Properties

Label 1.6.J.3.1a
  
Name \(J_1(E_3)\)
Weight $1$
Degree $6$
Real dimension $4$
Components $3$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)_2\)
Component group \(C_3\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$4$
Components:$3$
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:$C_3$
Order:$3$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J_1(E_1)$
Minimal supergroups:$J_1(J(E_3))$, $J(J(E_3),E_3)$, $J_2(E_3)$, $J_1(E_6)$, $J(E_6,E_3)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $42$ $0$ $670$ $0$ $13594$ $0$ $318276$ $0$ $8135028$
$a_2$ $1$ $2$ $11$ $77$ $661$ $6438$ $68173$ $764102$ $8917301$ $107237072$ $1319833201$ $16547138479$ $210623581495$
$a_3$ $1$ $0$ $16$ $0$ $1656$ $0$ $308840$ $0$ $72103864$ $0$ $18951590232$ $0$ $5364596229432$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $11$ $6$ $20$ $42$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $16$ $77$ $46$ $152$ $90$ $314$ $670$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $114$ $661$ $388$ $232$ $1370$ $806$ $2906$ $1700$ $6248$ $13594$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1036$ $6438$ $606$ $3732$ $2174$ $13790$ $7980$ $4640$ $29898$ $17252$ $65356$ $37590$ $143808$ $318276$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1656$ $10410$ $68173$ $6022$ $39074$ $22464$ $149224$ $12948$ $85332$ $48920$ $328966$ $187640$ $107300$ $729108$
$$ $414890$ $1623186$ $921564$ $3627576$ $8135028$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&3&0&2&0&3&0&0&4\\0&4&0&2&0&10&0&0&10&0&8&0&10&22&0\\1&0&8&0&7&0&16&17&0&9&0&28&0&0&52\\0&2&0&8&0&14&0&0&18&0&6&0&30&36&0\\2&0&7&0&13&0&15&22&0&15&0&39&0&0&64\\0&10&0&14&0&52&0&0&66&0&34&0&94&142&0\\1&0&16&0&15&0&54&51&0&37&0&90&0&0&196\\3&0&17&0&22&0&51&66&0&47&0&104&0&0&224\\0&10&0&18&0&66&0&0&108&0&40&0&146&214&0\\2&0&9&0&15&0&37&47&0&44&0&76&0&0&180\\0&8&0&6&0&34&0&0&40&0&34&0&60&104&0\\3&0&28&0&39&0&90&104&0&76&0&201&0&0&404\\0&10&0&30&0&94&0&0&146&0&60&0&234&316&0\\0&22&0&36&0&142&0&0&214&0&104&0&316&490&0\\4&0&52&0&64&0&196&224&0&180&0&404&0&0&932\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&8&8&13&52&54&66&108&44&34&201&234&490&932&438&431&970&780&210\end{bmatrix}$

Event probabilities

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