| 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: | $C_2^3$ |
| Order: | $8$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 &-1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(D_2)$, $L(J(C_2),C_2)$${}^{\times 3}$, $L(D_{2,1},C_2)$${}^{\times 3}$ |
| Minimal supergroups: | $L(J(D_4),D_4)$${}^{\times 2}$, $L(J(D_6),D_6)$, $L(J(D_4),D_{4,1})$, $L_2(J(D_2))$, $L(J(T),T)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$27$ |
$0$ |
$620$ |
$0$ |
$16835$ |
$0$ |
$489132$ |
$0$ |
$14773374$ |
| $a_2$ |
$1$ |
$2$ |
$10$ |
$68$ |
$670$ |
$7772$ |
$96757$ |
$1247108$ |
$16422490$ |
$219613460$ |
$2972099335$ |
$40611837488$ |
$559376011135$ |
| $a_3$ |
$1$ |
$0$ |
$10$ |
$0$ |
$1794$ |
$0$ |
$483400$ |
$0$ |
$146257874$ |
$0$ |
$47046603240$ |
$0$ |
$15736356314184$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$3$ |
$12$ |
$27$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$10$ |
$68$ |
$33$ |
$124$ |
$69$ |
$274$ |
$620$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$92$ |
$670$ |
$359$ |
$206$ |
$1429$ |
$805$ |
$3235$ |
$1810$ |
$7360$ |
$16835$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1064$ |
$7772$ |
$594$ |
$4287$ |
$2398$ |
$17502$ |
$9757$ |
$5460$ |
$40080$ |
$22310$ |
$92066$ |
$51135$ |
$211974$ |
$489132$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1794$ |
$12998$ |
$96757$ |
$7250$ |
$53513$ |
$29732$ |
$222062$ |
$16530$ |
$123009$ |
$68228$ |
$512405$ |
$283456$ |
$157042$ |
$1184646$ |
| $$ |
$654535$ |
$2743335$ |
$1513890$ |
$6361992$ |
$14773374$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&5&0&0&0&1&0&0&6\\0&2&0&1&0&7&0&0&11&0&6&0&13&25&0\\1&0&7&0&2&0&13&20&0&9&0&30&0&0&70\\0&1&0&6&0&13&0&0&18&0&7&0&38&46&0\\0&0&2&0&13&0&19&18&0&26&0&47&0&0&98\\0&7&0&13&0&52&0&0&83&0&40&0&132&202&0\\0&0&13&0&19&0&62&57&0&48&0&129&0&0&294\\5&0&20&0&18&0&57&98&0&60&0&144&0&0&358\\0&11&0&18&0&83&0&0&144&0&67&0&216&346&0\\0&0&9&0&26&0&48&60&0&75&0&130&0&0&302\\0&6&0&7&0&40&0&0&67&0&36&0&98&164&0\\1&0&30&0&47&0&129&144&0&130&0&310&0&0&710\\0&13&0&38&0&132&0&0&216&0&98&0&382&549&0\\0&25&0&46&0&202&0&0&346&0&164&0&549&866&0\\6&0&70&0&98&0&294&358&0&302&0&710&0&0&1712\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&7&6&13&52&62&98&144&75&36&310&382&866&1712&862&896&2094&1786&528\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
|---|
