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: | $C_6$ |
Order: | $6$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &0 & \zeta_{12}^{1} \\0 & 0 & 0 & 0 & \zeta_{12}^{5} & 0 \\0 & 0 & 0 & \zeta_{12}^{5} & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\\end{bmatrix}$ |
Maximal subgroups: | $L_1(C_{2,1})$, $L_1(C_3)$ |
Minimal supergroups: | $L_1(D_{6,1})$, $L(D_{6,1},C_{6,1})$, $L_1(J(C_6))$, $L_2(C_{6,1})$, $L(J(C_6),C_{6,1})$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$48$ |
$0$ |
$960$ |
$0$ |
$24080$ |
$0$ |
$673344$ |
$0$ |
$19982424$ |
$a_2$ |
$1$ |
$2$ |
$12$ |
$97$ |
$972$ |
$11007$ |
$133757$ |
$1696872$ |
$22140300$ |
$294570127$ |
$3975496707$ |
$54241850082$ |
$746516775605$ |
$a_3$ |
$1$ |
$0$ |
$18$ |
$0$ |
$2622$ |
$0$ |
$660080$ |
$0$ |
$196186158$ |
$0$ |
$62823369168$ |
$0$ |
$20989719099888$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$12$ |
$6$ |
$22$ |
$48$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$18$ |
$97$ |
$56$ |
$200$ |
$114$ |
$432$ |
$960$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$150$ |
$972$ |
$554$ |
$322$ |
$2124$ |
$1212$ |
$4730$ |
$2680$ |
$10630$ |
$24080$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1584$ |
$11007$ |
$900$ |
$6198$ |
$3506$ |
$24768$ |
$13922$ |
$7860$ |
$56190$ |
$31500$ |
$128112$ |
$71610$ |
$293216$ |
$673344$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2622$ |
$18416$ |
$133757$ |
$10350$ |
$74574$ |
$41684$ |
$305782$ |
$23348$ |
$170166$ |
$94892$ |
$701916$ |
$389852$ |
$216974$ |
$1616004$ |
$$ |
$895972$ |
$3729516$ |
$2064384$ |
$8624952$ |
$19982424$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&4&0&3&0&4&0&0&8\\0&4&0&2&0&12&0&0&16&0&10&0&18&36&0\\1&0&9&0&8&0&23&25&0&16&0&45&0&0&96\\0&2&0&10&0&20&0&0&28&0&10&0&52&64&0\\2&0&8&0&16&0&24&35&0&29&0&65&0&0&132\\0&12&0&20&0&78&0&0&116&0&58&0&182&278&0\\1&0&23&0&24&0&87&87&0&66&0&175&0&0&400\\4&0&25&0&35&0&87&116&0&91&0&209&0&0&484\\0&16&0&28&0&116&0&0&196&0&88&0&300&466&0\\3&0&16&0&29&0&66&91&0&85&0&168&0&0&408\\0&10&0&10&0&58&0&0&88&0&54&0&134&224&0\\4&0&45&0&65&0&175&209&0&168&0&420&0&0&952\\0&18&0&52&0&182&0&0&300&0&134&0&512&740&0\\0&36&0&64&0&278&0&0&466&0&224&0&740&1166&0\\8&0&96&0&132&0&400&484&0&408&0&952&0&0&2300\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&10&16&78&87&116&196&85&54&420&512&1166&2300&1148&1155&2764&2347&662\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $1/2$ | $0$ | $1/3$ | $0$ | $0$ | $1/6$ |
---|
$a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|