Properties

Label 1.4.D.1.1a
  
Name \(F\)
Weight $1$
Degree $4$
Real dimension $2$
Components $1$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)\times\mathrm{U}(1)\)
Component group \(C_1\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$2$
Components:$1$
Contained in:$\mathrm{USp}(4)$
Rational:yes

Identity component

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

Subgroups and supergroups

Maximal subgroups:
Minimal supergroups:$F_a$, $F_{ab}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $36$ $0$ $400$ $0$ $4900$ $0$ $63504$ $0$ $853776$
$a_2$ $1$ $2$ $8$ $32$ $148$ $712$ $3584$ $18496$ $97444$ $521096$ $2820448$ $15414016$ $84917584$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $8$ $16$ $36$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $32$ $72$ $168$ $400$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $148$ $344$ $824$ $2000$ $4900$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $712$ $1712$ $4176$ $10280$ $25480$ $63504$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $3584$ $8768$ $21672$ $53920$ $134848$ $338688$ $853776$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&2&0&2&0&3&0&2\\0&4&0&0&8&0&8&0&12&0\\1&0&5&6&0&6&0&15&0&10\\2&0&6&12&0&12&0&26&0&16\\0&8&0&0&24&0&24&0&40&0\\2&0&6&12&0&16&0&30&0&20\\0&8&0&0&24&0&28&0&44&0\\3&0&15&26&0&30&0&73&0&50\\0&12&0&0&40&0&44&0&80&0\\2&0&10&16&0&20&0&50&0&40\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&5&12&24&16&28&73&80&40\end{bmatrix}$

Event probabilities

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