| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &0 & i \\0 & 0 & 0 & 0 & i & 0 \\0 & 0 & 0 & i & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0& 0 & 0 & -1 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L(C_{2,1},C_1)$, $L(C_2,C_1)$, $L_1(C_{2,1})$ |
| Minimal supergroups: | $L(D_{6,1},D_{3,2})$, $L(D_{6,1},C_{6,1})$, $L_2(D_{2,1})$${}^{\times 2}$, $L(J(D_2),D_{2,1})$${}^{\times 2}$, $L(J(D_2),J(C_2))$${}^{\times 2}$, $L(D_{6,2},D_{3,2})$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$51$ |
$0$ |
$1230$ |
$0$ |
$33635$ |
$0$ |
$978138$ |
$0$ |
$29546286$ |
| $a_2$ |
$1$ |
$3$ |
$16$ |
$126$ |
$1310$ |
$15458$ |
$193261$ |
$2493473$ |
$32842786$ |
$439220430$ |
$5944179431$ |
$81223617863$ |
$1118751852515$ |
| $a_3$ |
$1$ |
$0$ |
$19$ |
$0$ |
$3585$ |
$0$ |
$966820$ |
$0$ |
$292516553$ |
$0$ |
$94093223364$ |
$0$ |
$31472712934212$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$16$ |
$7$ |
$24$ |
$51$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$19$ |
$126$ |
$66$ |
$248$ |
$141$ |
$548$ |
$1230$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$187$ |
$1310$ |
$721$ |
$409$ |
$2861$ |
$1610$ |
$6470$ |
$3630$ |
$14720$ |
$33635$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$2134$ |
$15458$ |
$1200$ |
$8583$ |
$4805$ |
$35016$ |
$19523$ |
$10910$ |
$80169$ |
$44620$ |
$184132$ |
$102305$ |
$423948$ |
$978138$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$3585$ |
$26020$ |
$193261$ |
$14509$ |
$107059$ |
$59482$ |
$444169$ |
$33100$ |
$246045$ |
$136486$ |
$1024846$ |
$566942$ |
$314049$ |
$2369322$ |
| $$ |
$1309070$ |
$5486670$ |
$3027906$ |
$12723984$ |
$29546286$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&2&6&0&1&0&4&0&0&12\\0&3&0&4&0&14&0&0&21&0&10&0&30&48&0\\2&0&11&0&8&0&27&38&0&23&0&60&0&0&140\\0&4&0&8&0&26&0&0&41&0&18&0&65&96&0\\0&0&8&0&19&0&40&39&0&41&0&92&0&0&196\\0&14&0&26&0&104&0&0&166&0&80&0&264&404&0\\2&0&27&0&40&0&113&125&0&107&0&255&0&0&588\\6&0&38&0&39&0&125&173&0&120&0&299&0&0&716\\0&21&0&41&0&166&0&0&281&0&129&0&445&687&0\\1&0&23&0&41&0&107&120&0&119&0&255&0&0&604\\0&10&0&18&0&80&0&0&129&0&67&0&209&323&0\\4&0&60&0&92&0&255&299&0&255&0&614&0&0&1420\\0&30&0&65&0&264&0&0&445&0&209&0&732&1111&0\\0&48&0&96&0&404&0&0&687&0&323&0&1111&1727&0\\12&0&140&0&196&0&588&716&0&604&0&1420&0&0&3424\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&11&8&19&104&113&173&281&119&67&614&732&1727&3424&1686&1745&4126&3516&979\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_3=0$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
