Properties

Label 1.6.E.2.1a
  
Name \(E_{t}\)
Weight $1$
Degree $6$
Real dimension $9$
Components $2$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{SU}(2)^3\)
Component group \(C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$9$
Components:$2$
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:$C_2$
Order:$2$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $13$ $0$ $145$ $0$ $2128$ $0$ $36624$ $0$ $701580$
$a_2$ $1$ $2$ $7$ $32$ $186$ $1287$ $10080$ $86249$ $788128$ $7578653$ $75926327$ $786836998$ $8389604459$
$a_3$ $1$ $0$ $7$ $0$ $366$ $0$ $36795$ $0$ $5116398$ $0$ $869147958$ $0$ $169257993960$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $7$ $3$ $8$ $13$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $7$ $32$ $17$ $46$ $27$ $79$ $145$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $38$ $186$ $106$ $64$ $311$ $187$ $572$ $345$ $1090$ $2128$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $252$ $1287$ $150$ $757$ $454$ $2364$ $1418$ $856$ $4561$ $2740$ $9002$ $5397$ $18046$ $36624$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $366$ $1877$ $10080$ $1124$ $5979$ $3576$ $19605$ $2145$ $11696$ $6997$ $39156$ $23336$ $13942$ $79384$
$$ $47215$ $162743$ $96558$ $336630$ $701580$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&0&0&1&5&0\\1&0&4&0&1&0&4&6&0&1&0&3&0&0&8\\0&1&0&3&0&4&0&0&4&0&0&0&4&6&0\\0&0&1&0&4&0&4&3&0&4&0&6&0&0&9\\0&3&0&4&0&11&0&0&12&0&4&0&12&19&0\\0&0&4&0&4&0&11&8&0&5&0&12&0&0&22\\2&0&6&0&3&0&8&15&0&5&0&10&0&0&24\\0&3&0&4&0&12&0&0&16&0&5&0&14&24&0\\0&0&1&0&4&0&5&5&0&7&0&9&0&0&16\\0&0&0&0&0&4&0&0&5&0&6&0&8&8&0\\0&0&3&0&6&0&12&10&0&9&0&23&0&0&36\\0&1&0&4&0&12&0&0&14&0&8&0&25&26&0\\0&5&0&6&0&19&0&0&24&0&8&0&26&45&0\\1&0&8&0&9&0&22&24&0&16&0&36&0&0&70\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&4&11&11&15&16&7&6&23&25&45&70&31&40&62&41&13\end{bmatrix}$

Event probabilities

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