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: | $D_4$ |
Order: | $8$ |
Abelian: | no |
Generators: | $\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$15$ |
$0$ |
$225$ |
$0$ |
$4739$ |
$0$ |
$114933$ |
$0$ |
$2996466$ |
$a_2$ |
$1$ |
$2$ |
$7$ |
$35$ |
$257$ |
$2392$ |
$25195$ |
$283467$ |
$3321771$ |
$40059818$ |
$493880885$ |
$6197928805$ |
$78933863871$ |
$a_3$ |
$1$ |
$0$ |
$8$ |
$0$ |
$630$ |
$0$ |
$114920$ |
$0$ |
$26956650$ |
$0$ |
$7100744184$ |
$0$ |
$2011274968824$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$7$ |
$3$ |
$8$ |
$15$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$8$ |
$35$ |
$19$ |
$56$ |
$33$ |
$109$ |
$225$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$46$ |
$257$ |
$145$ |
$89$ |
$495$ |
$292$ |
$1033$ |
$600$ |
$2195$ |
$4739$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$389$ |
$2392$ |
$228$ |
$1368$ |
$797$ |
$5020$ |
$2897$ |
$1686$ |
$10843$ |
$6234$ |
$23657$ |
$13531$ |
$51982$ |
$114933$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$630$ |
$3843$ |
$25195$ |
$2228$ |
$14365$ |
$8247$ |
$54896$ |
$4742$ |
$31316$ |
$17906$ |
$120997$ |
$68822$ |
$39250$ |
$268243$ |
$$ |
$152187$ |
$597435$ |
$338100$ |
$1335723$ |
$2996466$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&1&0&0&0&0&2\\0&2&0&1&0&3&0&0&5&0&2&0&3&7&0\\1&0&4&0&1&0&6&7&0&3&0&8&0&0&20\\0&1&0&4&0&5&0&0&6&0&1&0&13&11&0\\0&0&1&0&6&0&6&6&0&10&0&13&0&0&26\\0&3&0&5&0&18&0&0&25&0&11&0&35&52&0\\0&0&6&0&6&0&22&15&0&12&0&34&0&0&72\\2&0&7&0&6&0&15&31&0&17&0&36&0&0&84\\0&5&0&6&0&25&0&0&41&0&18&0&49&81&0\\1&0&3&0&10&0&12&17&0&25&0&29&0&0&66\\0&2&0&1&0&11&0&0&18&0&11&0&21&38&0\\0&0&8&0&13&0&34&36&0&29&0&74&0&0&154\\0&3&0&13&0&35&0&0&49&0&21&0&94&115&0\\0&7&0&11&0&52&0&0&81&0&38&0&115&187&0\\2&0&20&0&26&0&72&84&0&66&0&154&0&0&344\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&4&6&18&22&31&41&25&11&74&94&187&344&175&171&382&304&104\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=0$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_3=0$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|