Properties

Label 1.6.I.1.1a
  
Name \(\mathrm{SU}(2)\times E_1\)
Weight $1$
Degree $6$
Real dimension $6$
Components $1$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{SU}(2)\times\mathrm{SU}(2)_2\)
Component group \(C_1\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$6$
Components:$1$
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:$\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$

Subgroups and supergroups

Maximal subgroups:
Minimal supergroups:$\mathrm{SU}(2)\times E_2$, $\mathrm{SU}(2)\times J(E_1)$, $\mathrm{SU}(2)\times E_3$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $5$ $0$ $58$ $0$ $925$ $0$ $17598$ $0$ $374850$ $0$ $8638740$
$a_2$ $1$ $4$ $21$ $137$ $1059$ $9250$ $88075$ $892540$ $9478251$ $104393828$ $1184015783$ $13756977657$ $163109827813$
$a_3$ $1$ $0$ $26$ $0$ $2444$ $0$ $366740$ $0$ $70009940$ $0$ $15468784248$ $0$ $3769925528688$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $4$ $5$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $21$ $11$ $32$ $58$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $26$ $137$ $77$ $241$ $142$ $464$ $925$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $191$ $1059$ $612$ $360$ $2037$ $1198$ $4102$ $2415$ $8436$ $17598$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1586$ $9250$ $930$ $5382$ $3154$ $18775$ $10976$ $6434$ $39030$ $22793$ $82221$ $47922$ $174888$ $374850$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2444$ $14433$ $88075$ $8430$ $51127$ $29780$ $184830$ $17385$ $107300$ $62411$ $393102$ $227796$ $132232$ $843417$
$$ $487746$ $1821942$ $1051470$ $3957840$ $8638740$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&3&0&1&0&3&6&0&1&0&3&0&0&8\\0&5&0&6&0&16&0&0&18&0&5&0&17&33&0\\3&0&14&0&10&0&25&33&0&14&0&34&0&0&72\\0&6&0&9&0&24&0&0&30&0&9&0&32&54&0\\1&0&10&0&15&0&29&28&0&21&0&50&0&0&88\\0&16&0&24&0&76&0&0&96&0&40&0&120&184&0\\3&0&25&0&29&0&70&75&0&51&0&117&0&0&232\\6&0&33&0&28&0&75&96&0&52&0&124&0&0&264\\0&18&0&30&0&96&0&0&133&0&52&0&169&258&0\\1&0&14&0&21&0&51&52&0&44&0&94&0&0&192\\0&5&0&9&0&40&0&0&52&0&35&0&85&110&0\\3&0&34&0&50&0&117&124&0&94&0&235&0&0&448\\0&17&0&32&0&120&0&0&169&0&85&0&251&350&0\\0&33&0&54&0&184&0&0&258&0&110&0&350&535&0\\8&0&72&0&88&0&232&264&0&192&0&448&0&0&940\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&14&9&15&76&70&96&133&44&35&235&251&535&940&399&449&887&642&155\end{bmatrix}$

Event probabilities

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