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Properties

Label 1.4.F.8.3b
  
Name \(D_4\)
Weight $1$
Degree $4$
Real dimension $1$
Components $8$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)_2\)
Component group \(D_4\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$1$
Components:$8$
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:$D_4$
Order:$8$
Abelian:no
Generators:$\begin{bmatrix}\zeta_8&0&0&0\\0&\zeta_8^7&0&0\\0&0&\zeta_8^7&0\\0&0&0&\zeta_8\end{bmatrix}, \begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$C_4$, $D_2$${}^{\times 2}$
Minimal supergroups:$J(D_4)$, $O$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $18$ $0$ $200$ $0$ $2520$ $0$ $34272$ $0$ $487872$
$a_2$ $1$ $1$ $5$ $16$ $78$ $366$ $1898$ $10032$ $54694$ $302902$ $1700550$ $9636672$ $55009452$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $5$ $8$ $18$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $16$ $36$ $84$ $200$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $78$ $174$ $418$ $1020$ $2520$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $366$ $884$ $2172$ $5400$ $13552$ $34272$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1898$ $4656$ $11636$ $29336$ $74480$ $190176$ $487872$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&2&0&1&0&0\\0&2&0&0&4&0&4&0&6&0\\0&0&4&2&0&1&0&9&0&9\\1&0&2&7&0&7&0&12&0&7\\0&4&0&0&12&0&12&0&22&0\\2&0&1&7&0&13&0&13&0&7\\0&4&0&0&12&0&14&0&24&0\\1&0&9&12&0&13&0&42&0&35\\0&6&0&0&22&0&24&0&48&0\\0&0&9&7&0&7&0&35&0&35\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&7&12&13&14&42&48&35\end{bmatrix}$

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$$0$
$a_1=0$$5/8$$0$$0$$0$$0$$0$$0$