Properties

Label 1.4.D.4.2a
  
Name \(F_{a,b}\)
Weight $1$
Degree $4$
Real dimension $2$
Components $4$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)\times\mathrm{U}(1)\)
Component group \(C_2^2\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$2$
Components:$4$
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:$u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$

Component group

Name:$C_2^2$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}, \begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$F_a$${}^{\times 2}$, $F_{ab}$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $12$ $0$ $110$ $0$ $1260$ $0$ $16002$ $0$ $213906$
$a_2$ $1$ $2$ $5$ $14$ $49$ $202$ $944$ $4720$ $24553$ $130658$ $705880$ $3855040$ $21232468$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $5$ $6$ $12$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $14$ $22$ $48$ $110$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $49$ $94$ $218$ $520$ $1260$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $202$ $444$ $1068$ $2610$ $6440$ $16002$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $944$ $2224$ $5466$ $13560$ $33852$ $84924$ $213906$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&1&0&0&0&1\\0&2&0&0&2&0&2&0&2&0\\1&0&2&1&0&2&0&3&0&3\\0&0&1&5&0&1&0&7&0&2\\0&2&0&0&6&0&6&0&10&0\\1&0&2&1&0&6&0&7&0&7\\0&2&0&0&6&0&8&0&10&0\\0&0&3&7&0&7&0&19&0&12\\0&2&0&0&10&0&10&0&22&0\\1&0&3&2&0&7&0&12&0&12\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&2&5&6&6&8&19&22&12\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-2$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$
$-$$1$$3/4$$0$$0$$0$$0$$3/4$
$a_1=0$$1/4$$1/4$$0$$0$$0$$0$$1/4$