# Properties

 Label 1.4.E.4.2a Name $$J(E_2)$$ Weight $1$ Degree $4$ Real dimension $3$ Components $4$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{SU}(2)_2$$ Component group $$C_2^2$$

## Invariants

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

## Identity component

 Name: $\mathrm{SU}(2)_2$ Index: $4$ $\mathbb{R}$-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: $C_2^2$ Order: $4$ Abelian: yes Generators: $\begin{bmatrix}i&0&0&0\\0&i&0&0\\0&0&-i&0\\0&0&0&-i\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

## Subgroups and supergroups

 Maximal subgroups: $E_2$, $J(E_1)$${}^{\times 2}$ Minimal supergroups: $J(E_4)$, $J(E_6)$

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $8$ $0$ $80$ $0$ $896$ $0$ $10752$ $0$ $135168$
$a_2$ $1$ $1$ $4$ $10$ $42$ $166$ $768$ $3620$ $17902$ $90310$ $465096$ $2429164$ $12843988$

## Moment simplex

 $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1$ $1$ $4$ $4$ $8$ $10$ $17$ $36$ $80$ $42$ $76$ $168$ $384$ $896$ $166$ $354$ $808$ $1888$ $4480$ $10752$ $768$ $1704$ $3984$ $9472$ $22784$ $55296$ $135168$

## Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&2&0&0&0&0\\0&1&0&0&2&0&1&0&3&0\\0&0&3&0&0&0&0&4&0&6\\0&0&0&4&0&3&0&4&0&1\\0&2&0&0&5&0&4&0&8&0\\2&0&0&3&0&8&0&2&0&0\\0&1&0&0&4&0&5&0&7&0\\0&0&4&4&0&2&0&15&0&12\\0&3&0&0&8&0&7&0&14&0\\0&0&6&1&0&0&0&12&0&16\end{bmatrix},\qquad\mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&3&4&5&8&5&15&14&16\end{bmatrix}$

## Event probabilities

 $\mathrm{Pr}[a_1=0]=\frac{3}{4}$