Properties

Label 1.6.E.3.1a
  
Name \(E_{s}\)
Weight $1$
Degree $6$
Real dimension $9$
Components $3$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{SU}(2)^3\)
Component group \(C_3\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$9$
Components:$3$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{SU}(2)^3$
$\mathbb{R}$-dimension:$9$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A,B,C\in\mathrm{SU}(2)\right\}$ Symplectic form:$\begin{bmatrix}J_2&0&0\\0&J_2&0\\0&0&J_2\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
Hodge circle:$u\mapsto\mathrm{diag}(u,\bar u, u, \bar u, u, \bar u)$

Component group

Name:$C_3$
Order:$3$
Abelian:yes
Generators:$\begin{bmatrix}0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 &0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$E$
Minimal supergroups:$E_{s,t}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $8$ $0$ $95$ $0$ $1414$ $0$ $24402$ $0$ $467676$
$a_2$ $1$ $1$ $4$ $20$ $121$ $851$ $6703$ $57457$ $525311$ $5052157$ $50616822$ $524556066$ $5593064469$
$a_3$ $1$ $0$ $5$ $0$ $244$ $0$ $24525$ $0$ $3410876$ $0$ $579431412$ $0$ $112838656920$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $4$ $2$ $5$ $8$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $5$ $20$ $11$ $30$ $18$ $52$ $95$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $25$ $121$ $70$ $42$ $206$ $124$ $380$ $230$ $725$ $1414$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $167$ $851$ $100$ $503$ $302$ $1573$ $944$ $569$ $3038$ $1825$ $5998$ $3598$ $12026$ $24402$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $244$ $1249$ $6703$ $748$ $3982$ $2382$ $13063$ $1430$ $7794$ $4663$ $26098$ $15554$ $9290$ $52916$
$$ $31472$ $108486$ $64372$ $224406$ $467676$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&1&1&0&0&0&0&0&0&0\\0&1&0&1&0&2&0&0&2&0&0&0&1&3&0\\0&0&3&0&1&0&2&4&0&1&0&2&0&0&6\\0&1&0&2&0&2&0&0&3&0&1&0&2&5&0\\0&0&1&0&2&0&3&2&0&2&0&4&0&0&6\\0&2&0&2&0&8&0&0&8&0&2&0&8&12&0\\1&0&2&0&3&0&7&6&0&4&0&8&0&0&14\\1&0&4&0&2&0&6&9&0&3&0&7&0&0&16\\0&2&0&3&0&8&0&0&10&0&3&0&10&16&0\\0&0&1&0&2&0&4&3&0&4&0&6&0&0&10\\0&0&0&1&0&2&0&0&3&0&4&0&6&6&0\\0&0&2&0&4&0&8&7&0&6&0&15&0&0&24\\0&1&0&2&0&8&0&0&10&0&6&0&15&18&0\\0&3&0&5&0&12&0&0&16&0&6&0&18&30&0\\0&0&6&0&6&0&14&16&0&10&0&24&0&0&48\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&3&2&2&8&7&9&10&4&4&15&15&30&48&19&27&40&26&8\end{bmatrix}$

Event probabilities

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