Properties

Label 1.4.E.2.1a
  
Name \(E_2\)
Weight $1$
Degree $4$
Real dimension $3$
Components $2$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{SU}(2)_2\)
Component group \(C_2\)

Learn more

Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$3$
Components:$2$
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:$u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$

Component group

Name:$C_2$
Order:$2$
Abelian:yes
Generators:$\begin{bmatrix}i&0&0&0\\0&i&0&0\\0&0&-i&0\\0&0&0&-i\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$E_1$
Minimal supergroups:$E_4$, $J(E_2)$, $E_6$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $16$ $0$ $160$ $0$ $1792$ $0$ $21504$ $0$ $270336$
$a_2$ $1$ $1$ $6$ $17$ $78$ $322$ $1516$ $7205$ $35734$ $180494$ $929940$ $4857866$ $25687052$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $6$ $8$ $16$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $17$ $34$ $72$ $160$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $78$ $152$ $336$ $768$ $1792$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $322$ $708$ $1616$ $3776$ $8960$ $21504$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1516$ $3408$ $7968$ $18944$ $45568$ $110592$ $270336$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&3&0&0&0&0\\0&2&0&0&4&0&2&0&6&0\\0&0&5&1&0&0&0&8&0&11\\1&0&1&6&0&7&0&8&0&3\\0&4&0&0&10&0&8&0&16&0\\3&0&0&7&0&14&0&5&0&0\\0&2&0&0&8&0&10&0&14&0\\0&0&8&8&0&5&0&29&0&24\\0&6&0&0&16&0&14&0&28&0\\0&0&11&3&0&0&0&24&0&30\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&6&10&14&10&29&28&30\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-2$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$
$-$$1$$0$$0$$0$$0$$0$$0$
$a_1=0$$1/2$$0$$0$$0$$0$$0$$0$