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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{1} \\0 & 0 & 0 & 0 & \zeta_{12}^{5} & 0 \\0 & 0 & 0 & \zeta_{12}^{5} & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\\end{bmatrix}$ |
Maximal subgroups: | $L_1(D_3)$, $L(D_{2,1},C_2)$, $L(D_{3,2},C_3)$, $L(C_{6,1},C_3)$ |
Minimal supergroups: | $L_2(D_{6,1})$, $L(J(D_6),D_6)$${}^{\times 2}$, $L(J(D_6),J(D_3))$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$24$ |
$0$ |
$480$ |
$0$ |
$12040$ |
$0$ |
$336672$ |
$0$ |
$9991212$ |
$a_2$ |
$1$ |
$2$ |
$9$ |
$57$ |
$511$ |
$5577$ |
$67096$ |
$849081$ |
$11072067$ |
$147290769$ |
$1987765354$ |
$27120975741$ |
$373258539110$ |
$a_3$ |
$1$ |
$0$ |
$9$ |
$0$ |
$1311$ |
$0$ |
$330030$ |
$0$ |
$98092799$ |
$0$ |
$31411678914$ |
$0$ |
$10494859447842$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$3$ |
$11$ |
$24$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$9$ |
$57$ |
$28$ |
$100$ |
$57$ |
$216$ |
$480$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$74$ |
$511$ |
$276$ |
$161$ |
$1061$ |
$606$ |
$2365$ |
$1340$ |
$5315$ |
$12040$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$790$ |
$5577$ |
$447$ |
$3096$ |
$1750$ |
$12380$ |
$6958$ |
$3930$ |
$28092$ |
$15750$ |
$64056$ |
$35805$ |
$146608$ |
$336672$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1311$ |
$9200$ |
$67096$ |
$5172$ |
$37276$ |
$20836$ |
$152876$ |
$11664$ |
$85074$ |
$47436$ |
$350946$ |
$194916$ |
$108487$ |
$807992$ |
$$ |
$447986$ |
$1864758$ |
$1032192$ |
$4312476$ |
$9991212$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&1&0&0&4\\0&2&0&1&0&6&0&0&8&0&5&0&9&18&0\\1&0&6&0&2&0&10&16&0&5&0&21&0&0&48\\0&1&0&5&0&10&0&0&13&0&5&0&27&32&0\\0&0&2&0&11&0&14&12&0&19&0&35&0&0&66\\0&6&0&10&0&39&0&0&58&0&29&0&91&139&0\\0&0&10&0&14&0&45&39&0&34&0&90&0&0&200\\4&0&16&0&12&0&39&71&0&40&0&98&0&0&242\\0&8&0&13&0&58&0&0&100&0&45&0&147&234&0\\0&0&5&0&19&0&34&40&0&52&0&87&0&0&204\\0&5&0&5&0&29&0&0&45&0&27&0&66&112&0\\1&0&21&0&35&0&90&98&0&87&0&213&0&0&476\\0&9&0&27&0&91&0&0&147&0&66&0&260&369&0\\0&18&0&32&0&139&0&0&234&0&112&0&369&583&0\\4&0&48&0&66&0&200&242&0&204&0&476&0&0&1150\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&11&39&45&71&100&52&27&213&260&583&1150&580&607&1401&1196&355\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $1/2$ | $0$ | $1/6$ | $0$ | $0$ | $1/3$ |
---|
$a_1=0$ | $1/2$ | $1/2$ | $0$ | $1/6$ | $0$ | $0$ | $1/3$ |
---|
$a_3=0$ | $1/2$ | $1/2$ | $0$ | $1/6$ | $0$ | $0$ | $1/3$ |
---|
$a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $1/6$ | $0$ | $0$ | $1/3$ |
---|