Properties

Label 1.4.F.8.3a
  
Name \(D_{4,1}\)
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:$\begin{bmatrix}0&0&0&\zeta_8\\0&0&-\zeta_8^7&0\\0&-\zeta_8^7&0&0\\\zeta_8&0&0&0\end{bmatrix}, \begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$C_{4,1}$, $D_2$, $D_{2,1}$
Minimal supergroups:$J(D_4)$, $O_1$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $12$ $0$ $160$ $0$ $2240$ $0$ $32256$ $0$ $473088$
$a_2$ $1$ $1$ $4$ $13$ $63$ $311$ $1678$ $9206$ $51523$ $290875$ $1654554$ $9460826$ $54334206$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $4$ $5$ $12$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $13$ $26$ $64$ $160$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $63$ $139$ $348$ $880$ $2240$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $311$ $758$ $1920$ $4896$ $12544$ $32256$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1678$ $4194$ $10712$ $27488$ $70784$ $182784$ $473088$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&2&0&1&0&1\\0&1&0&0&3&0&3&0&6&0\\0&0&3&1&0&2&0&7&0&8\\0&0&1&6&0&5&0&12&0&7\\0&3&0&0&10&0&11&0&21&0\\2&0&2&5&0&11&0&13&0&10\\0&3&0&0&11&0&13&0&24&0\\1&0&7&12&0&13&0&40&0&32\\0&6&0&0&21&0&24&0&46&0\\1&0&8&7&0&10&0&32&0&31\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&3&6&10&11&13&40&46&31\end{bmatrix}$

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$$0$$0$$1/4$$0$$1/4$
$a_1=0$$7/8$$1/2$$0$$0$$1/4$$0$$1/4$