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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{11} & 0 & 0 \\0 & 0 & 0 & 0& \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$32$ |
$0$ |
$535$ |
$0$ |
$10864$ |
$0$ |
$246204$ |
$0$ |
$6003360$ |
$a_2$ |
$1$ |
$3$ |
$13$ |
$75$ |
$567$ |
$5133$ |
$51640$ |
$553752$ |
$6199847$ |
$71689077$ |
$850556430$ |
$10309312170$ |
$127245697160$ |
$a_3$ |
$1$ |
$0$ |
$14$ |
$0$ |
$1316$ |
$0$ |
$223525$ |
$0$ |
$47677084$ |
$0$ |
$11592760410$ |
$0$ |
$3082004790084$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$3$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$13$ |
$6$ |
$17$ |
$32$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$14$ |
$75$ |
$40$ |
$127$ |
$76$ |
$256$ |
$535$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$98$ |
$567$ |
$322$ |
$192$ |
$1113$ |
$660$ |
$2346$ |
$1390$ |
$5025$ |
$10864$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$840$ |
$5133$ |
$496$ |
$2970$ |
$1748$ |
$10807$ |
$6328$ |
$3722$ |
$23374$ |
$13660$ |
$50973$ |
$29708$ |
$111776$ |
$246204$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1316$ |
$8076$ |
$51640$ |
$4724$ |
$29854$ |
$17376$ |
$112311$ |
$10148$ |
$65048$ |
$37782$ |
$246922$ |
$142702$ |
$82684$ |
$545507$ |
$$ |
$314552$ |
$1209670$ |
$696024$ |
$2691024$ |
$6003360$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&4&0&0&0&1&0&0&4\\0&3&0&3&0&8&0&0&9&0&4&0&10&17&0\\2&0&8&0&4&0&12&18&0&6&0&20&0&0&40\\0&3&0&5&0&12&0&0&15&0&6&0&20&29&0\\0&0&4&0&11&0&17&12&0&14&0&34&0&0&52\\0&8&0&12&0&42&0&0&52&0&28&0&76&108&0\\1&0&12&0&17&0&40&38&0&30&0&75&0&0&140\\4&0&18&0&12&0&38&58&0&26&0&76&0&0&160\\0&9&0&15&0&52&0&0&75&0&34&0&106&155&0\\0&0&6&0&14&0&30&26&0&30&0&60&0&0&120\\0&4&0&6&0&28&0&0&34&0&25&0&57&76&0\\1&0&20&0&34&0&75&76&0&60&0&161&0&0&296\\0&10&0&20&0&76&0&0&106&0&57&0&169&232&0\\0&17&0&29&0&108&0&0&155&0&76&0&232&347&0\\4&0&40&0&52&0&140&160&0&120&0&296&0&0&620\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&8&5&11&42&40&58&75&30&25&161&169&347&620&271&292&592&457&114\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
---|
$a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|