| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L(C_{2,1},C_1)$, $L(J(C_1),C_1)$, $L_1(C_2)$ |
| Minimal supergroups: | $L(J(C_6),C_6)$, $L(J(D_3),D_3)$, $L(J(D_2),D_{2,1})$, $L(J(D_2),D_2)$${}^{\times 3}$, $L(J(C_4),C_4)$, $L_2(J(C_2))$, $L(J(C_4),C_{4,1})$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$51$ |
$0$ |
$1230$ |
$0$ |
$33635$ |
$0$ |
$978138$ |
$0$ |
$29546286$ |
| $a_2$ |
$1$ |
$2$ |
$14$ |
$119$ |
$1290$ |
$15397$ |
$193079$ |
$2492926$ |
$32841146$ |
$439215509$ |
$5944164669$ |
$81223573576$ |
$1118751719655$ |
| $a_3$ |
$1$ |
$0$ |
$18$ |
$0$ |
$3570$ |
$0$ |
$966600$ |
$0$ |
$292513298$ |
$0$ |
$94093174728$ |
$0$ |
$31472712201480$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$14$ |
$6$ |
$23$ |
$51$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$18$ |
$119$ |
$64$ |
$245$ |
$138$ |
$545$ |
$1230$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$182$ |
$1290$ |
$714$ |
$406$ |
$2851$ |
$1604$ |
$6461$ |
$3620$ |
$14710$ |
$33635$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$2120$ |
$15397$ |
$1188$ |
$8562$ |
$4790$ |
$34985$ |
$19502$ |
$10900$ |
$80139$ |
$44600$ |
$184102$ |
$102270$ |
$423913$ |
$978138$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$3570$ |
$25976$ |
$193079$ |
$14482$ |
$106994$ |
$59440$ |
$444073$ |
$33060$ |
$245982$ |
$136436$ |
$1024753$ |
$566872$ |
$314014$ |
$2369222$ |
| $$ |
$1309000$ |
$5486565$ |
$3027780$ |
$12723858$ |
$29546286$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&2&6&0&1&0&4&0&0&12\\0&3&0&3&0&14&0&0&21&0&11&0&29&49&0\\1&0&11&0&8&0&27&35&0&24&0&63&0&0&140\\0&3&0&9&0&26&0&0&39&0&17&0&69&95&0\\1&0&8&0&19&0&38&43&0&41&0&90&0&0&196\\0&14&0&26&0&104&0&0&166&0&80&0&264&404&0\\2&0&27&0&38&0&116&125&0&100&0&254&0&0&588\\6&0&35&0&43&0&125&172&0&125&0&299&0&0&716\\0&21&0&39&0&166&0&0&283&0&131&0&441&689&0\\1&0&24&0&41&0&100&125&0&127&0&256&0&0&604\\0&11&0&17&0&80&0&0&131&0&68&0&204&325&0\\4&0&63&0&90&0&254&299&0&256&0&614&0&0&1420\\0&29&0&69&0&264&0&0&441&0&204&0&742&1107&0\\0&49&0&95&0&404&0&0&689&0&325&0&1107&1727&0\\12&0&140&0&196&0&588&716&0&604&0&1420&0&0&3424\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&11&9&19&104&116&172&283&127&68&614&742&1727&3424&1697&1742&4146&3530&1003\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
