Properties

Label 1.6.J.2.1b
  
Name \(J_1(E_2)\)
Weight $1$
Degree $6$
Real dimension $4$
Components $2$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)_2\)
Component group \(C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$4$
Components:$2$
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:$C_2$
Order:$2$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & -i \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J_1(E_1)$
Minimal supergroups:$J(E_4,E_2)$, $J_1(E_4)$, $J(J(E_2),E_2)$, $J_2(E_2)$, $J_1(J(E_2))$, $J_1(E_6)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $46$ $0$ $840$ $0$ $18662$ $0$ $458136$ $0$ $11976492$
$a_2$ $1$ $2$ $13$ $99$ $907$ $9188$ $99515$ $1129418$ $13270603$ $160175358$ $1975265743$ $24790641351$ $315730866109$
$a_3$ $1$ $0$ $20$ $0$ $2356$ $0$ $456720$ $0$ $107764356$ $0$ $28401536976$ $0$ $8045064418512$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $13$ $6$ $22$ $46$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $20$ $99$ $58$ $194$ $110$ $396$ $840$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $150$ $907$ $524$ $312$ $1882$ $1094$ $3998$ $2300$ $8588$ $18662$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1448$ $9188$ $830$ $5284$ $3046$ $19736$ $11332$ $6544$ $42890$ $24572$ $93928$ $53606$ $206892$ $458136$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2356$ $15006$ $99515$ $8634$ $56790$ $32488$ $218306$ $18588$ $124324$ $70916$ $482154$ $273928$ $155964$ $1070272$
$$ $606746$ $2385710$ $1349460$ $5336820$ $11976492$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&5&0&4&0&3&0&0&8\\0&4&0&2&0&12&0&0&18&0&8&0&14&32&0\\1&0&10&0&7&0&24&23&0&13&0&38&0&0&80\\0&2&0&12&0&20&0&0&22&0&6&0&50&50&0\\2&0&7&0&15&0&21&34&0&31&0&49&0&0&104\\0&12&0&20&0&70&0&0&100&0&44&0&140&208&0\\1&0&24&0&21&0&80&69&0&43&0&136&0&0&288\\5&0&23&0&34&0&69&102&0&81&0&150&0&0&336\\0&18&0&22&0&100&0&0&160&0&72&0&198&328&0\\4&0&13&0&31&0&43&81&0&82&0&116&0&0&264\\0&8&0&6&0&44&0&0&72&0&40&0&86&152&0\\3&0&38&0&49&0&136&150&0&116&0&287&0&0&616\\0&14&0&50&0&140&0&0&198&0&86&0&368&464&0\\0&32&0&50&0&208&0&0&328&0&152&0&464&732&0\\8&0&80&0&104&0&288&336&0&264&0&616&0&0&1376\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&10&12&15&70&80&102&160&82&40&287&368&732&1376&680&649&1520&1188&396\end{bmatrix}$

Event probabilities

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