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: | $C_6$ |
Order: | $6$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &0 & \zeta_{12}^{1} \\0 & 0 & 0 & 0 & \zeta_{12}^{5} & 0 \\0 & 0 & 0 & \zeta_{12}^{5} & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$32$ |
$0$ |
$555$ |
$0$ |
$11984$ |
$0$ |
$291144$ |
$0$ |
$7601880$ |
$a_2$ |
$1$ |
$2$ |
$10$ |
$67$ |
$564$ |
$5507$ |
$58937$ |
$668754$ |
$7896028$ |
$95946967$ |
$1191641907$ |
$15058194362$ |
$192989406221$ |
$a_3$ |
$1$ |
$0$ |
$13$ |
$0$ |
$1370$ |
$0$ |
$263720$ |
$0$ |
$63318710$ |
$0$ |
$17000040288$ |
$0$ |
$4888747851504$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$5$ |
$16$ |
$32$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$13$ |
$67$ |
$38$ |
$126$ |
$74$ |
$260$ |
$555$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$95$ |
$564$ |
$325$ |
$192$ |
$1163$ |
$680$ |
$2496$ |
$1455$ |
$5440$ |
$11984$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$866$ |
$5507$ |
$504$ |
$3169$ |
$1838$ |
$11910$ |
$6872$ |
$3981$ |
$26206$ |
$15090$ |
$58150$ |
$33404$ |
$129808$ |
$291144$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1370$ |
$8820$ |
$58937$ |
$5082$ |
$33673$ |
$19310$ |
$130715$ |
$11102$ |
$74630$ |
$42711$ |
$292276$ |
$166555$ |
$95132$ |
$656757$ |
$$ |
$373576$ |
$1481340$ |
$841176$ |
$3351432$ |
$7601880$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&3&0&1&0&2&0&0&4\\0&3&0&2&0&8&0&0&9&0&5&0&10&19&0\\1&0&7&0&5&0&13&16&0&7&0&23&0&0&44\\0&2&0&6&0&12&0&0&15&0&6&0&23&32&0\\1&0&5&0&10&0&15&17&0&13&0&35&0&0&58\\0&8&0&12&0&44&0&0&56&0&30&0&84&126&0\\1&0&13&0&15&0&43&43&0&30&0&82&0&0&164\\3&0&16&0&17&0&43&58&0&35&0&92&0&0&192\\0&9&0&15&0&56&0&0&85&0&38&0&124&187&0\\1&0&7&0&13&0&30&35&0&33&0&67&0&0&148\\0&5&0&6&0&30&0&0&38&0&28&0&63&94&0\\2&0&23&0&35&0&82&92&0&67&0&188&0&0&364\\0&10&0&23&0&84&0&0&124&0&63&0&204&288&0\\0&19&0&32&0&126&0&0&187&0&94&0&288&440&0\\4&0&44&0&58&0&164&192&0&148&0&364&0&0&800\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&6&10&44&43&58&85&33&28&188&204&440&800&360&381&825&643&165\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $1/2$ | $0$ | $1/3$ | $0$ | $0$ | $1/6$ |
---|
$a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|