| 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: | $C_2\times C_4$ |
| Order: | $8$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$33$ |
$0$ |
$660$ |
$0$ |
$17115$ |
$0$ |
$491148$ |
$0$ |
$14788158$ |
| $a_2$ |
$1$ |
$2$ |
$10$ |
$71$ |
$694$ |
$7917$ |
$97537$ |
$1251084$ |
$16442202$ |
$219709781$ |
$2972566555$ |
$40614096294$ |
$559386920143$ |
| $a_3$ |
$1$ |
$0$ |
$15$ |
$0$ |
$1905$ |
$0$ |
$486180$ |
$0$ |
$146332809$ |
$0$ |
$47048716260$ |
$0$ |
$15736417602180$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$5$ |
$16$ |
$33$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$15$ |
$71$ |
$42$ |
$142$ |
$81$ |
$300$ |
$660$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$109$ |
$694$ |
$395$ |
$233$ |
$1507$ |
$860$ |
$3354$ |
$1890$ |
$7540$ |
$17115$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1138$ |
$7917$ |
$642$ |
$4445$ |
$2505$ |
$17852$ |
$9995$ |
$5630$ |
$40617$ |
$22680$ |
$92900$ |
$51695$ |
$213262$ |
$491148$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1905$ |
$13314$ |
$97537$ |
$7473$ |
$54215$ |
$30214$ |
$223655$ |
$16850$ |
$124089$ |
$68958$ |
$514876$ |
$285124$ |
$158197$ |
$1188496$ |
| $$ |
$657146$ |
$2749362$ |
$1517922$ |
$6371400$ |
$14788158$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&3&0&3&0&2&0&0&6\\0&3&0&2&0&8&0&0&13&0&6&0&12&24&0\\1&0&7&0&5&0&17&16&0&11&0&31&0&0&70\\0&2&0&8&0&14&0&0&17&0&6&0&41&44&0\\1&0&5&0&11&0&17&26&0&25&0&43&0&0&98\\0&8&0&14&0&54&0&0&84&0&40&0&132&202&0\\1&0&17&0&17&0&63&59&0&43&0&128&0&0&294\\3&0&16&0&26&0&59&89&0&74&0&150&0&0&358\\0&13&0&17&0&84&0&0&149&0&69&0&209&347&0\\3&0&11&0&25&0&43&74&0&79&0&121&0&0&302\\0&6&0&6&0&40&0&0&69&0&37&0&95&165&0\\2&0&31&0&43&0&128&150&0&121&0&308&0&0&710\\0&12&0&41&0&132&0&0&209&0&95&0&394&545&0\\0&24&0&44&0&202&0&0&347&0&165&0&545&871&0\\6&0&70&0&98&0&294&358&0&302&0&710&0&0&1712\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&8&11&54&63&89&149&79&37&308&394&871&1712&882&877&2136&1800&587\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/8$ | $0$ | $1/4$ | $0$ | $1/8$ |
|---|
| $a_1=0$ | $1/4$ | $1/4$ | $0$ | $0$ | $1/4$ | $0$ | $0$ |
|---|
| $a_3=0$ | $1/4$ | $1/4$ | $0$ | $0$ | $1/4$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $1/4$ | $1/4$ | $0$ | $0$ | $1/4$ | $0$ | $0$ |
|---|
