| 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: | $C_2^4$ |
| Order: | $16$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1& 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 &0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(J(D_2))$, $L(J(D_2),J(C_2))$${}^{\times 3}$, $L(J(D_2),D_{2,1})$${}^{\times 3}$, $L_2(D_2)$, $L(J(D_2),D_2)$, $L_2(D_{2,1})$${}^{\times 3}$, $L_2(J(C_2))$${}^{\times 3}$ |
| Minimal supergroups: | $L_2(J(T))$, $L_2(J(D_4))$${}^{\times 2}$, $L_2(J(D_6))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$21$ |
$0$ |
$395$ |
$0$ |
$9555$ |
$0$ |
$260757$ |
$0$ |
$7623462$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$50$ |
$412$ |
$4302$ |
$50832$ |
$638808$ |
$8309070$ |
$110445620$ |
$1490269504$ |
$20334000018$ |
$279875880445$ |
| $a_3$ |
$1$ |
$0$ |
$9$ |
$0$ |
$1017$ |
$0$ |
$247140$ |
$0$ |
$73420137$ |
$0$ |
$23539881204$ |
$0$ |
$7869147987492$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$3$ |
$10$ |
$21$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$9$ |
$50$ |
$25$ |
$84$ |
$48$ |
$178$ |
$395$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$62$ |
$412$ |
$223$ |
$133$ |
$841$ |
$486$ |
$1872$ |
$1070$ |
$4210$ |
$9555$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$620$ |
$4302$ |
$351$ |
$2397$ |
$1363$ |
$9527$ |
$5386$ |
$3070$ |
$21654$ |
$12220$ |
$49476$ |
$27825$ |
$113414$ |
$260757$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1017$ |
$7005$ |
$50832$ |
$3959$ |
$28308$ |
$15884$ |
$115952$ |
$8925$ |
$64734$ |
$36234$ |
$266668$ |
$148609$ |
$83071$ |
$614911$ |
| $$ |
$342139$ |
$1420846$ |
$789264$ |
$3288642$ |
$7623462$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&0&0&0&3\\0&2&0&1&0&5&0&0&7&0&4&0&6&14&0\\1&0&6&0&1&0&8&12&0&3&0&17&0&0&35\\0&1&0&5&0&8&0&0&8&0&3&0&23&23&0\\0&0&1&0&10&0&11&10&0&16&0&28&0&0&49\\0&5&0&8&0&31&0&0&43&0&24&0&70&106&0\\0&0&8&0&11&0&36&27&0&20&0&70&0&0&147\\4&0&12&0&10&0&27&58&0&31&0&73&0&0&179\\0&7&0&8&0&43&0&0&76&0&36&0&103&176&0\\0&0&3&0&16&0&20&31&0&49&0&63&0&0&151\\0&4&0&3&0&24&0&0&36&0&24&0&50&87&0\\0&0&17&0&28&0&70&73&0&63&0&170&0&0&355\\0&6&0&23&0&70&0&0&103&0&50&0&207&274&0\\0&14&0&23&0&106&0&0&176&0&87&0&274&442&0\\3&0&35&0&49&0&147&179&0&151&0&355&0&0&856\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&10&31&36&58&76&49&24&170&207&442&856&453&464&1083&933&312\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/8$ | $0$ | $0$ | $0$ | $3/8$ |
|---|
| $a_1=0$ | $7/16$ | $1/4$ | $1/16$ | $0$ | $0$ | $0$ | $3/16$ |
|---|
| $a_3=0$ | $7/16$ | $1/4$ | $1/16$ | $0$ | $0$ | $0$ | $3/16$ |
|---|
| $a_1=a_3=0$ | $7/16$ | $1/4$ | $1/16$ | $0$ | $0$ | $0$ | $3/16$ |
|---|
