Properties

Label 1.6.J.12.4a
  
Name \(J(J(E_6),J(E_3))\)
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:$\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{11} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J_1(J(E_3))$, $J(J(E_2),J(E_1))$, $J(J(E_3),E_3)$, $J(E_6,E_3)$
Minimal supergroups:$J_2(J(E_6))$${}^{\times 2}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $15$ $0$ $195$ $0$ $3605$ $0$ $81333$ $0$ $2050026$
$a_2$ $1$ $2$ $7$ $32$ $208$ $1767$ $17680$ $193811$ $2242294$ $26872496$ $330276940$ $4138428978$ $52664529604$
$a_3$ $1$ $0$ $7$ $0$ $471$ $0$ $78750$ $0$ $18075575$ $0$ $4739655258$ $0$ $1341214962450$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $7$ $3$ $8$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $7$ $32$ $17$ $49$ $30$ $95$ $195$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $39$ $208$ $118$ $73$ $387$ $233$ $797$ $475$ $1680$ $3605$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $300$ $1767$ $180$ $1020$ $605$ $3642$ $2133$ $1260$ $7797$ $4544$ $16895$ $9793$ $36932$ $81333$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $471$ $2774$ $17680$ $1627$ $10153$ $5889$ $38206$ $3432$ $21972$ $12687$ $83788$ $48012$ $27616$ $185006$
$$ $105665$ $410690$ $233856$ $915768$ $2050026$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&2&0&2&5&0\\1&0&4&0&1&0&4&6&0&2&0&6&0&0&13\\0&1&0&3&0&4&0&0&5&0&1&0&8&8&0\\0&0&1&0&6&0&5&3&0&6&0&11&0&0&16\\0&3&0&4&0&14&0&0&17&0&9&0&24&35&0\\0&0&4&0&5&0&16&11&0&10&0&23&0&0&49\\2&0&6&0&3&0&11&22&0&9&0&24&0&0&56\\0&3&0&5&0&17&0&0&28&0&10&0&36&53&0\\0&0&2&0&6&0&10&9&0&15&0&20&0&0&45\\0&2&0&1&0&9&0&0&10&0&10&0&15&26&0\\0&0&6&0&11&0&23&24&0&20&0&53&0&0&101\\0&2&0&8&0&24&0&0&36&0&15&0&61&78&0\\0&5&0&8&0&35&0&0&53&0&26&0&78&125&0\\1&0&13&0&16&0&49&56&0&45&0&101&0&0&233\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&6&14&16&22&28&15&10&53&61&125&233&113&117&245&202&56\end{bmatrix}$

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$$0$
$a_1=0$$1/3$$0$$0$$0$$0$$0$$0$
$a_3=0$$1/3$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$1/3$$0$$0$$0$$0$$0$$0$