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Properties

Label 1.6.J.16.11a
  
Name \(J_2(J(E_4))\)
Weight $1$
Degree $6$
Real dimension $4$
Components $16$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)_2\)
Component group \(C_2\times D_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$4$
Components:$16$
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:$C_2\times D_4$
Order:$16$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $15$ $0$ $195$ $0$ $3479$ $0$ $74193$ $0$ $1769856$
$a_2$ $1$ $2$ $7$ $32$ $203$ $1647$ $15685$ $164761$ $1843287$ $21539126$ $259843505$ $3212105448$ $40479053697$
$a_3$ $1$ $0$ $7$ $0$ $438$ $0$ $66580$ $0$ $14390250$ $0$ $3657874068$ $0$ $1019594782008$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $7$ $3$ $8$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $7$ $32$ $17$ $49$ $30$ $95$ $195$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $38$ $203$ $115$ $72$ $375$ $228$ $772$ $465$ $1625$ $3479$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $284$ $1647$ $171$ $956$ $572$ $3368$ $1995$ $1194$ $7187$ $4245$ $15527$ $9128$ $33824$ $74193$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $438$ $2518$ $15685$ $1492$ $9087$ $5328$ $33676$ $3137$ $19588$ $11444$ $73541$ $42650$ $24849$ $161731$
$$ $93534$ $357580$ $206220$ $794028$ $1769856$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&2&0&2&5&0\\1&0&4&0&1&0&4&6&0&1&0&6&0&0&11\\0&1&0&3&0&4&0&0&4&0&1&0&8&7&0\\0&0&1&0&6&0&5&3&0&6&0&11&0&0&14\\0&3&0&4&0&14&0&0&15&0&9&0&22&32&0\\0&0&4&0&5&0&15&9&0&7&0&22&0&0&40\\2&0&6&0&3&0&9&22&0&7&0&21&0&0&46\\0&3&0&4&0&15&0&0&24&0&10&0&28&45&0\\0&0&1&0&6&0&7&7&0&16&0&16&0&0&36\\0&2&0&1&0&9&0&0&10&0&10&0&14&23&0\\0&0&6&0&11&0&22&21&0&16&0&50&0&0&83\\0&2&0&8&0&22&0&0&28&0&14&0&55&64&0\\0&5&0&7&0&32&0&0&45&0&23&0&64&105&0\\1&0&11&0&14&0&40&46&0&36&0&83&0&0&184\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&6&14&15&22&24&16&10&50&55&105&184&96&97&199&165&57\end{bmatrix}$

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$$0$
$a_1=0$$5/16$$0$$0$$0$$0$$0$$0$
$a_3=0$$5/16$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$5/16$$0$$0$$0$$0$$0$$0$