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Properties

Label 1.4.F.2.1b
  
Name \(C_2\)
Weight $1$
Degree $4$
Real dimension $1$
Components $2$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)_2\)
Component group \(C_2\)

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Invariants

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

Subgroups and supergroups

Maximal subgroups:$C_1$
Minimal supergroups:$C_4$, $C_{4,1}$, $D_2$, $J(C_2)$, $D_{2,1}$, $D_3$, $C_6$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $48$ $0$ $640$ $0$ $8960$ $0$ $129024$ $0$ $1892352$
$a_2$ $1$ $2$ $10$ $44$ $230$ $1212$ $6628$ $36696$ $205766$ $1162988$ $6616940$ $37841256$ $217331804$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $10$ $20$ $48$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $44$ $104$ $256$ $640$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $230$ $556$ $1392$ $3520$ $8960$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $1212$ $3032$ $7680$ $19584$ $50176$ $129024$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $6628$ $16776$ $42848$ $109952$ $283136$ $731136$ $1892352$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&2&0&4&0&5&0&4\\0&4&0&0&12&0&12&0&24&0\\1&0&7&8&0&10&0&27&0&24\\2&0&8&18&0&22&0&48&0&36\\0&12&0&0&40&0&44&0&84&0\\4&0&10&22&0&32&0&60&0&46\\0&12&0&0&44&0&52&0&96&0\\5&0&27&48&0&60&0&153&0&122\\0&24&0&0&84&0&96&0&184&0\\4&0&24&36&0&46&0&122&0&104\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&7&18&40&32&52&153&184&104\end{bmatrix}$

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