Properties

Label 1.6.J.8.2a
  
Name \(J_2(E_4)\)
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 \(C_2\times C_4\)

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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:$C_2\times C_4$
Order:$8$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(E_4,E_2)$, $J_1(E_4)$, $J_2(E_2)$
Minimal supergroups:$J_2(J(E_4))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $27$ $0$ $380$ $0$ $6923$ $0$ $148260$ $0$ $3539250$
$a_2$ $1$ $2$ $9$ $51$ $371$ $3198$ $31103$ $328772$ $3684451$ $43072206$ $519669707$ $6424161175$ $80957964029$
$a_3$ $1$ $0$ $12$ $0$ $858$ $0$ $132960$ $0$ $28778050$ $0$ $7315716384$ $0$ $2039189137128$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $5$ $14$ $27$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $12$ $51$ $31$ $93$ $57$ $184$ $380$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $71$ $371$ $222$ $138$ $737$ $447$ $1529$ $920$ $3230$ $6923$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $554$ $3198$ $333$ $1890$ $1129$ $6701$ $3966$ $2368$ $14335$ $8460$ $31004$ $18221$ $67578$ $148260$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $858$ $4997$ $31103$ $2957$ $18113$ $10614$ $67256$ $6244$ $39110$ $22838$ $146977$ $85220$ $49628$ $323332$
$$ $186963$ $714985$ $412314$ $1587804$ $3539250$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&2&0&1&0&1&0&0&2\\0&3&0&2&0&6&0&0&6&0&4&0&5&10&0\\1&0&6&0&4&0&9&9&0&4&0&14&0&0&22\\0&2&0&5&0&8&0&0&8&0&3&0&15&15&0\\1&0&4&0&8&0&9&11&0&8&0&20&0&0&28\\0&6&0&8&0&28&0&0&30&0&18&0&44&64&0\\1&0&9&0&9&0&27&22&0&13&0&43&0&0&80\\2&0&9&0&11&0&22&34&0&19&0&46&0&0&92\\0&6&0&8&0&30&0&0&47&0&20&0&57&90&0\\1&0&4&0&8&0&13&19&0&26&0&30&0&0&72\\0&4&0&3&0&18&0&0&20&0&18&0&29&46&0\\1&0&14&0&20&0&43&46&0&30&0&97&0&0&166\\0&5&0&15&0&44&0&0&57&0&29&0&106&130&0\\0&10&0&15&0&64&0&0&90&0&46&0&130&207&0\\2&0&22&0&28&0&80&92&0&72&0&166&0&0&368\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&5&8&28&27&34&47&26&18&97&106&207&368&187&178&394&321&109\end{bmatrix}$

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$$0$
$a_1=0$$1/8$$0$$0$$0$$0$$0$$0$
$a_3=0$$1/8$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$1/8$$0$$0$$0$$0$$0$$0$