Properties

Label 1.4.E.6.2a
  
Name \(E_6\)
Weight $1$
Degree $4$
Real dimension $3$
Components $6$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{SU}(2)_2\)
Component group \(C_6\)

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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:$\mathrm{SU}(2)_2$
$\mathbb{R}$-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:$C_6$
Order:$6$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_{12}&0&0&0\\0&\zeta_{12}&0&0\\0&0&\zeta_{12}^{11}&0\\0&0&0&\zeta_{12}^{11}\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$E_2$, $E_3$
Minimal supergroups:$J(E_6)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $12$ $0$ $100$ $0$ $980$ $0$ $10584$ $0$ $122232$
$a_2$ $1$ $1$ $4$ $11$ $44$ $172$ $754$ $3397$ $16020$ $77516$ $384578$ $1944626$ $9997970$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $4$ $6$ $12$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $11$ $22$ $46$ $100$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $44$ $88$ $192$ $430$ $980$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $172$ $376$ $846$ $1940$ $4508$ $10584$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $754$ $1686$ $3884$ $9074$ $21420$ $50988$ $122232$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&1&0&0&0&0\\0&2&0&0&2&0&2&0&2&0\\0&0&3&1&0&0&0&4&0&3\\1&0&1&4&0&3&0&4&0&1\\0&2&0&0&6&0&4&0&6&0\\1&0&0&3&0&6&0&3&0&0\\0&2&0&0&4&0&6&0&6&0\\0&0&4&4&0&3&0&13&0&8\\0&2&0&0&6&0&6&0&12&0\\0&0&3&1&0&0&0&8&0&12\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&4&6&6&6&13&12&12\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$
$-$$1$$0$$0$$0$$0$$0$
$a_1=0$$1/6$$0$$0$$0$$0$$0$