Properties

Label 1.6.J.8.5a
  
Name \(J_2(J(E_2))\)
Weight $1$
Degree $6$
Real dimension $4$
Components $8$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)_2\)
Component group \(C_2^3\)

Learn more

Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$4$
Components:$8$
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:$C_2^3$
Order:$8$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & -i \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $17$ $0$ $255$ $0$ $5131$ $0$ $119973$ $0$ $3061938$
$a_2$ $1$ $2$ $8$ $41$ $291$ $2582$ $26276$ $289739$ $3358823$ $40282082$ $495231516$ $6206226743$ $78985328065$
$a_3$ $1$ $0$ $8$ $0$ $662$ $0$ $116840$ $0$ $27057002$ $0$ $7105905144$ $0$ $2011543142136$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $3$ $9$ $17$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $8$ $41$ $21$ $64$ $37$ $124$ $255$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $50$ $291$ $161$ $97$ $547$ $322$ $1131$ $660$ $2389$ $5131$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $421$ $2582$ $244$ $1472$ $857$ $5340$ $3093$ $1806$ $11456$ $6622$ $24875$ $14315$ $54446$ $119973$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $662$ $4051$ $26276$ $2348$ $15005$ $8639$ $56837$ $4982$ $32542$ $18682$ $124779$ $71258$ $40818$ $275817$
$$ $157115$ $612835$ $348180$ $1367355$ $3061938$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&0&0&0&2\\0&2&0&1&0&4&0&0&5&0&2&0&3&9&0\\1&0&5&0&1&0&6&9&0&2&0&9&0&0&20\\0&1&0&4&0&6&0&0&6&0&1&0&13&13&0\\0&0&1&0&7&0&7&6&0&10&0&15&0&0&26\\0&4&0&6&0&20&0&0&26&0&12&0&36&54&0\\0&0&6&0&7&0&23&16&0&11&0&36&0&0&72\\3&0&9&0&6&0&16&34&0&16&0&35&0&0&84\\0&5&0&6&0&26&0&0&41&0&18&0&49&83&0\\0&0&2&0&10&0&11&16&0&25&0&31&0&0&66\\0&2&0&1&0&12&0&0&18&0&13&0&23&38&0\\0&0&9&0&15&0&36&35&0&31&0&78&0&0&154\\0&3&0&13&0&36&0&0&49&0&23&0&96&115&0\\0&9&0&13&0&54&0&0&83&0&38&0&115&187&0\\2&0&20&0&26&0&72&84&0&66&0&154&0&0&344\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&7&20&23&34&41&25&13&78&96&187&344&175&178&384&309&104\end{bmatrix}$

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$$0$
$a_1=0$$3/8$$0$$0$$0$$0$$0$$0$
$a_3=0$$3/8$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$3/8$$0$$0$$0$$0$$0$$0$