Properties

Label 1.6.J.2.1d
  
Name \(J_1(J(E_1))\)
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:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J_1(E_1)$
Minimal supergroups:$J_1(J(E_3))$, $J(J(E_2),J(E_1))$, $J_1(J(E_2))$${}^{\times 2}$, $J_2(J(E_1))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $46$ $0$ $840$ $0$ $18662$ $0$ $458136$ $0$ $11976492$
$a_2$ $1$ $3$ $15$ $105$ $923$ $9233$ $99641$ $1129775$ $13271619$ $160178265$ $1975274093$ $24790665419$ $315730935685$
$a_3$ $1$ $0$ $20$ $0$ $2356$ $0$ $456720$ $0$ $107764356$ $0$ $28401536976$ $0$ $8045064418512$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $15$ $8$ $24$ $46$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $20$ $105$ $60$ $198$ $116$ $402$ $840$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $156$ $923$ $532$ $312$ $1894$ $1100$ $4010$ $2320$ $8608$ $18662$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1460$ $9233$ $848$ $5304$ $3064$ $19768$ $11356$ $6544$ $42926$ $24592$ $93968$ $53676$ $206962$ $458136$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2356$ $15044$ $99641$ $8652$ $56848$ $32524$ $218396$ $18648$ $124384$ $70976$ $482250$ $274008$ $155964$ $1070392$
$$ $606816$ $2385850$ $1349712$ $5337072$ $11976492$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&1&0&2&4&0&2&0&3&0&0&8\\0&4&0&4&0&12&0&0&16&0&6&0&18&30&0\\2&0&10&0&8&0&22&26&0&16&0&36&0&0&80\\0&4&0&8&0&20&0&0&28&0&10&0&38&54&0\\1&0&8&0&13&0&26&28&0&24&0&51&0&0&104\\0&12&0&20&0&70&0&0&100&0&44&0&140&208&0\\2&0&22&0&26&0&72&76&0&58&0&134&0&0&288\\4&0&26&0&28&0&76&96&0&66&0&152&0&0&336\\0&16&0&28&0&100&0&0&152&0&66&0&214&322&0\\2&0&16&0&24&0&58&66&0&58&0&120&0&0&264\\0&6&0&10&0&44&0&0&66&0&36&0&100&146&0\\3&0&36&0&51&0&134&152&0&120&0&287&0&0&616\\0&18&0&38&0&140&0&0&214&0&100&0&332&478&0\\0&30&0&54&0&208&0&0&322&0&146&0&478&728&0\\8&0&80&0&104&0&288&336&0&264&0&616&0&0&1376\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&10&8&13&70&72&96&152&58&36&287&332&728&1376&636&639&1444&1148&300\end{bmatrix}$

Event probabilities

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