Properties

Label 1.6.F.1.1a
  
Name \(F\)
Weight $1$
Degree $6$
Real dimension $7$
Components $1$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)^2\)
Component group \(C_1\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$7$
Components:$1$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)\times\mathrm{SU}(2)^2$
$\mathbb{R}$-dimension:$7$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B,C\in\mathrm{SU}(2)\right\}$ 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:$F_{at}$, $F_{a}$, $F_{t}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $40$ $0$ $570$ $0$ $9898$ $0$ $195216$ $0$ $4209084$
$a_2$ $1$ $3$ $14$ $84$ $607$ $4993$ $45009$ $433807$ $4399453$ $46442487$ $506453976$ $5673429798$ $65011948659$
$a_3$ $1$ $0$ $18$ $0$ $1384$ $0$ $181460$ $0$ $31197880$ $0$ $6324402672$ $0$ $1432380307776$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $14$ $8$ $22$ $40$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $18$ $84$ $50$ $150$ $92$ $288$ $570$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $118$ $607$ $364$ $222$ $1172$ $710$ $2348$ $1420$ $4790$ $9898$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $906$ $4993$ $548$ $2982$ $1794$ $10112$ $6048$ $3632$ $20884$ $12460$ $43634$ $25956$ $91966$ $195216$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1384$ $7726$ $45009$ $4620$ $26656$ $15850$ $93972$ $9456$ $55574$ $32960$ $198426$ $117040$ $69218$ $422262$
$$ $248402$ $904218$ $530544$ $1946448$ $4209084$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&1&0&2&3&0&1&0&2&0&0&4\\0&4&0&4&0&10&0&0&10&0&4&0&10&16&0\\2&0&9&0&7&0&15&17&0&8&0&20&0&0&36\\0&4&0&6&0&14&0&0&16&0&6&0&18&26&0\\1&0&7&0&10&0&17&16&0&11&0&28&0&0&44\\0&10&0&14&0&44&0&0&50&0&24&0&64&90&0\\2&0&15&0&17&0&39&39&0&26&0&62&0&0&112\\3&0&17&0&16&0&39&47&0&26&0&65&0&0&124\\0&10&0&16&0&50&0&0&66&0&28&0&84&120&0\\1&0&8&0&11&0&26&26&0&22&0&45&0&0&88\\0&4&0&6&0&24&0&0&28&0&20&0&44&56&0\\2&0&20&0&28&0&62&65&0&45&0&121&0&0&212\\0&10&0&18&0&64&0&0&84&0&44&0&124&166&0\\0&16&0&26&0&90&0&0&120&0&56&0&166&244&0\\4&0&36&0&44&0&112&124&0&88&0&212&0&0&420\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&6&10&44&39&47&66&22&20&121&124&244&420&174&179&358&258&62\end{bmatrix}$

Event probabilities

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