Name: | $\mathrm{U}(1)\times\mathrm{SU}(2)_2$ |
$\mathbb{R}$-dimension: | $4$ |
Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B\in\mathrm{SU}(2)\right\}$ |
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: | $\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$ |
Name: | $C_2\times C_4$ |
Order: | $8$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$27$ |
$0$ |
$380$ |
$0$ |
$6923$ |
$0$ |
$148260$ |
$0$ |
$3539250$ |
$a_2$ |
$1$ |
$2$ |
$9$ |
$51$ |
$371$ |
$3198$ |
$31103$ |
$328772$ |
$3684451$ |
$43072206$ |
$519669707$ |
$6424161175$ |
$80957964029$ |
$a_3$ |
$1$ |
$0$ |
$12$ |
$0$ |
$858$ |
$0$ |
$132960$ |
$0$ |
$28778050$ |
$0$ |
$7315716384$ |
$0$ |
$2039189137128$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$5$ |
$14$ |
$27$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$12$ |
$51$ |
$31$ |
$93$ |
$57$ |
$184$ |
$380$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$71$ |
$371$ |
$222$ |
$138$ |
$737$ |
$447$ |
$1529$ |
$920$ |
$3230$ |
$6923$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$554$ |
$3198$ |
$333$ |
$1890$ |
$1129$ |
$6701$ |
$3966$ |
$2368$ |
$14335$ |
$8460$ |
$31004$ |
$18221$ |
$67578$ |
$148260$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$858$ |
$4997$ |
$31103$ |
$2957$ |
$18113$ |
$10614$ |
$67256$ |
$6244$ |
$39110$ |
$22838$ |
$146977$ |
$85220$ |
$49628$ |
$323332$ |
$$ |
$186963$ |
$714985$ |
$412314$ |
$1587804$ |
$3539250$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&2&0&1&0&1&0&0&2\\0&3&0&2&0&6&0&0&6&0&4&0&5&10&0\\1&0&6&0&4&0&9&9&0&4&0&14&0&0&22\\0&2&0&5&0&8&0&0&8&0&3&0&15&15&0\\1&0&4&0&8&0&9&11&0&8&0&20&0&0&28\\0&6&0&8&0&28&0&0&30&0&18&0&44&64&0\\1&0&9&0&9&0&27&22&0&13&0&43&0&0&80\\2&0&9&0&11&0&22&34&0&19&0&46&0&0&92\\0&6&0&8&0&30&0&0&47&0&20&0&57&90&0\\1&0&4&0&8&0&13&19&0&26&0&30&0&0&72\\0&4&0&3&0&18&0&0&20&0&18&0&29&46&0\\1&0&14&0&20&0&43&46&0&30&0&97&0&0&166\\0&5&0&15&0&44&0&0&57&0&29&0&106&130&0\\0&10&0&15&0&64&0&0&90&0&46&0&130&207&0\\2&0&22&0&28&0&80&92&0&72&0&166&0&0&368\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&5&8&28&27&34&47&26&18&97&106&207&368&187&178&394&321&109\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=0$ | $1/8$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_3=0$ | $1/8$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $1/8$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|