Properties

Label 1.6.J.6.1a
  
Name \(J(J(E_3),E_3)\)
Weight $1$
Degree $6$
Real dimension $4$
Components $6$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)_2\)
Component group \(S_3\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$4$
Components:$6$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)\times\mathrm{SU}(2)_2$
$\mathbb{R}$-dimension:$4$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B\in\mathrm{SU}(2)\right\}$ 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:$\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$

Component group

Name:$S_3$
Order:$6$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(J(E_1),E_1)$, $J_1(E_3)$
Minimal supergroups:$J(J(E_6),J(E_3))$, $J_2(J(E_3))$, $J(J(E_6),E_6)$${}^{\times 2}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $21$ $0$ $335$ $0$ $6797$ $0$ $159138$ $0$ $4067514$
$a_2$ $1$ $2$ $8$ $45$ $348$ $3267$ $34220$ $382426$ $4459712$ $53621559$ $659925252$ $8273594100$ $105311862430$
$a_3$ $1$ $0$ $8$ $0$ $828$ $0$ $154420$ $0$ $36051932$ $0$ $9475795116$ $0$ $2682298114716$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $3$ $10$ $21$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $8$ $45$ $23$ $76$ $45$ $157$ $335$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $57$ $348$ $194$ $116$ $685$ $403$ $1453$ $850$ $3124$ $6797$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $518$ $3267$ $303$ $1866$ $1087$ $6895$ $3990$ $2320$ $14949$ $8626$ $32678$ $18795$ $71904$ $159138$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $828$ $5205$ $34220$ $3011$ $19537$ $11232$ $74612$ $6474$ $42666$ $24460$ $164483$ $93820$ $53650$ $364554$
$$ $207445$ $811593$ $460782$ $1813788$ $4067514$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&1&0&0&2\\0&2&0&1&0&5&0&0&5&0&4&0&5&11&0\\1&0&5&0&2&0&7&11&0&3&0&13&0&0&26\\0&1&0&4&0&7&0&0&9&0&3&0&15&18&0\\0&0&2&0&9&0&9&7&0&10&0&21&0&0&32\\0&5&0&7&0&26&0&0&33&0&17&0&47&71&0\\0&0&7&0&9&0&28&23&0&20&0&46&0&0&98\\3&0&11&0&7&0&23&40&0&19&0&49&0&0&112\\0&5&0&9&0&33&0&0&54&0&20&0&73&107&0\\0&0&3&0&10&0&20&19&0&25&0&40&0&0&90\\0&4&0&3&0&17&0&0&20&0&17&0&30&52&0\\1&0&13&0&21&0&46&49&0&40&0&102&0&0&202\\0&5&0&15&0&47&0&0&73&0&30&0&117&158&0\\0&11&0&18&0&71&0&0&107&0&52&0&158&245&0\\2&0&26&0&32&0&98&112&0&90&0&202&0&0&466\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&9&26&28&40&54&25&17&102&117&245&466&219&227&485&395&105\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$0$$0$$0$$0$$0$$0$
$a_1=0$$1/2$$0$$0$$0$$0$$0$$0$
$a_3=0$$1/2$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$1/2$$0$$0$$0$$0$$0$$0$