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Properties

Label 1.6.J.24.14a
  
Name \(J_2(J(E_6))\)
Weight $1$
Degree $6$
Real dimension $4$
Components $24$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)_2\)
Component group \(C_2\times D_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$4$
Components:$24$
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:$C_2\times D_6$
Order:$24$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{11} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J_2(J(E_2))$, $J_2(E_6)$, $J(J(E_6),J(E_3))$${}^{\times 2}$, $J_1(J(E_6))$, $J_2(J(E_3))$${}^{\times 2}$, $J(J(E_6),E_6)$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $15$ $0$ $195$ $0$ $3465$ $0$ $73143$ $0$ $1712766$
$a_2$ $1$ $2$ $7$ $32$ $202$ $1627$ $15305$ $157936$ $1728046$ $19689614$ $231266725$ $2782618093$ $34152435256$
$a_3$ $1$ $0$ $7$ $0$ $435$ $0$ $63950$ $0$ $13066375$ $0$ $3124258578$ $0$ $823343695746$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $7$ $3$ $8$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $7$ $32$ $17$ $49$ $30$ $95$ $195$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $38$ $202$ $115$ $72$ $374$ $228$ $770$ $465$ $1620$ $3465$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $283$ $1627$ $171$ $950$ $571$ $3333$ $1984$ $1192$ $7113$ $4220$ $15355$ $9065$ $33404$ $73143$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $435$ $2485$ $15305$ $1480$ $8924$ $5263$ $32876$ $3116$ $19239$ $11305$ $71750$ $41860$ $24525$ $157584$
$$ $91672$ $347760$ $201726$ $770448$ $1712766$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&2&0&2&5&0\\1&0&4&0&1&0&4&6&0&1&0&6&0&0&11\\0&1&0&3&0&4&0&0&4&0&1&0&8&7&0\\0&0&1&0&6&0&5&3&0&6&0&11&0&0&14\\0&3&0&4&0&14&0&0&15&0&9&0&22&31&0\\0&0&4&0&5&0&15&9&0&7&0&22&0&0&39\\2&0&6&0&3&0&9&21&0&7&0&21&0&0&44\\0&3&0&4&0&15&0&0&23&0&10&0&28&43&0\\0&0&1&0&6&0&7&7&0&14&0&16&0&0&33\\0&2&0&1&0&9&0&0&10&0&10&0&14&23&0\\0&0&6&0&11&0&22&21&0&16&0&49&0&0&81\\0&2&0&8&0&22&0&0&28&0&14&0&54&62&0\\0&5&0&7&0&31&0&0&43&0&23&0&62&99&0\\1&0&11&0&14&0&39&44&0&33&0&81&0&0&170\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&6&14&15&21&23&14&10&49&54&99&170&82&86&169&134&44\end{bmatrix}$

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$$0$
$a_1=0$$7/24$$0$$0$$0$$0$$0$$0$
$a_3=0$$7/24$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$7/24$$0$$0$$0$$0$$0$$0$