Properties

Label 1.4.E.12.4a
  
Name \(J(E_6)\)
Weight $1$
Degree $4$
Real dimension $3$
Components $12$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{SU}(2)_2\)
Component group \(D_6\)

Learn more

Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$3$
Components:$12$
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:$D_6$
Order:$12$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{12}&0&0&0\\0&\zeta_{12}&0&0\\0&0&\zeta_{12}^{11}&0\\0&0&0&\zeta_{12}^{11}\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(E_2)$, $J(E_3)$${}^{\times 2}$, $E_6$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $6$ $0$ $50$ $0$ $490$ $0$ $5292$ $0$ $61116$
$a_2$ $1$ $1$ $3$ $7$ $25$ $91$ $387$ $1716$ $8045$ $38821$ $192415$ $972544$ $4999447$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $3$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $7$ $11$ $23$ $50$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $25$ $44$ $96$ $215$ $490$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $91$ $188$ $423$ $970$ $2254$ $5292$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $387$ $843$ $1942$ $4537$ $10710$ $25494$ $61116$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&1&0&0&0&0\\0&1&0&0&1&0&1&0&1&0\\0&0&2&0&0&0&0&2&0&2\\0&0&0&3&0&1&0&2&0&0\\0&1&0&0&3&0&2&0&3&0\\1&0&0&1&0&4&0&1&0&0\\0&1&0&0&2&0&3&0&3&0\\0&0&2&2&0&1&0&7&0&4\\0&1&0&0&3&0&3&0&6&0\\0&0&2&0&0&0&0&4&0&7\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&3&3&4&3&7&6&7\end{bmatrix}$

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$$0$
$a_1=0$$7/12$$0$$0$$0$$0$$0$$0$