Name: | $\mathrm{SU}(2)\times\mathrm{U}(1)_2$ |
$\mathbb{R}$-dimension: | $4$ |
Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$17$ |
$0$ |
$280$ |
$0$ |
$5999$ |
$0$ |
$145593$ |
$0$ |
$3801006$ |
$a_2$ |
$1$ |
$2$ |
$8$ |
$41$ |
$303$ |
$2812$ |
$29636$ |
$334861$ |
$3949427$ |
$47977634$ |
$595833204$ |
$7529133460$ |
$96494810814$ |
$a_3$ |
$1$ |
$0$ |
$8$ |
$0$ |
$696$ |
$0$ |
$131965$ |
$0$ |
$31660496$ |
$0$ |
$8500033542$ |
$0$ |
$2444374091412$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$3$ |
$9$ |
$17$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$8$ |
$41$ |
$21$ |
$66$ |
$38$ |
$132$ |
$280$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$50$ |
$303$ |
$167$ |
$99$ |
$589$ |
$344$ |
$1254$ |
$730$ |
$2725$ |
$5999$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$442$ |
$2812$ |
$254$ |
$1598$ |
$924$ |
$5976$ |
$3445$ |
$1998$ |
$13118$ |
$7555$ |
$29090$ |
$16709$ |
$64918$ |
$145593$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$696$ |
$4434$ |
$29636$ |
$2554$ |
$16874$ |
$9673$ |
$65416$ |
$5556$ |
$37342$ |
$21368$ |
$146180$ |
$83300$ |
$47587$ |
$328416$ |
$$ |
$186816$ |
$740712$ |
$420609$ |
$1675758$ |
$3801006$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&0&0&0&2\\0&2&0&1&0&4&0&0&5&0&2&0&4&10&0\\1&0&5&0&1&0&6&9&0&2&0&11&0&0&22\\0&1&0&4&0&6&0&0&6&0&2&0&13&16&0\\0&0&1&0&7&0&8&7&0&9&0&18&0&0&29\\0&4&0&6&0&22&0&0&28&0&15&0&42&63&0\\0&0&6&0&8&0&23&19&0&14&0&42&0&0&82\\3&0&9&0&7&0&19&36&0&17&0&42&0&0&96\\0&5&0&6&0&28&0&0&46&0&20&0&58&94&0\\0&0&2&0&9&0&14&17&0&23&0&34&0&0&74\\0&2&0&2&0&15&0&0&20&0&16&0&31&46&0\\0&0&11&0&18&0&42&42&0&34&0&97&0&0&182\\0&4&0&13&0&42&0&0&58&0&31&0&108&142&0\\0&10&0&16&0&63&0&0&94&0&46&0&142&222&0\\2&0&22&0&29&0&82&96&0&74&0&182&0&0&400\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&7&22&23&36&46&23&16&97&108&222&400&186&205&429&336&99\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $1/2$ | $1/12$ | $0$ | $0$ | $1/6$ | $1/4$ |
---|
$a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|