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: | $C_2^2\times C_4$ |
Order: | $16$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
Maximal subgroups: | $L_2(C_{4,1})$, $L_2(C_4)$, $L(J(C_4),J(C_2))$, $L(J(C_4),C_{4,1})$, $L_2(J(C_2))$, $L_1(J(C_4))$, $L(J(C_4),C_4)$ |
Minimal supergroups: | $L_2(J(D_4))$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$33$ |
$0$ |
$570$ |
$0$ |
$12495$ |
$0$ |
$314748$ |
$0$ |
$8668968$ |
$a_2$ |
$1$ |
$2$ |
$10$ |
$65$ |
$556$ |
$5587$ |
$62317$ |
$743542$ |
$9284198$ |
$119681921$ |
$1578940855$ |
$21194313472$ |
$288293777053$ |
$a_3$ |
$1$ |
$0$ |
$13$ |
$0$ |
$1389$ |
$0$ |
$291280$ |
$0$ |
$79531305$ |
$0$ |
$24452644608$ |
$0$ |
$8011397470692$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$5$ |
$16$ |
$33$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$13$ |
$65$ |
$38$ |
$126$ |
$75$ |
$264$ |
$570$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$93$ |
$556$ |
$323$ |
$195$ |
$1171$ |
$690$ |
$2552$ |
$1490$ |
$5620$ |
$12495$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$864$ |
$5587$ |
$504$ |
$3213$ |
$1865$ |
$12296$ |
$7075$ |
$4100$ |
$27429$ |
$15730$ |
$61580$ |
$35175$ |
$138922$ |
$314748$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1389$ |
$9058$ |
$62317$ |
$5211$ |
$35343$ |
$20150$ |
$140335$ |
$11522$ |
$79455$ |
$45140$ |
$318002$ |
$179550$ |
$101745$ |
$723412$ |
$$ |
$407456$ |
$1651020$ |
$927738$ |
$3778530$ |
$8668968$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&3&0&1&0&2&0&0&4\\0&3&0&2&0&8&0&0&9&0&6&0&10&18&0\\1&0&7&0&5&0&13&14&0&7&0&25&0&0&44\\0&2&0&6&0&12&0&0&13&0&6&0&27&30&0\\1&0&5&0&11&0&15&18&0&15&0&35&0&0&60\\0&8&0&12&0&44&0&0&56&0&32&0&88&130&0\\1&0&13&0&15&0&45&41&0&27&0&86&0&0&174\\3&0&14&0&18&0&41&61&0&40&0&96&0&0&208\\0&9&0&13&0&56&0&0&91&0&43&0&127&203&0\\1&0&7&0&15&0&27&40&0&47&0&71&0&0&170\\0&6&0&6&0&32&0&0&43&0&29&0&63&103&0\\2&0&25&0&35&0&86&96&0&71&0&202&0&0&404\\0&10&0&27&0&88&0&0&127&0&63&0&232&317&0\\0&18&0&30&0&130&0&0&203&0&103&0&317&497&0\\4&0&44&0&60&0&174&208&0&170&0&404&0&0&942\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&6&11&44&45&61&91&47&29&202&232&497&942&478&477&1128&958&307\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $1/2$ | $1/8$ | $0$ | $1/4$ | $0$ | $1/8$ |
---|
$a_1=0$ | $5/16$ | $1/4$ | $1/16$ | $0$ | $1/8$ | $0$ | $1/16$ |
---|
$a_3=0$ | $7/16$ | $3/8$ | $1/16$ | $0$ | $1/4$ | $0$ | $1/16$ |
---|
$a_1=a_3=0$ | $5/16$ | $1/4$ | $1/16$ | $0$ | $1/8$ | $0$ | $1/16$ |
---|