Properties

Label 1.4.E.1.1a
  
Name \(E_1\)
Weight $1$
Degree $4$
Real dimension $3$
Components $1$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{SU}(2)_2\)
Component group \(C_1\)

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Invariants

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

Identity component

Name:$\mathrm{SU}(2)_2$
$\mathbb{R}$-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:$u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$

Subgroups and supergroups

Maximal subgroups:
Minimal supergroups:$E_2$, $J(E_1)$, $E_3$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $32$ $0$ $320$ $0$ $3584$ $0$ $43008$ $0$ $540672$
$a_2$ $1$ $3$ $10$ $37$ $150$ $654$ $3012$ $14445$ $71398$ $361114$ $1859628$ $9716194$ $51373180$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $3$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $10$ $16$ $32$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $37$ $68$ $144$ $320$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $150$ $304$ $672$ $1536$ $3584$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $654$ $1416$ $3232$ $7552$ $17920$ $43008$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $3012$ $6816$ $15936$ $37888$ $91136$ $221184$ $540672$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&1&0&3&0&2&0&4\\0&4&0&0&8&0&4&0&12&0\\2&0&5&5&0&8&0&10&0&11\\1&0&5&10&0&9&0&20&0&13\\0&8&0&0&20&0&16&0&32&0\\3&0&8&9&0&14&0&21&0&20\\0&4&0&0&16&0&20&0&28&0\\2&0&10&20&0&21&0&49&0&32\\0&12&0&0&32&0&28&0&56&0\\4&0&11&13&0&20&0&32&0&30\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&5&10&20&14&20&49&56&30\end{bmatrix}$

Event probabilities

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

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