Properties

Label 1.4.E.3.1a
  
Name \(E_3\)
Weight $1$
Degree $4$
Real dimension $3$
Components $3$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{SU}(2)_2\)
Component group \(C_3\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$3$
Components:$3$
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:$C_3$
Order:$3$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_6&0&0&0\\0&\zeta_6&0&0\\0&0&\zeta_6^5&0\\0&0&0&\zeta_6^5\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$E_1$
Minimal supergroups:$J(E_3)$, $E_6$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $12$ $0$ $110$ $0$ $1204$ $0$ $14364$ $0$ $180312$
$a_2$ $1$ $1$ $4$ $13$ $52$ $222$ $1014$ $4839$ $23860$ $120526$ $620278$ $3239788$ $17127202$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $4$ $6$ $12$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $13$ $24$ $50$ $110$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $52$ $104$ $228$ $518$ $1204$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $222$ $478$ $1086$ $2530$ $5992$ $14364$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1014$ $2286$ $5332$ $12658$ $30420$ $73788$ $180312$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&1&0&0&0&2\\0&2&0&0&2&0&2&0&4&0\\0&0&3&1&0&2&0&4&0&3\\1&0&1&4&0&3&0&6&0&5\\0&2&0&0&8&0&4&0&10&0\\1&0&2&3&0&6&0&7&0&6\\0&2&0&0&4&0&8&0&10&0\\0&0&4&6&0&7&0&17&0&10\\0&4&0&0&10&0&10&0&20&0\\2&0&3&5&0&6&0&10&0&12\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&4&8&6&8&17&20&12\end{bmatrix}$

Event probabilities

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

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