Properties

Label 1.6.J.4.2d
  
Name \(J_2(E_2)\)
Weight $1$
Degree $6$
Real dimension $4$
Components $4$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)_2\)
Component group \(C_2^2\)

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Invariants

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

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $31$ $0$ $500$ $0$ $10227$ $0$ $239820$ $0$ $6123414$
$a_2$ $1$ $2$ $11$ $69$ $547$ $5068$ $52285$ $578728$ $6715523$ $80558118$ $990445729$ $12412403765$ $157970512765$
$a_3$ $1$ $0$ $14$ $0$ $1306$ $0$ $233480$ $0$ $54111554$ $0$ $14211778536$ $0$ $4023085857384$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $11$ $5$ $16$ $31$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $14$ $69$ $39$ $123$ $71$ $242$ $500$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $95$ $547$ $314$ $188$ $1081$ $635$ $2247$ $1310$ $4758$ $10227$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $828$ $5068$ $479$ $2922$ $1699$ $10645$ $6162$ $3592$ $22873$ $13214$ $49700$ $28595$ $108822$ $239820$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1306$ $8063$ $52285$ $4669$ $29949$ $17236$ $113578$ $9934$ $65018$ $37314$ $249453$ $142436$ $81566$ $551504$
$$ $314125$ $1225495$ $696234$ $2734458$ $6123414$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&4&0&1&0&1&0&0&4\\0&3&0&2&0&8&0&0&10&0&4&0&7&18&0\\1&0&8&0&4&0&13&15&0&6&0&20&0&0&40\\0&2&0&7&0&12&0&0&12&0&3&0&25&27&0\\1&0&4&0&10&0&13&17&0&16&0&28&0&0&52\\0&8&0&12&0&40&0&0&52&0&24&0&72&108&0\\1&0&13&0&13&0&43&36&0&21&0&71&0&0&144\\4&0&15&0&17&0&36&58&0&37&0&74&0&0&168\\0&10&0&12&0&52&0&0&81&0&36&0&99&166&0\\1&0&6&0&16&0&21&37&0&44&0&60&0&0&132\\0&4&0&3&0&24&0&0&36&0&24&0&47&76&0\\1&0&20&0&28&0&71&74&0&60&0&153&0&0&308\\0&7&0&25&0&72&0&0&99&0&47&0&188&232&0\\0&18&0&27&0&108&0&0&166&0&76&0&232&371&0\\4&0&40&0&52&0&144&168&0&132&0&308&0&0&688\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&8&7&10&40&43&58&81&44&24&153&188&371&688&345&340&764&609&203\end{bmatrix}$

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$$0$
$a_1=0$$1/4$$0$$0$$0$$0$$0$$0$
$a_3=0$$1/4$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$1/4$$0$$0$$0$$0$$0$$0$