Properties

Label 1.4.E.4.2a
  
Name \(J(E_2)\)
Weight $1$
Degree $4$
Real dimension $3$
Components $4$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{SU}(2)_2\)
Component group \(C_2^2\)

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Invariants

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

Identity component

Name:$\mathrm{SU}(2)_2$
Index:$4$
$\mathbb{R}$-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:$C_2^2$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}i&0&0&0\\0&i&0&0\\0&0&-i&0\\0&0&0&-i\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$E_2$, $J(E_1)$${}^{\times 2}$
Minimal supergroups:$J(E_4)$, $J(E_6)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $8$ $0$ $80$ $0$ $896$ $0$ $10752$ $0$ $135168$
$a_2$ $1$ $1$ $4$ $10$ $42$ $166$ $768$ $3620$ $17902$ $90310$ $465096$ $2429164$ $12843988$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $4$ $4$ $8$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $10$ $17$ $36$ $80$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $42$ $76$ $168$ $384$ $896$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $166$ $354$ $808$ $1888$ $4480$ $10752$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $768$ $1704$ $3984$ $9472$ $22784$ $55296$ $135168$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&2&0&0&0&0\\0&1&0&0&2&0&1&0&3&0\\0&0&3&0&0&0&0&4&0&6\\0&0&0&4&0&3&0&4&0&1\\0&2&0&0&5&0&4&0&8&0\\2&0&0&3&0&8&0&2&0&0\\0&1&0&0&4&0&5&0&7&0\\0&0&4&4&0&2&0&15&0&12\\0&3&0&0&8&0&7&0&14&0\\0&0&6&1&0&0&0&12&0&16\end{bmatrix},\qquad\mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&3&4&5&8&5&15&14&16\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_1=0]=\frac{3}{4}$