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: | $D_6$ |
Order: | $12$ |
Abelian: | no |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &0 & \zeta_{12}^{1} \\0 & 0 & 0 & 0 & \zeta_{12}^{5} & 0 \\0 & 0 & 0 & \zeta_{12}^{5} & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0& 0 & 0 & -1 & 0 \\\end{bmatrix}$ |
Maximal subgroups: | $L(D_3,C_3)$, $L(D_{2,1},C_{2,1})$, $L_1(C_{6,1})$, $L(D_{3,2},C_3)$ |
Minimal supergroups: | $L_2(D_{6,1})$, $L(J(D_6),J(C_6))$${}^{\times 2}$, $L(J(D_6),D_{6,1})$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$24$ |
$0$ |
$480$ |
$0$ |
$12040$ |
$0$ |
$336672$ |
$0$ |
$9991212$ |
$a_2$ |
$1$ |
$2$ |
$9$ |
$57$ |
$511$ |
$5577$ |
$67096$ |
$849081$ |
$11072067$ |
$147290769$ |
$1987765354$ |
$27120975741$ |
$373258539110$ |
$a_3$ |
$1$ |
$0$ |
$9$ |
$0$ |
$1311$ |
$0$ |
$330040$ |
$0$ |
$98093079$ |
$0$ |
$31411684584$ |
$0$ |
$10494859549944$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$3$ |
$11$ |
$24$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$9$ |
$57$ |
$28$ |
$100$ |
$57$ |
$216$ |
$480$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$75$ |
$511$ |
$277$ |
$161$ |
$1062$ |
$606$ |
$2365$ |
$1340$ |
$5315$ |
$12040$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$792$ |
$5577$ |
$450$ |
$3099$ |
$1753$ |
$12384$ |
$6961$ |
$3930$ |
$28095$ |
$15750$ |
$64056$ |
$35805$ |
$146608$ |
$336672$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1311$ |
$9208$ |
$67096$ |
$5175$ |
$37287$ |
$20842$ |
$152891$ |
$11674$ |
$85083$ |
$47446$ |
$350958$ |
$194926$ |
$108487$ |
$808002$ |
$$ |
$447986$ |
$1864758$ |
$1032192$ |
$4312476$ |
$9991212$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&1&0&0&4\\0&2&0&1&0&6&0&0&8&0&5&0&9&18&0\\1&0&6&0&2&0&10&16&0&6&0&21&0&0&48\\0&1&0&5&0&10&0&0&14&0&5&0&26&32&0\\0&0&2&0&11&0&14&12&0&18&0&35&0&0&66\\0&6&0&10&0&39&0&0&58&0&29&0&91&139&0\\0&0&10&0&14&0&45&40&0&35&0&89&0&0&200\\4&0&16&0&12&0&40&69&0&38&0&99&0&0&242\\0&8&0&14&0&58&0&0&98&0&44&0&150&233&0\\0&0&6&0&18&0&35&38&0&48&0&88&0&0&204\\0&5&0&5&0&29&0&0&44&0&27&0&67&112&0\\1&0&21&0&35&0&89&99&0&88&0&213&0&0&476\\0&9&0&26&0&91&0&0&150&0&67&0&256&370&0\\0&18&0&32&0&139&0&0&233&0&112&0&370&583&0\\4&0&48&0&66&0&200&242&0&204&0&476&0&0&1150\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&11&39&45&69&98&48&27&213&256&583&1150&574&601&1382&1185&331\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $1/2$ | $0$ | $1/6$ | $0$ | $0$ | $1/3$ |
---|
$a_1=0$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
---|
$a_3=0$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
---|
$a_1=a_3=0$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
---|