0&29&0&37&0&115&125&0&102&0&254&0&0&588\\5&0&35&0&44&0&125&168&0&129&0&302&0&0&716\\0&22&0&40&0&166&0&0&284&0&130&0&440&688&0\\3&0&24&0&41&0&102&129&0&123&0&250&0&0&604\\0&10&0&16&0&80&0&0&130&0&70&0&206&324&0\\4&0&60&0&90&0&254&302&0&250&0&614&0&0&1420\\0&28&0&68&0&264&0&0&440&0&206&0&744&1106&0\\0&48&0&94&0&404&0&0&688&0&324&0&1106&1732&0\\12&0&140&0&196&0&588&716&0&604&0&1420&0&0&3424\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&11&10&18&104&115&168&284&123&70&614&744&1732&3424&1702&1733&4148&3521&1008\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
