Properties

Label 1.6.J.4.1b
  
Name \(J_1(E_4)\)
Weight $1$
Degree $6$
Real dimension $4$
Components $4$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)_2\)
Component group \(C_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$4$
Components:$4$
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:$C_4$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $42$ $0$ $660$ $0$ $12838$ $0$ $285096$ $0$ $6939108$
$a_2$ $1$ $2$ $11$ $75$ $623$ $5828$ $59313$ $642050$ $7282563$ $85648062$ $1036414961$ $12830752047$ $161808697025$
$a_3$ $1$ $0$ $16$ $0$ $1524$ $0$ $259520$ $0$ $57298052$ $0$ $14619734592$ $0$ $4077807324624$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $11$ $6$ $20$ $42$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $16$ $75$ $46$ $150$ $90$ $310$ $660$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $110$ $623$ $372$ $228$ $1298$ $778$ $2758$ $1640$ $5920$ $12838$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $964$ $5828$ $570$ $3428$ $2026$ $12488$ $7332$ $4336$ $27040$ $15840$ $58972$ $34426$ $129332$ $285096$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1524$ $9290$ $59313$ $5450$ $34398$ $20028$ $129544$ $11688$ $74960$ $43516$ $284766$ $164368$ $95224$ $629076$
$$ $362278$ $1395490$ $801780$ $3106776$ $6939108$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&3&0&2&0&3&0&0&4\\0&4&0&2&0&10&0&0&10&0&8&0&10&20&0\\1&0&8&0&7&0&16&15&0&7&0&28&0&0&44\\0&2&0&8&0&14&0&0&14&0&6&0&30&30&0\\2&0&7&0&13&0&15&22&0&15&0&37&0&0&56\\0&10&0&14&0&50&0&0&58&0&34&0&86&124&0\\1&0&16&0&15&0&50&43&0&25&0&84&0&0&160\\3&0&15&0&22&0&43&60&0&43&0&92&0&0&184\\0&10&0&14&0&58&0&0&92&0&40&0&114&180&0\\2&0&7&0&15&0&25&43&0&46&0&60&0&0&144\\0&8&0&6&0&34&0&0&40&0&32&0&54&92&0\\3&0&28&0&37&0&84&92&0&60&0&183&0&0&332\\0&10&0&30&0&86&0&0&114&0&54&0&208&260&0\\0&20&0&30&0&124&0&0&180&0&92&0&260&404&0\\4&0&44&0&56&0&160&184&0&144&0&332&0&0&736\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&8&8&13&50&50&60&92&46&32&183&208&404&736&364&339&784&622&208\end{bmatrix}$

Event probabilities

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