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Properties

Label 1.4.F.4.2b
  
Name \(J(C_2)\)
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:$\begin{bmatrix}i&0&0&0\\0&-i&0&0\\0&0&-i&0\\0&0&0&i\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$C_2$, $J(C_1)$, $C_{2,1}$
Minimal supergroups:$J(C_4)$, $J(D_2)$, $J(D_3)$, $J(C_6)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $24$ $0$ $320$ $0$ $4480$ $0$ $64512$ $0$ $946176$
$a_2$ $1$ $1$ $7$ $22$ $123$ $606$ $3346$ $18348$ $103011$ $581494$ $3308982$ $18920628$ $108667950$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $7$ $10$ $24$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $22$ $52$ $128$ $320$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $123$ $278$ $696$ $1760$ $4480$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $606$ $1516$ $3840$ $9792$ $25088$ $64512$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $3346$ $8388$ $21424$ $54976$ $141568$ $365568$ $946176$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&4&0&1&0&0\\0&2&0&0&6&0&6&0&12&0\\0&0&6&2&0&1&0&17&0&20\\1&0&2&11&0&13&0&22&0&12\\0&6&0&0&20&0&22&0&42&0\\4&0&1&13&0&26&0&22&0&9\\0&6&0&0&22&0&26&0&48&0\\1&0&17&22&0&22&0&83&0&73\\0&12&0&0&42&0&48&0&92&0\\0&0&20&12&0&9&0&73&0&78\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&11&20&26&26&83&92&78\end{bmatrix}$

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$$1/4$$0$$0$$0$$1/4$
$a_1=0$$3/4$$1/2$$1/4$$0$$0$$0$$1/4$