| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &0 & i \\0 & 0 & 0 & 0 & i & 0 \\0 & 0 & 0 & i & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0& 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0& 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L(D_{2,1},C_{2,1})$${}^{\times 2}$, $L_1(D_{2,1})$, $L(J(C_2),C_2)$, $L(D_2,C_2)$, $L(J(C_2),C_{2,1})$${}^{\times 2}$ |
| Minimal supergroups: | $L(J(D_4),D_{4,1})$, $L(J(D_6),D_{6,2})$, $L_2(J(D_2))$${}^{\times 3}$, $L(J(D_4),D_{4,2})$${}^{\times 2}$, $L(J(D_6),D_{6,1})$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$27$ |
$0$ |
$620$ |
$0$ |
$16835$ |
$0$ |
$489132$ |
$0$ |
$14773374$ |
| $a_2$ |
$1$ |
$2$ |
$10$ |
$68$ |
$670$ |
$7772$ |
$96757$ |
$1247108$ |
$16422490$ |
$219613460$ |
$2972099335$ |
$40611837488$ |
$559376011135$ |
| $a_3$ |
$1$ |
$0$ |
$11$ |
$0$ |
$1809$ |
$0$ |
$483620$ |
$0$ |
$146261129$ |
$0$ |
$47046651876$ |
$0$ |
$15736357046916$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$4$ |
$13$ |
$27$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$11$ |
$68$ |
$35$ |
$127$ |
$72$ |
$277$ |
$620$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$97$ |
$670$ |
$366$ |
$209$ |
$1439$ |
$811$ |
$3244$ |
$1820$ |
$7370$ |
$16835$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1078$ |
$7772$ |
$606$ |
$4308$ |
$2413$ |
$17533$ |
$9778$ |
$5470$ |
$40110$ |
$22330$ |
$92096$ |
$51170$ |
$212009$ |
$489132$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1809$ |
$13042$ |
$96757$ |
$7277$ |
$53578$ |
$29774$ |
$222158$ |
$16570$ |
$123072$ |
$68278$ |
$512498$ |
$283526$ |
$157077$ |
$1184746$ |
| $$ |
$654605$ |
$2743440$ |
$1514016$ |
$6362118$ |
$14773374$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&1&4&0&0&0&1&0&0&6\\0&2&0&2&0&7&0&0&11&0&5&0&14&24&0\\1&0&7&0&3&0&13&19&0&12&0&30&0&0&70\\0&2&0&5&0&13&0&0&20&0&8&0&34&47&0\\0&0&3&0&11&0&20&19&0&22&0&46&0&0&98\\0&7&0&13&0&52&0&0&83&0&40&0&132&202&0\\1&0&13&0&20&0&59&61&0&51&0&127&0&0&294\\4&0&19&0&19&0&61&91&0&59&0&147&0&0&358\\0&11&0&20&0&83&0&0&142&0&65&0&220&344&0\\0&0&12&0&22&0&51&59&0&67&0&129&0&0&302\\0&5&0&8&0&40&0&0&65&0&35&0&103&162&0\\1&0&30&0&46&0&127&147&0&129&0&310&0&0&710\\0&14&0&34&0&132&0&0&220&0&103&0&372&553&0\\0&24&0&47&0&202&0&0&344&0&162&0&553&866&0\\6&0&70&0&98&0&294&358&0&302&0&710&0&0&1712\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&7&5&11&52&59&91&142&67&35&310&372&866&1712&851&881&2074&1772&504\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
|---|
| $a_1=0$ | $1/2$ | $1/4$ | $1/8$ | $0$ | $0$ | $0$ | $1/8$ |
|---|
| $a_3=0$ | $1/2$ | $1/4$ | $1/8$ | $0$ | $0$ | $0$ | $1/8$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/4$ | $1/8$ | $0$ | $0$ | $0$ | $1/8$ |
|---|
