Properties

Label 1.6.I.4.1a
  
Name \(\mathrm{SU}(2)\times E_4\)
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 \(C_4\)

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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:$C_4$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $26$ $0$ $345$ $0$ $5782$ $0$ $112686$ $0$ $2442132$
$a_2$ $1$ $2$ $9$ $51$ $359$ $2908$ $26103$ $252822$ $2595083$ $27872238$ $310353971$ $3558078967$ $41781089633$
$a_3$ $1$ $0$ $12$ $0$ $812$ $0$ $105080$ $0$ $18674068$ $0$ $3986830992$ $0$ $956002577904$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $5$ $14$ $26$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $12$ $51$ $31$ $91$ $56$ $174$ $345$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $71$ $359$ $216$ $134$ $689$ $420$ $1376$ $835$ $2800$ $5782$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $532$ $2908$ $322$ $1736$ $1046$ $5861$ $3506$ $2112$ $12072$ $7205$ $25187$ $14980$ $53060$ $112686$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $812$ $4477$ $26103$ $2678$ $15411$ $9148$ $54346$ $5451$ $32050$ $18971$ $114622$ $67414$ $39794$ $243903$
$$ $143052$ $522702$ $305718$ $1126896$ $2442132$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&2&0&1&0&1&0&0&2\\0&3&0&2&0&6&0&0&6&0&3&0&5&9&0\\1&0&6&0&4&0&9&9&0&4&0&12&0&0&20\\0&2&0&5&0&8&0&0&8&0&3&0&12&14&0\\1&0&4&0&7&0&9&10&0&7&0&16&0&0&24\\0&6&0&8&0&26&0&0&28&0&14&0&36&52&0\\1&0&9&0&9&0&24&21&0&13&0&35&0&0&64\\2&0&9&0&10&0&21&30&0&16&0&36&0&0&72\\0&6&0&8&0&28&0&0&41&0&16&0&45&70&0\\1&0&4&0&7&0&13&16&0&18&0&24&0&0&52\\0&3&0&3&0&14&0&0&16&0&13&0&23&32&0\\1&0&12&0&16&0&35&36&0&24&0&71&0&0&120\\0&5&0&12&0&36&0&0&45&0&23&0&75&94&0\\0&9&0&14&0&52&0&0&70&0&32&0&94&147&0\\2&0&20&0&24&0&64&72&0&52&0&120&0&0&250\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&5&7&26&24&30&41&18&13&71&75&147&250&113&121&239&176&53\end{bmatrix}$

Event probabilities

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