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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}$ |
Maximal subgroups: | $L(C_4,C_2)$, $L_1(D_2)$, $L(D_2,C_2)$ |
Minimal supergroups: | $L(J(D_4),D_{4,1})$, $L(O,T)$, $L_2(D_4)$${}^{\times 2}$, $L(J(D_4),J(D_2))$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$33$ |
$0$ |
$660$ |
$0$ |
$17115$ |
$0$ |
$491148$ |
$0$ |
$14788158$ |
$a_2$ |
$1$ |
$2$ |
$10$ |
$71$ |
$693$ |
$7912$ |
$97516$ |
$1251007$ |
$16441935$ |
$219708890$ |
$2972563650$ |
$40614086977$ |
$559386890607$ |
$a_3$ |
$1$ |
$0$ |
$14$ |
$0$ |
$1890$ |
$0$ |
$485960$ |
$0$ |
$146329554$ |
$0$ |
$47048667624$ |
$0$ |
$15736416869448$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$5$ |
$16$ |
$33$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$14$ |
$71$ |
$41$ |
$141$ |
$81$ |
$300$ |
$660$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$108$ |
$693$ |
$393$ |
$230$ |
$1504$ |
$857$ |
$3351$ |
$1890$ |
$7540$ |
$17115$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1132$ |
$7912$ |
$642$ |
$4437$ |
$2502$ |
$17841$ |
$9989$ |
$5620$ |
$40608$ |
$22670$ |
$92890$ |
$51695$ |
$213262$ |
$491148$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1890$ |
$13298$ |
$97516$ |
$7458$ |
$54191$ |
$30196$ |
$223620$ |
$16850$ |
$124065$ |
$68948$ |
$514843$ |
$285104$ |
$158162$ |
$1188466$ |
$$ |
$657111$ |
$2749327$ |
$1517922$ |
$6371400$ |
$14788158$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&3&0&2&0&2&0&0&6\\0&3&0&2&0&8&0&0&12&0&6&0&13&24&0\\1&0&7&0&5&0&16&17&0&12&0&31&0&0&70\\0&2&0&7&0&14&0&0&19&0&7&0&38&45&0\\1&0&5&0&11&0&18&24&0&23&0&44&0&0&98\\0&8&0&14&0&54&0&0&84&0&40&0&132&202&0\\1&0&16&0&18&0&62&61&0&47&0&127&0&0&294\\3&0&17&0&24&0&61&87&0&68&0&151&0&0&358\\0&12&0&19&0&84&0&0&145&0&67&0&216&345&0\\2&0&12&0&23&0&47&68&0&70&0&124&0&0&302\\0&6&0&7&0&40&0&0&67&0&36&0&98&164&0\\2&0&31&0&44&0&127&151&0&124&0&307&0&0&710\\0&13&0&38&0&132&0&0&216&0&98&0&382&549&0\\0&24&0&45&0&202&0&0&345&0&164&0&549&869&0\\6&0&70&0&98&0&294&358&0&302&0&710&0&0&1712\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&7&11&54&62&87&145&70&36&307&382&869&1712&865&869&2094&1774&531\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=0$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_3=0$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|