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 \(C_2^2\)

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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

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

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 $n\in\mathbb{Z}$.