Properties

Label 1.4.F.4.2a
  
Name \(D_2\)
Weight $1$
Degree $4$
Real dimension $1$
Components $4$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)_2\)
Component group \(C_2^2\)

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Invariants

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

Identity component

Name:$\mathrm{U}(1)_2$
$\mathbb{R}$-dimension:$1$
Description:$\left\{\begin{bmatrix}\alpha I_2&0\\0&\bar\alpha I_2\end{bmatrix}: \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form:$\begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}$
Hodge circle:$u\mapsto\mathrm{diag}(u, u,\bar u,\bar u)$

Component group

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

Subgroups and supergroups

Maximal subgroups:$C_2$${}^{\times 3}$
Minimal supergroups:$D_4$, $D_{4,1}$, $J(D_2)$, $D_6$, $T$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $24$ $0$ $320$ $0$ $4480$ $0$ $64512$ $0$ $946176$
$a_2$ $1$ $1$ $6$ $22$ $118$ $606$ $3324$ $18348$ $102918$ $581494$ $3308596$ $18920628$ $108666364$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $6$ $10$ $24$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $22$ $52$ $128$ $320$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $118$ $278$ $696$ $1760$ $4480$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $606$ $1516$ $3840$ $9792$ $25088$ $64512$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $3324$ $8388$ $21424$ $54976$ $141568$ $365568$ $946176$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&3&0&2&0&1\\0&2&0&0&6&0&6&0&12&0\\0&0&5&3&0&3&0&15&0&15\\1&0&3&10&0&12&0&23&0&16\\0&6&0&0&20&0&22&0&42&0\\3&0&3&12&0&20&0&27&0&18\\0&6&0&0&22&0&26&0&48&0\\2&0&15&23&0&27&0&79&0&65\\0&12&0&0&42&0&48&0&92&0\\1&0&15&16&0&18&0&65&0&60\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&10&20&20&26&79&92&60\end{bmatrix}$

Event probabilities

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