Properties

Label 1.6.J.8.3b
  
Name \(J(J(E_4),E_4)\)
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 \(D_4\)

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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:$D_4$
Order:$8$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $21$ $0$ $330$ $0$ $6419$ $0$ $142548$ $0$ $3469554$
$a_2$ $1$ $2$ $8$ $44$ $329$ $2962$ $29790$ $321400$ $3642343$ $42827054$ $518216132$ $6415400884$ $80904420195$
$a_3$ $1$ $0$ $8$ $0$ $762$ $0$ $129760$ $0$ $28649026$ $0$ $7309867296$ $0$ $2038903662312$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $3$ $10$ $21$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $8$ $44$ $23$ $75$ $45$ $155$ $330$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $55$ $329$ $186$ $114$ $649$ $389$ $1379$ $820$ $2960$ $6419$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $482$ $2962$ $285$ $1714$ $1013$ $6244$ $3666$ $2168$ $13520$ $7920$ $29486$ $17213$ $64666$ $142548$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $762$ $4645$ $29790$ $2725$ $17199$ $10014$ $64772$ $5844$ $37480$ $21758$ $142383$ $82184$ $47612$ $314538$
$$ $181139$ $697745$ $400890$ $1553388$ $3469554$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&1&0&0&2\\0&2&0&1&0&5&0&0&5&0&4&0&5&10&0\\1&0&5&0&2&0&7&10&0&2&0&13&0&0&22\\0&1&0&4&0&7&0&0&7&0&3&0&15&15&0\\0&0&2&0&9&0&9&7&0&10&0&20&0&0&28\\0&5&0&7&0&25&0&0&29&0&17&0&43&62&0\\0&0&7&0&9&0&26&19&0&14&0&43&0&0&80\\3&0&10&0&7&0&19&37&0&17&0&43&0&0&92\\0&5&0&7&0&29&0&0&46&0&20&0&57&90&0\\0&0&2&0&10&0&14&17&0&26&0&32&0&0&72\\0&4&0&3&0&17&0&0&20&0&16&0&27&46&0\\1&0&13&0&20&0&43&43&0&32&0&93&0&0&166\\0&5&0&15&0&43&0&0&57&0&27&0&104&130&0\\0&10&0&15&0&62&0&0&90&0&46&0&130&202&0\\2&0&22&0&28&0&80&92&0&72&0&166&0&0&368\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&9&25&26&37&46&26&16&93&104&202&368&182&181&392&316&104\end{bmatrix}$

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$$0$
$a_1=0$$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$