Properties

Label 1.6.J.12.5a
  
Name \(J_2(E_6)\)
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 \(C_2\times C_6\)

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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:$C_2\times C_6$
Order:$12$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{11} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $27$ $0$ $380$ $0$ $6895$ $0$ $146160$ $0$ $3425070$
$a_2$ $1$ $2$ $9$ $51$ $369$ $3158$ $30343$ $315122$ $3453969$ $39373182$ $462516147$ $5565186465$ $68304727147$
$a_3$ $1$ $0$ $12$ $0$ $852$ $0$ $127700$ $0$ $26130300$ $0$ $6248485404$ $0$ $1646686964604$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $5$ $14$ $27$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $12$ $51$ $31$ $93$ $57$ $184$ $380$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $71$ $369$ $222$ $138$ $735$ $447$ $1525$ $920$ $3220$ $6895$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $552$ $3158$ $333$ $1878$ $1127$ $6631$ $3944$ $2364$ $14187$ $8410$ $30660$ $18095$ $66738$ $146160$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $852$ $4931$ $30343$ $2933$ $17787$ $10484$ $65656$ $6202$ $38412$ $22560$ $143395$ $83640$ $48980$ $315038$
$$ $183239$ $695345$ $403326$ $1540644$ $3425070$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&2&0&1&0&1&0&0&2\\0&3&0&2&0&6&0&0&6&0&4&0&5&10&0\\1&0&6&0&4&0&9&9&0&4&0&14&0&0&22\\0&2&0&5&0&8&0&0&8&0&3&0&15&15&0\\1&0&4&0&8&0&9&11&0&8&0&20&0&0&28\\0&6&0&8&0&28&0&0&30&0&18&0&44&62&0\\1&0&9&0&9&0&27&22&0&13&0&43&0&0&78\\2&0&9&0&11&0&22&32&0&19&0&46&0&0&88\\0&6&0&8&0&30&0&0&45&0&20&0&57&86&0\\1&0&4&0&8&0&13&19&0&22&0&30&0&0&66\\0&4&0&3&0&18&0&0&20&0&18&0&29&46&0\\1&0&14&0&20&0&43&46&0&30&0&95&0&0&162\\0&5&0&15&0&44&0&0&57&0&29&0&104&126&0\\0&10&0&15&0&62&0&0&86&0&46&0&126&195&0\\2&0&22&0&28&0&78&88&0&66&0&162&0&0&340\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&5&8&28&27&32&45&22&18&95&104&195&340&159&156&334&259&83\end{bmatrix}$

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$$0$
$a_1=0$$1/12$$0$$0$$0$$0$$0$$0$
$a_3=0$$1/12$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$1/12$$0$$0$$0$$0$$0$$0$