Properties

Label 1.6.J.12.4b
  
Name \(J_2(J(E_3))\)
Weight $1$
Degree $6$
Real dimension $4$
Components $12$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)_2\)
Component group \(D_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$4$
Components:$12$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)\times\mathrm{SU}(2)_2$
$\mathbb{R}$-dimension:$4$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B\in\mathrm{SU}(2)\right\}$ Symplectic form:$\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
Hodge circle:$\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$

Component group

Name:$D_6$
Order:$12$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J_1(J(E_3))$, $J(J(E_3),E_3)$, $J_2(J(E_1))$, $J_2(E_3)$
Minimal supergroups:$J_2(J(E_6))$${}^{\times 2}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $15$ $0$ $200$ $0$ $3717$ $0$ $83223$ $0$ $2079066$
$a_2$ $1$ $2$ $7$ $33$ $216$ $1822$ $18040$ $196142$ $2257358$ $26969895$ $330907320$ $4142513509$ $52691027944$
$a_3$ $1$ $0$ $7$ $0$ $471$ $0$ $79070$ $0$ $18104247$ $0$ $4741590618$ $0$ $1341333910290$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $7$ $3$ $8$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $7$ $33$ $17$ $50$ $30$ $97$ $200$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $39$ $216$ $120$ $73$ $398$ $237$ $819$ $485$ $1729$ $3717$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $304$ $1822$ $180$ $1042$ $613$ $3731$ $2177$ $1280$ $7977$ $4642$ $17283$ $10017$ $37786$ $83223$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $471$ $2818$ $18040$ $1643$ $10331$ $5977$ $38851$ $3472$ $22332$ $12883$ $85108$ $48788$ $28064$ $187820$
$$ $107373$ $416786$ $237636$ $929058$ $2079066$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&2&0&2&6&0\\1&0&4&0&1&0&4&7&0&1&0&6&0&0&13\\0&1&0&3&0&4&0&0&5&0&1&0&8&9&0\\0&0&1&0&6&0&5&3&0&6&0&12&0&0&16\\0&3&0&4&0&15&0&0&17&0&9&0&24&37&0\\0&0&4&0&5&0&16&11&0&10&0&24&0&0&49\\2&0&7&0&3&0&11&24&0&8&0&24&0&0&56\\0&3&0&5&0&17&0&0&28&0&10&0&36&54&0\\0&0&1&0&6&0&10&8&0&15&0&20&0&0&45\\0&2&0&1&0&9&0&0&10&0&11&0&16&26&0\\0&0&6&0&12&0&24&24&0&20&0&55&0&0&101\\0&2&0&8&0&24&0&0&36&0&16&0&62&78&0\\0&6&0&9&0&37&0&0&54&0&26&0&78&126&0\\1&0&13&0&16&0&49&56&0&45&0&101&0&0&233\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&6&15&16&24&28&15&11&55&62&126&233&114&121&246&204&57\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$0$$0$$0$$0$$0$$0$
$a_1=0$$1/4$$0$$0$$0$$0$$0$$0$
$a_3=0$$1/4$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$1/4$$0$$0$$0$$0$$0$$0$