Properties

Label 1.6.J.6.2c
  
Name \(J_1(E_6)\)
Weight $1$
Degree $6$
Real dimension $4$
Components $6$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)_2\)
Component group \(C_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$4$
Components:$6$
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:$C_6$
Order:$6$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{11} \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J_1(E_2)$, $J_1(E_3)$
Minimal supergroups:$J_2(E_6)$, $J_1(J(E_6))$, $J(J(E_6),E_6)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $42$ $0$ $660$ $0$ $12810$ $0$ $281736$ $0$ $6727908$
$a_2$ $1$ $2$ $11$ $75$ $621$ $5768$ $57953$ $615940$ $6830181$ $78310746$ $922531581$ $11115725877$ $136522207423$
$a_3$ $1$ $0$ $16$ $0$ $1512$ $0$ $249000$ $0$ $52009720$ $0$ $12486132792$ $0$ $3292873266936$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $11$ $6$ $20$ $42$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $16$ $75$ $46$ $150$ $90$ $310$ $660$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $110$ $621$ $372$ $228$ $1296$ $778$ $2754$ $1640$ $5910$ $12810$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $960$ $5768$ $570$ $3408$ $2022$ $12376$ $7296$ $4328$ $26802$ $15760$ $58420$ $34230$ $127988$ $281736$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1512$ $9166$ $57953$ $5402$ $33802$ $19784$ $126614$ $11604$ $73680$ $43000$ $278174$ $161480$ $94040$ $613792$
$$ $355502$ $1359274$ $785484$ $3019716$ $6727908$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&3&0&2&0&3&0&0&4\\0&4&0&2&0&10&0&0&10&0&8&0&10&20&0\\1&0&8&0&7&0&16&15&0&7&0&28&0&0&44\\0&2&0&8&0&14&0&0&14&0&6&0&30&30&0\\2&0&7&0&13&0&15&22&0&15&0&37&0&0&56\\0&10&0&14&0&50&0&0&58&0&34&0&86&122&0\\1&0&16&0&15&0&50&43&0&25&0&84&0&0&156\\3&0&15&0&22&0&43&58&0&41&0&92&0&0&176\\0&10&0&14&0&58&0&0&88&0&40&0&114&172&0\\2&0&7&0&15&0&25&41&0&40&0&60&0&0&132\\0&8&0&6&0&34&0&0&40&0&32&0&54&92&0\\3&0&28&0&37&0&84&92&0&60&0&181&0&0&324\\0&10&0&30&0&86&0&0&114&0&54&0&204&252&0\\0&20&0&30&0&122&0&0&172&0&92&0&252&384&0\\4&0&44&0&56&0&156&176&0&132&0&324&0&0&680\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&8&8&13&50&50&58&88&40&32&181&204&384&680&312&299&664&504&160\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.