Properties

Label 1.4.D.2.1b
  
Name \(F_a\)
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 \(C_2\)

Learn more

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:$\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $21$ $0$ $210$ $0$ $2485$ $0$ $31878$ $0$ $427350$
$a_2$ $1$ $2$ $6$ $20$ $82$ $372$ $1824$ $9312$ $48850$ $260804$ $1410736$ $7708032$ $42460840$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $6$ $10$ $21$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $20$ $40$ $90$ $210$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $82$ $180$ $424$ $1020$ $2485$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $372$ $872$ $2112$ $5180$ $12810$ $31878$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1824$ $4416$ $10884$ $27040$ $67564$ $169596$ $427350$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&1&0&1&0&1&0&1\\0&3&0&0&4&0&4&0&5&0\\1&0&3&3&0&3&0&7&0&5\\1&0&3&7&0&5&0&13&0&7\\0&4&0&0&12&0&12&0&20&0\\1&0&3&5&0&9&0&15&0&11\\0&4&0&0&12&0&15&0&21&0\\1&0&7&13&0&15&0&37&0&25\\0&5&0&0&20&0&21&0&42&0\\1&0&5&7&0&11&0&25&0&21\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&3&7&12&9&15&37&42&21\end{bmatrix}$

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$$1/2$
$a_1=0$$0$$0$$0$$0$$0$$0$$0$