Properties

Label 1.4.D.2.1a
  
Name \(F_{ab}\)
Weight $1$
Degree $4$
Real dimension $2$
Components $2$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)\times\mathrm{U}(1)\)
Component group \(C_2\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$2$
Components:$2$
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:$C_2$
Order:$2$
Abelian:yes
Generators:$\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$F$
Minimal supergroups:$F_{a,b}$, $F_{ac}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $18$ $0$ $200$ $0$ $2450$ $0$ $31752$ $0$ $426888$
$a_2$ $1$ $2$ $6$ $20$ $82$ $372$ $1824$ $9312$ $48850$ $260804$ $1410736$ $7708032$ $42460840$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $6$ $8$ $18$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $20$ $36$ $84$ $200$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $82$ $172$ $412$ $1000$ $2450$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $372$ $856$ $2088$ $5140$ $12740$ $31752$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1824$ $4384$ $10836$ $26960$ $67424$ $169344$ $426888$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&2&0&1&0&2\\0&2&0&0&4&0&4&0&6&0\\1&0&3&2&0&4&0&7&0&6\\0&0&2&8&0&4&0&14&0&6\\0&4&0&0&12&0&12&0&20&0\\2&0&4&4&0&10&0&14&0&12\\0&4&0&0&12&0&14&0&22&0\\1&0&7&14&0&14&0&37&0&24\\0&6&0&0&20&0&22&0&40&0\\2&0&6&6&0&12&0&24&0&22\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&8&12&10&14&37&40&22\end{bmatrix}$

Event probabilities

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