Properties

Label 1.6.E.1.1a
  
Name \(E\)
Weight $1$
Degree $6$
Real dimension $9$
Components $1$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{SU}(2)^3\)
Component group \(C_1\)

Learn more

Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$9$
Components:$1$
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:$u\mapsto\mathrm{diag}(u,\bar u, u, \bar u, u, \bar u)$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $24$ $0$ $285$ $0$ $4242$ $0$ $73206$ $0$ $1403028$
$a_2$ $1$ $3$ $12$ $60$ $363$ $2553$ $20109$ $172371$ $1575933$ $15156471$ $151850466$ $1573668198$ $16779193407$
$a_3$ $1$ $0$ $13$ $0$ $728$ $0$ $73565$ $0$ $10232600$ $0$ $1738294152$ $0$ $338515970496$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $12$ $6$ $15$ $24$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $13$ $60$ $33$ $90$ $54$ $156$ $285$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $75$ $363$ $210$ $126$ $618$ $372$ $1140$ $690$ $2175$ $4242$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $501$ $2553$ $300$ $1509$ $906$ $4719$ $2832$ $1707$ $9114$ $5475$ $17994$ $10794$ $36078$ $73206$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $728$ $3747$ $20109$ $2244$ $11946$ $7146$ $39189$ $4290$ $23382$ $13989$ $78294$ $46662$ $27870$ $158748$
$$ $94416$ $325458$ $193116$ $673218$ $1403028$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&3&0&0&0&0&0&0&2\\0&3&0&3&0&6&0&0&6&0&0&0&3&9&0\\2&0&7&0&3&0&8&12&0&3&0&6&0&0&16\\0&3&0&4&0&8&0&0&9&0&1&0&6&13&0\\0&0&3&0&6&0&9&6&0&6&0&12&0&0&18\\0&6&0&8&0&22&0&0&24&0&8&0&24&38&0\\1&0&8&0&9&0&19&18&0&12&0&24&0&0&44\\3&0&12&0&6&0&18&27&0&9&0&21&0&0&48\\0&6&0&9&0&24&0&0&30&0&9&0&30&48&0\\0&0&3&0&6&0&12&9&0&10&0&18&0&0&32\\0&0&0&1&0&8&0&0&9&0&10&0&18&16&0\\0&0&6&0&12&0&24&21&0&18&0&45&0&0&72\\0&3&0&6&0&24&0&0&30&0&18&0&45&54&0\\0&9&0&13&0&38&0&0&48&0&16&0&54&88&0\\2&0&16&0&18&0&44&48&0&32&0&72&0&0&140\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&4&6&22&19&27&30&10&10&45&45&88&140&57&77&118&78&20\end{bmatrix}$

Event probabilities

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