Properties

Label 1.6.J.6.1b
  
Name \(J_1(J(E_3))\)
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:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $24$ $0$ $345$ $0$ $6832$ $0$ $159264$ $0$ $4067976$
$a_2$ $1$ $2$ $8$ $45$ $348$ $3267$ $34220$ $382426$ $4459712$ $53621559$ $659925252$ $8273594100$ $105311862430$
$a_3$ $1$ $0$ $10$ $0$ $846$ $0$ $154620$ $0$ $36054382$ $0$ $9475826868$ $0$ $2682298541604$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $4$ $12$ $24$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $10$ $45$ $26$ $81$ $48$ $163$ $345$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $62$ $348$ $202$ $122$ $698$ $412$ $1468$ $860$ $3144$ $6832$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $532$ $3267$ $312$ $1888$ $1102$ $6930$ $4014$ $2340$ $14988$ $8656$ $32728$ $18830$ $71974$ $159264$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $846$ $5244$ $34220$ $3038$ $19598$ $11274$ $74708$ $6504$ $42732$ $24510$ $164588$ $93900$ $53720$ $364684$
$$ $207550$ $811768$ $460908$ $1814040$ $4067976$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&0&2&0&1&0&1&0&0&2\\0&3&0&1&0&5&0&0&5&0&4&0&4&11&0\\1&0&5&0&3&0&8&9&0&4&0&13&0&0&26\\0&1&0&5&0&7&0&0&9&0&2&0&16&17&0\\1&0&3&0&8&0&7&10&0&9&0&20&0&0&32\\0&5&0&7&0&26&0&0&33&0&17&0&47&71&0\\0&0&8&0&7&0&29&24&0&18&0&45&0&0&98\\2&0&9&0&10&0&24&36&0&23&0&51&0&0&112\\0&5&0&9&0&33&0&0&55&0&20&0&72&107&0\\1&0&4&0&9&0&18&23&0&25&0&38&0&0&90\\0&4&0&2&0&17&0&0&20&0&19&0&29&52&0\\1&0&13&0&20&0&45&51&0&38&0&102&0&0&202\\0&4&0&16&0&47&0&0&72&0&29&0&121&156&0\\0&11&0&17&0&71&0&0&107&0&52&0&156&248&0\\2&0&26&0&32&0&98&112&0&90&0&202&0&0&466\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&5&5&8&26&29&36&55&25&19&102&121&248&466&224&220&489&394&110\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.