| 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: | $C_2$ |
| Order: | $2$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 &0 \\0 & 0 & 0 & 0 & 0 & i \\0 & 0 & 0 & 0 & i & 0 \\0 & 0 & 0 & i & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(C_1)$ |
| Minimal supergroups: | $L(C_{6,1},C_3)$, $L(J(C_2),J(C_1))$, $L(D_{2,1},C_{2,1})$, $L(D_{3,2},C_3)$, $L_2(C_{2,1})$, $L(J(C_2),C_2)$, $L(D_{2,1},C_2)$${}^{\times 2}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$5$ |
$0$ |
$99$ |
$0$ |
$2450$ |
$0$ |
$67235$ |
$0$ |
$1956150$ |
$0$ |
$59092110$ |
| $a_2$ |
$1$ |
$4$ |
$26$ |
$235$ |
$2570$ |
$30769$ |
$386087$ |
$4985656$ |
$65681738$ |
$878429449$ |
$11888324861$ |
$162447134326$ |
$2237503402415$ |
| $a_3$ |
$1$ |
$0$ |
$34$ |
$0$ |
$7122$ |
$0$ |
$1933000$ |
$0$ |
$585024146$ |
$0$ |
$188186317704$ |
$0$ |
$62945423976072$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$4$ |
$5$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$26$ |
$12$ |
$45$ |
$99$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$34$ |
$235$ |
$126$ |
$487$ |
$276$ |
$1087$ |
$2450$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$362$ |
$2570$ |
$1424$ |
$806$ |
$5695$ |
$3202$ |
$12913$ |
$7240$ |
$29410$ |
$67235$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$4232$ |
$30769$ |
$2376$ |
$17112$ |
$9574$ |
$69951$ |
$38992$ |
$21780$ |
$160257$ |
$89180$ |
$368174$ |
$204540$ |
$847791$ |
$1956150$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$7122$ |
$51932$ |
$386087$ |
$28946$ |
$213956$ |
$118856$ |
$888095$ |
$66120$ |
$491928$ |
$272852$ |
$2049449$ |
$1133704$ |
$627958$ |
$4738374$ |
| $$ |
$2617930$ |
$10973025$ |
$6055560$ |
$25447590$ |
$59092110$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&3&0&1&0&4&10&0&3&0&10&0&0&24\\0&5&0&7&0&28&0&0&41&0&21&0&61&97&0\\3&0&19&0&18&0&55&73&0&46&0&123&0&0&280\\0&7&0&15&0&52&0&0&81&0&37&0&131&193&0\\1&0&18&0&35&0&78&83&0&79&0&182&0&0&392\\0&28&0&52&0&208&0&0&332&0&160&0&528&808&0\\4&0&55&0&78&0&224&253&0&212&0&510&0&0&1176\\10&0&73&0&83&0&253&336&0&247&0&603&0&0&1432\\0&41&0&81&0&332&0&0&561&0&259&0&891&1375&0\\3&0&46&0&79&0&212&247&0&231&0&508&0&0&1208\\0&21&0&37&0&160&0&0&259&0&132&0&416&647&0\\10&0&123&0&182&0&510&603&0&508&0&1222&0&0&2840\\0&61&0&131&0&528&0&0&891&0&416&0&1462&2223&0\\0&97&0&193&0&808&0&0&1375&0&647&0&2223&3449&0\\24&0&280&0&392&0&1176&1432&0&1208&0&2840&0&0&6848\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&19&15&35&208&224&336&561&231&132&1222&1462&3449&6848&3367&3470&8250&7018&1953\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
