Properties

Label 1.4.E.4.1a
  
Name \(E_4\)
Weight $1$
Degree $4$
Real dimension $3$
Components $4$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{SU}(2)_2\)
Component group \(C_4\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$3$
Components:$4$
Contained in:$\mathrm{USp}(4)$
Rational:yes

Identity component

Name:$\mathrm{SU}(2)_2$
$\mathbb{R}$-dimension:$3$
Description:$\left\{\begin{bmatrix}A&0\\0&\bar{A}\end{bmatrix}: A\in \mathrm{SU}(2)\right\}$ Symplectic form:$\begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}$
Hodge circle:$u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$

Component group

Name:$C_4$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_8&0&0&0\\0&\zeta_8&0&0\\0&0&\zeta_8^7&0\\0&0&0&\zeta_8^7\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$E_2$
Minimal supergroups:$J(E_4)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $12$ $0$ $100$ $0$ $1008$ $0$ $11424$ $0$ $139392$
$a_2$ $1$ $1$ $4$ $11$ $46$ $182$ $824$ $3817$ $18582$ $92678$ $473368$ $2458326$ $12947532$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $4$ $6$ $12$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $11$ $22$ $46$ $100$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $46$ $90$ $196$ $440$ $1008$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $182$ $396$ $892$ $2056$ $4816$ $11424$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $824$ $1836$ $4248$ $10000$ $23840$ $57408$ $139392$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&1&0&0&0&0\\0&2&0&0&2&0&2&0&2&0\\0&0&3&1&0&0&0&4&0&5\\1&0&1&4&0&3&0&4&0&1\\0&2&0&0&6&0&4&0&8&0\\1&0&0&3&0&8&0&3&0&0\\0&2&0&0&4&0&6&0&6&0\\0&0&4&4&0&3&0&15&0&12\\0&2&0&0&8&0&6&0&16&0\\0&0&5&1&0&0&0&12&0&16\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&4&6&8&6&15&16&16\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-2$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$
$-$$1$$0$$0$$0$$0$$0$$0$
$a_1=0$$1/4$$0$$0$$0$$0$$0$$0$