| 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: | $S_4$ |
| Order: | $24$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\0 & 0 & 0 & 0 & \zeta_{8}^{3} & 0 \\0 & 0 & 0 & \zeta_{8}^{3} & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$24$ |
$0$ |
$350$ |
$0$ |
$7280$ |
$0$ |
$184338$ |
$0$ |
$5209512$ |
| $a_2$ |
$1$ |
$2$ |
$8$ |
$45$ |
$349$ |
$3372$ |
$37405$ |
$450963$ |
$5721371$ |
$74967258$ |
$1003428913$ |
$13630106613$ |
$187141044942$ |
| $a_3$ |
$1$ |
$0$ |
$10$ |
$0$ |
$852$ |
$0$ |
$176980$ |
$0$ |
$49932204$ |
$0$ |
$15776819100$ |
$0$ |
$5253363527676$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$4$ |
$12$ |
$24$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$10$ |
$45$ |
$26$ |
$81$ |
$48$ |
$164$ |
$350$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$62$ |
$349$ |
$202$ |
$122$ |
$707$ |
$416$ |
$1510$ |
$880$ |
$3290$ |
$7280$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$534$ |
$3372$ |
$312$ |
$1932$ |
$1120$ |
$7299$ |
$4186$ |
$2420$ |
$16146$ |
$9220$ |
$36096$ |
$20510$ |
$81312$ |
$184338$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$852$ |
$5456$ |
$37405$ |
$3130$ |
$21120$ |
$11994$ |
$83812$ |
$6838$ |
$47220$ |
$26704$ |
$189610$ |
$106466$ |
$60002$ |
$431478$ |
| $$ |
$241542$ |
$986356$ |
$550620$ |
$2263212$ |
$5209512$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&0&2&0&1&0&1&0&0&2\\0&3&0&1&0&5&0&0&5&0&4&0&4&11&0\\1&0&5&0&3&0&8&9&0&4&0&13&0&0&26\\0&1&0&5&0&7&0&0&9&0&2&0&16&17&0\\1&0&3&0&8&0&7&10&0&9&0&21&0&0&34\\0&5&0&7&0&26&0&0&33&0&18&0&50&76&0\\0&0&8&0&7&0&29&24&0&18&0&48&0&0&106\\2&0&9&0&10&0&24&37&0&24&0&55&0&0&126\\0&5&0&9&0&33&0&0&57&0&22&0&78&121&0\\1&0&4&0&9&0&18&24&0&28&0&42&0&0&106\\0&4&0&2&0&18&0&0&22&0&20&0&34&60&0\\1&0&13&0&21&0&48&55&0&42&0&117&0&0&242\\0&4&0&16&0&50&0&0&78&0&34&0&140&188&0\\0&11&0&17&0&76&0&0&121&0&60&0&188&303&0\\2&0&26&0&34&0&106&126&0&106&0&242&0&0&588\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&5&5&8&26&29&37&57&28&20&117&140&303&588&301&303&710&606&185\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $1/4$ | $1/4$ | $0$ | $0$ | $1/4$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
