| Name: | $\mathrm{U}(1)\times\mathrm{U}(1)_2$ |
| $\mathbb{R}$-dimension: | $6$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
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: | $D_4$ |
| Order: | $8$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0& 0 & 0 & -1 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$45$ |
$0$ |
$930$ |
$0$ |
$22435$ |
$0$ |
$595098$ |
$0$ |
$16849602$ |
| $a_2$ |
$1$ |
$2$ |
$12$ |
$95$ |
$933$ |
$10192$ |
$118926$ |
$1452523$ |
$18352767$ |
$237988850$ |
$3148971912$ |
$42330196273$ |
$576200865807$ |
| $a_3$ |
$1$ |
$0$ |
$16$ |
$0$ |
$2466$ |
$0$ |
$569440$ |
$0$ |
$158413010$ |
$0$ |
$48870130176$ |
$0$ |
$16020795671496$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$12$ |
$6$ |
$21$ |
$45$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$16$ |
$95$ |
$54$ |
$195$ |
$114$ |
$423$ |
$930$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$144$ |
$933$ |
$534$ |
$312$ |
$2027$ |
$1170$ |
$4491$ |
$2580$ |
$10010$ |
$22435$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1502$ |
$10192$ |
$864$ |
$5796$ |
$3316$ |
$22741$ |
$12930$ |
$7380$ |
$51141$ |
$29000$ |
$115490$ |
$65310$ |
$261737$ |
$595098$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2466$ |
$16858$ |
$118926$ |
$9582$ |
$66986$ |
$37854$ |
$269391$ |
$21444$ |
$151476$ |
$85380$ |
$612755$ |
$343780$ |
$193340$ |
$1397958$ |
| $$ |
$782698$ |
$3197831$ |
$1786932$ |
$7332486$ |
$16849602$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&2&4&0&1&0&4&0&0&8\\0&3&0&3&0&12&0&0&15&0&9&0&21&33&0\\1&0&9&0&8&0&21&25&0&16&0&45&0&0&88\\0&3&0&7&0&20&0&0&27&0&13&0&45&61&0\\1&0&8&0&15&0&28&31&0&25&0&62&0&0&120\\0&12&0&20&0&76&0&0&108&0&56&0&168&250&0\\2&0&21&0&28&0&78&83&0&62&0&162&0&0&348\\4&0&25&0&31&0&83&107&0&76&0&188&0&0&416\\0&15&0&27&0&108&0&0&173&0&81&0&263&401&0\\1&0&16&0&25&0&62&76&0&72&0&147&0&0&340\\0&9&0&13&0&56&0&0&81&0&46&0&126&195&0\\4&0&45&0&62&0&162&188&0&147&0&375&0&0&808\\0&21&0&45&0&168&0&0&263&0&126&0&432&635&0\\0&33&0&61&0&250&0&0&401&0&195&0&635&975&0\\8&0&88&0&120&0&348&416&0&340&0&808&0&0&1884\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&9&7&15&76&78&107&173&72&46&375&432&975&1884&911&925&2184&1834&517\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
