Properties

Label 1.4.E.6.1a
  
Name \(J(E_3)\)
Weight $1$
Degree $4$
Real dimension $3$
Components $6$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{SU}(2)_2\)
Component group \(S_3\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$3$
Components:$6$
Contained in:$\mathrm{USp}(4)$
Rational:yes

Identity component

Name:$\mathrm{SU}(2)_2$
Index:$6$
$\mathbb{R}$-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:$u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$

Component group

Name:$S_3$
Order:$6$
Abelian:no
Generators:$\begin{bmatrix}\zeta_6&0&0&0\\0&\zeta_6&0&0\\0&0&\zeta_6^5&0\\0&0&0&\zeta_6^5\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(E_1)$, $E_3$
Minimal supergroups:$J(E_6)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $6$ $0$ $55$ $0$ $602$ $0$ $7182$ $0$ $90156$
$a_2$ $1$ $1$ $3$ $8$ $29$ $116$ $517$ $2437$ $11965$ $60326$ $310265$ $1620125$ $8564063$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $3$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $8$ $12$ $25$ $55$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $29$ $52$ $114$ $259$ $602$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $116$ $239$ $543$ $1265$ $2996$ $7182$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $517$ $1143$ $2666$ $6329$ $15210$ $36894$ $90156$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&1&0&0&0&1\\0&1&0&0&1&0&1&0&2&0\\0&0&2&0&0&1&0&2&0&2\\0&0&0&3&0&1&0&3&0&2\\0&1&0&0&4&0&2&0&5&0\\1&0&1&1&0&4&0&3&0&3\\0&1&0&0&2&0&4&0&5&0\\0&0&2&3&0&3&0&9&0&5\\0&2&0&0&5&0&5&0&10&0\\1&0&2&2&0&3&0&5&0&7\end{bmatrix},\qquad\mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&3&4&4&4&9&10&7\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_1=0]=\frac{3}{6}$