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: | $C_4$ |
Order: | $4$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}$ |
Maximal subgroups: | $L_1(C_2)$ |
Minimal supergroups: | $L_2(C_4)$, $L(J(C_4),J(C_2))$, $L(J(C_4),C_{4,1})$, $L(D_4,D_2)$, $L(D_{4,2},D_{2,1})$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$5$ |
$0$ |
$63$ |
$0$ |
$1310$ |
$0$ |
$34195$ |
$0$ |
$982170$ |
$0$ |
$29575854$ |
$a_2$ |
$1$ |
$3$ |
$17$ |
$135$ |
$1367$ |
$15773$ |
$194891$ |
$2501621$ |
$32882763$ |
$439414641$ |
$5945118347$ |
$81228148301$ |
$1118773707425$ |
$a_3$ |
$1$ |
$0$ |
$26$ |
$0$ |
$3762$ |
$0$ |
$971720$ |
$0$ |
$292656658$ |
$0$ |
$94097303496$ |
$0$ |
$31472833312008$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$5$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$17$ |
$10$ |
$31$ |
$63$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$26$ |
$135$ |
$80$ |
$279$ |
$162$ |
$597$ |
$1310$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$214$ |
$1367$ |
$782$ |
$454$ |
$3001$ |
$1708$ |
$6693$ |
$3780$ |
$15070$ |
$34195$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$2256$ |
$15773$ |
$1284$ |
$8862$ |
$4998$ |
$35663$ |
$19966$ |
$11220$ |
$81195$ |
$45320$ |
$185750$ |
$103390$ |
$426489$ |
$982170$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$3762$ |
$26576$ |
$194891$ |
$14898$ |
$108350$ |
$60368$ |
$447189$ |
$33700$ |
$248094$ |
$137876$ |
$1029629$ |
$570168$ |
$316254$ |
$2376862$ |
$$ |
$1314152$ |
$5498549$ |
$3035844$ |
$12742674$ |
$29575854$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&2&0&3&4&0&4&0&5&0&0&12\\0&5&0&5&0&16&0&0&23&0&11&0&29&47&0\\2&0&12&0&12&0&32&34&0&26&0&62&0&0&140\\0&5&0&11&0&28&0&0&41&0&17&0&69&93&0\\2&0&12&0&18&0&38&48&0&40&0&88&0&0&196\\0&16&0&28&0&108&0&0&168&0&80&0&264&404&0\\3&0&32&0&38&0&117&128&0&102&0&253&0&0&588\\4&0&34&0&48&0&128&164&0&132&0&306&0&0&716\\0&23&0&41&0&168&0&0&285&0&131&0&441&687&0\\4&0&26&0&40&0&102&132&0&122&0&248&0&0&604\\0&11&0&17&0&80&0&0&131&0&68&0&204&325&0\\5&0&62&0&88&0&253&306&0&248&0&611&0&0&1420\\0&29&0&69&0&264&0&0&441&0&204&0&742&1107&0\\0&47&0&93&0&404&0&0&687&0&325&0&1107&1733&0\\12&0&140&0&196&0&588&716&0&604&0&1420&0&0&3424\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&12&11&18&108&117&164&285&122&68&611&742&1733&3424&1703&1722&4146&3519&1009\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.