| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &0 & \zeta_{8}^{1} \\0 & 0 & 0 & 0 & \zeta_{8}^{3} & 0 \\0 & 0 & 0 & \zeta_{8}^{3} & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0& 0 & 0 & -1 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(D_2)$, $L_1(D_{2,1})$, $L_1(C_{4,1})$ |
| Minimal supergroups: | $L_2(D_{4,1})$, $L(J(D_4),D_{4,1})$, $L_1(J(D_4))$${}^{\times 2}$, $L_1(O_1)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$30$ |
$0$ |
$630$ |
$0$ |
$16870$ |
$0$ |
$489258$ |
$0$ |
$14773836$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$65$ |
$660$ |
$7742$ |
$96666$ |
$1246835$ |
$16421670$ |
$219611000$ |
$2972091954$ |
$40611815345$ |
$559375944705$ |
| $a_3$ |
$1$ |
$0$ |
$12$ |
$0$ |
$1818$ |
$0$ |
$483720$ |
$0$ |
$146262354$ |
$0$ |
$47046667752$ |
$0$ |
$15736357260360$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$4$ |
$14$ |
$30$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$12$ |
$65$ |
$36$ |
$129$ |
$72$ |
$280$ |
$630$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$98$ |
$660$ |
$368$ |
$212$ |
$1443$ |
$814$ |
$3250$ |
$1820$ |
$7380$ |
$16870$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1082$ |
$7742$ |
$606$ |
$4314$ |
$2416$ |
$17543$ |
$9784$ |
$5480$ |
$40122$ |
$22340$ |
$92116$ |
$51170$ |
$212044$ |
$489258$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1818$ |
$13052$ |
$96666$ |
$7286$ |
$53594$ |
$29786$ |
$222184$ |
$16570$ |
$123090$ |
$68288$ |
$512528$ |
$283546$ |
$157112$ |
$1184786$ |
| $$ |
$654640$ |
$2743510$ |
$1514016$ |
$6362244$ |
$14773836$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&0&3&0&2&0&2&0&0&6\\0&3&0&1&0&7&0&0&11&0&6&0&12&25&0\\1&0&6&0&4&0&15&17&0&11&0&30&0&0&70\\0&1&0&7&0&13&0&0&19&0&6&0&38&45&0\\1&0&4&0&11&0&16&23&0&23&0&45&0&0&98\\0&7&0&13&0&52&0&0&83&0&40&0&132&202&0\\0&0&15&0&16&0&62&60&0&47&0&127&0&0&294\\3&0&17&0&23&0&60&87&0&68&0&150&0&0&358\\0&11&0&19&0&83&0&0&145&0&66&0&216&345&0\\2&0&11&0&23&0&47&68&0&69&0&124&0&0&302\\0&6&0&6&0&40&0&0&66&0&38&0&98&164&0\\2&0&30&0&45&0&127&150&0&124&0&308&0&0&710\\0&12&0&38&0&132&0&0&216&0&98&0&384&548&0\\0&25&0&45&0&202&0&0&345&0&164&0&548&869&0\\6&0&70&0&98&0&294&358&0&302&0&710&0&0&1712\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&7&11&52&62&87&145&69&38&308&384&869&1712&865&872&2096&1774&531\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $1/4$ | $1/4$ | $0$ | $0$ | $1/4$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
