Properties

Label 1.6.J.8.3c
  
Name \(J_1(J(E_4))\)
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:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $24$ $0$ $340$ $0$ $6454$ $0$ $142674$ $0$ $3470016$
$a_2$ $1$ $2$ $8$ $44$ $329$ $2962$ $29790$ $321400$ $3642343$ $42827054$ $518216132$ $6415400884$ $80904420195$
$a_3$ $1$ $0$ $10$ $0$ $780$ $0$ $129960$ $0$ $28651476$ $0$ $7309899048$ $0$ $2038904089200$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $4$ $12$ $24$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $10$ $44$ $26$ $80$ $48$ $161$ $340$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $60$ $329$ $194$ $120$ $662$ $398$ $1394$ $830$ $2980$ $6454$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $496$ $2962$ $294$ $1736$ $1028$ $6279$ $3690$ $2188$ $13559$ $7950$ $29536$ $17248$ $64736$ $142674$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $780$ $4684$ $29790$ $2752$ $17260$ $10056$ $64868$ $5874$ $37546$ $21808$ $142488$ $82264$ $47682$ $314668$
$$ $181244$ $697920$ $401016$ $1553640$ $3470016$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&0&2&0&1&0&1&0&0&2\\0&3&0&1&0&5&0&0&5&0&4&0&4&10&0\\1&0&5&0&3&0&8&8&0&3&0&13&0&0&22\\0&1&0&5&0&7&0&0&7&0&2&0&16&14&0\\1&0&3&0&8&0&7&10&0&9&0&19&0&0&28\\0&5&0&7&0&25&0&0&29&0&17&0&43&62&0\\0&0&8&0&7&0&27&20&0&12&0&42&0&0&80\\2&0&8&0&10&0&20&33&0&21&0&45&0&0&92\\0&5&0&7&0&29&0&0&47&0&20&0&56&90&0\\1&0&3&0&9&0&12&21&0&26&0&30&0&0&72\\0&4&0&2&0&17&0&0&20&0&18&0&26&46&0\\1&0&13&0&19&0&42&45&0&30&0&93&0&0&166\\0&4&0&16&0&43&0&0&56&0&26&0&108&128&0\\0&10&0&14&0&62&0&0&90&0&46&0&128&205&0\\2&0&22&0&28&0&80&92&0&72&0&166&0&0&368\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&5&5&8&25&27&33&47&26&18&93&108&205&368&187&174&396&315&109\end{bmatrix}$

Event probabilities

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