Properties

Label 1.4.F.1.1a
  
Name \(C_1\)
Weight $1$
Degree $4$
Real dimension $1$
Components $1$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)_2\)
Component group \(C_1\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$1$
Components:$1$
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:$u\mapsto\mathrm{diag}(u, u,\bar u,\bar u)$

Subgroups and supergroups

Maximal subgroups:
Minimal supergroups:$C_2$, $J(C_1)$, $C_{2,1}$, $C_3$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $8$ $0$ $96$ $0$ $1280$ $0$ $17920$ $0$ $258048$ $0$ $3784704$
$a_2$ $1$ $4$ $18$ $88$ $454$ $2424$ $13236$ $73392$ $411462$ $2325976$ $13233628$ $75682512$ $434662684$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $4$ $8$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $18$ $40$ $96$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $88$ $208$ $512$ $1280$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $454$ $1112$ $2784$ $7040$ $17920$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $2424$ $6064$ $15360$ $39168$ $100352$ $258048$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $13236$ $33552$ $85696$ $219904$ $566272$ $1462272$ $3784704$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&3&4&0&6&0&11&0&10\\0&8&0&0&24&0&24&0&48&0\\3&0&11&18&0&24&0&51&0&42\\4&0&18&34&0&42&0&98&0&76\\0&24&0&0&80&0&88&0&168&0\\6&0&24&42&0&56&0&126&0&102\\0&24&0&0&88&0&104&0&192&0\\11&0&51&98&0&126&0&301&0&236\\0&48&0&0&168&0&192&0&368&0\\10&0&42&76&0&102&0&236&0&192\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&8&11&34&80&56&104&301&368&192\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2$ and $n\in\mathbb{Z}$.