Properties

Label 1.6.J.4.2a
  
Name \(J(J(E_2),E_2)\)
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:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & -i \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J_1(E_2)$, $J(J(E_1),E_1)$${}^{\times 2}$
Minimal supergroups:$J_2(J(E_2))$, $J(J(E_4),J(E_2))$, $J(J(E_4),E_4)$${}^{\times 2}$, $J(J(E_6),E_6)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $23$ $0$ $420$ $0$ $9331$ $0$ $229068$ $0$ $5988246$
$a_2$ $1$ $2$ $9$ $56$ $471$ $4642$ $49891$ $565084$ $6636363$ $80090702$ $987641523$ $12395345536$ $157865504737$
$a_3$ $1$ $0$ $10$ $0$ $1178$ $0$ $228360$ $0$ $53882178$ $0$ $14200768488$ $0$ $4022532209256$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $3$ $11$ $23$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $10$ $56$ $29$ $97$ $55$ $198$ $420$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $75$ $471$ $262$ $156$ $941$ $547$ $1999$ $1150$ $4294$ $9331$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $724$ $4642$ $415$ $2642$ $1523$ $9868$ $5666$ $3272$ $21445$ $12286$ $46964$ $26803$ $103446$ $229068$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1178$ $7503$ $49891$ $4317$ $28395$ $16244$ $109153$ $9294$ $62162$ $35458$ $241077$ $136964$ $77982$ $535136$
$$ $303373$ $1192855$ $674730$ $2668410$ $5988246$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&1&0&1&0&0&4\\0&2&0&1&0&6&0&0&9&0&4&0&7&16&0\\1&0&6&0&2&0&11&14&0&5&0&18&0&0&40\\0&1&0&6&0&10&0&0&11&0&3&0&25&25&0\\0&0&2&0&10&0&12&13&0&18&0&26&0&0&52\\0&6&0&10&0&35&0&0&50&0&22&0&70&104&0\\0&0&11&0&12&0&41&32&0&23&0&69&0&0&144\\4&0&14&0&13&0&32&58&0&36&0&72&0&0&168\\0&9&0&11&0&50&0&0&80&0&36&0&99&164&0\\1&0&5&0&18&0&23&36&0&44&0&60&0&0&132\\0&4&0&3&0&22&0&0&36&0&20&0&43&76&0\\1&0&18&0&26&0&69&72&0&60&0&145&0&0&308\\0&7&0&25&0&70&0&0&99&0&43&0&184&232&0\\0&16&0&25&0&104&0&0&164&0&76&0&232&366&0\\4&0&40&0&52&0&144&168&0&132&0&308&0&0&688\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&6&10&35&41&58&80&44&20&145&184&366&688&340&336&760&599&198\end{bmatrix}$

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$$0$
$a_1=0$$1/2$$0$$0$$0$$0$$0$$0$
$a_3=0$$1/2$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$1/2$$0$$0$$0$$0$$0$$0$