# Properties

 Label 1.6.I.4.2a Name $$\mathrm{SU}(2)\times J(E_2)$$ Weight $1$ Degree $6$ Real dimension $6$ Components $4$ Contained in $$\mathrm{USp}(6)$$ Identity component $$\mathrm{SU}(2)\times\mathrm{SU}(2)_2$$ Component group $$C_2^2$$

## Invariants

 Weight: $1$ Degree: $6$ $\mathbb{R}$-dimension: $6$ Components: $4$ Contained in: $\mathrm{USp}(6)$ Rational: yes

## Identity component

 Name: $\mathrm{SU}(2)\times\mathrm{SU}(2)_2$ $\mathbb{R}$-dimension: $6$ Description: $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A,B\in\mathrm{SU}(2)\right\}$ 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: $C_2^2$ Order: $4$ Abelian: yes Generators: $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & -i \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

## Subgroups and supergroups

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

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $16$ $0$ $235$ $0$ $4410$ $0$ $93744$ $0$ $2159784$
$a_2$ $1$ $2$ $8$ $41$ $283$ $2362$ $22156$ $223519$ $2370647$ $26101538$ $296012748$ $3439269671$ $40777529689$
$a_3$ $1$ $0$ $8$ $0$ $620$ $0$ $91760$ $0$ $17503220$ $0$ $3867204000$ $0$ $942481473648$

## Moment simplex

 $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$  $2$ $2$ $8$ $3$ $9$ $16$ $8$ $41$ $21$ $63$ $36$ $118$ $235$ $50$ $283$ $157$ $93$ $516$ $303$ $1031$ $605$ $2114$ $4410$ $404$ $2362$ $234$ $1357$ $793$ $4712$ $2752$ $1616$ $9771$ $5707$ $20569$ $11984$ $43736$ $93744$ $620$ $3628$ $22156$ $2118$ $12813$ $7460$ $46257$ $4350$ $26848$ $15614$ $98312$ $56969$ $33079$ $210888$ $121961$ $455524$ $262878$ $989502$ $2159784$

## Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&0&0&0&2\\0&2&0&1&0&4&0&0&5&0&1&0&3&9&0\\1&0&5&0&1&0&6&9&0&2&0&8&0&0&18\\0&1&0&4&0&6&0&0&6&0&1&0&10&13&0\\0&0&1&0&6&0&7&6&0&8&0&13&0&0&22\\0&4&0&6&0&19&0&0&24&0&10&0&30&46&0\\0&0&6&0&7&0&20&16&0&11&0&30&0&0&58\\3&0&9&0&6&0&16&30&0&13&0&28&0&0&66\\0&5&0&6&0&24&0&0&36&0&14&0&39&65&0\\0&0&2&0&8&0&11&13&0&17&0&24&0&0&48\\0&1&0&1&0&10&0&0&14&0&11&0&20&27&0\\0&0&8&0&13&0&30&28&0&24&0&61&0&0&112\\0&3&0&10&0&30&0&0&39&0&20&0&69&85&0\\0&9&0&13&0&46&0&0&65&0&27&0&85&136&0\\2&0&18&0&22&0&58&66&0&48&0&112&0&0&235\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&6&19&20&30&36&17&11&61&69&136&235&106&122&233&170&50\end{bmatrix}$

## Event probabilities

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