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Properties

Label 1.4.F.16.11a
  
Name \(J(D_4)\)
Weight $1$
Degree $4$
Real dimension $1$
Components $16$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)_2\)
Component group \(C_2\times D_4\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$1$
Components:$16$
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:$C_2\times D_4$
Order:$16$
Abelian:no
Generators:$\begin{bmatrix}\zeta_8&0&0&0\\0&\zeta_8^7&0&0\\0&0&\zeta_8^7&0\\0&0&0&\zeta_8\end{bmatrix}, \begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(C_4)$, $D_4$, $D_{4,1}$${}^{\times 2}$, $D_{4,2}$, $J(D_2)$${}^{\times 2}$
Minimal supergroups:$J(O)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $9$ $0$ $100$ $0$ $1260$ $0$ $17136$ $0$ $243936$
$a_2$ $1$ $1$ $4$ $10$ $45$ $191$ $973$ $5048$ $27443$ $151579$ $850659$ $4818848$ $27506262$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $4$ $4$ $9$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $10$ $18$ $42$ $100$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $45$ $87$ $209$ $510$ $1260$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $191$ $442$ $1086$ $2700$ $6776$ $17136$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $973$ $2328$ $5818$ $14668$ $37240$ $95088$ $243936$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&2&0&0&0&0\\0&1&0&0&2&0&2&0&3&0\\0&0&3&0&0&0&0&5&0&7\\0&0&0&5&0&3&0&6&0&1\\0&2&0&0&6&0&6&0&11&0\\2&0&0&3&0&10&0&4&0&1\\0&2&0&0&6&0&7&0&12&0\\0&0&5&6&0&4&0&23&0&20\\0&3&0&0&11&0&12&0&24&0\\0&0&7&1&0&1&0&20&0&25\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&3&5&6&10&7&23&24&25\end{bmatrix}$

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$$1/16$$0$$1/8$$0$$5/16$
$a_1=0$$13/16$$1/2$$1/16$$0$$1/8$$0$$5/16$