# 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 $$C_3$$

## 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: $\begin{bmatrix}\zeta_6&0&0&0\\0&\zeta_6&0&0\\0&0&\zeta_6^5&0\\0&0&0&\zeta_6^5\end{bmatrix}$

## 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

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

## 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 $n\in\mathbb{Z}$.