Properties

Label 1.4.F.8.5a
  
Name \(J(D_2)\)
Weight $1$
Degree $4$
Real dimension $1$
Components $8$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)_2\)
Component group \(C_2^3\)

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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:$C_2^3$
Order:$8$
Abelian:yes
Generators:$\begin{bmatrix}i&0&0&0\\0&-i&0&0\\0&0&-i&0\\0&0&0&i\end{bmatrix}, \begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$D_2$, $J(C_2)$${}^{\times 3}$, $D_{2,1}$${}^{\times 3}$
Minimal supergroups:$J(D_4)$, $J(D_6)$, $J(T)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $12$ $0$ $160$ $0$ $2240$ $0$ $32256$ $0$ $473088$
$a_2$ $1$ $1$ $5$ $13$ $67$ $311$ $1694$ $9206$ $51587$ $290875$ $1654810$ $9460826$ $54335230$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $5$ $5$ $12$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $13$ $26$ $64$ $160$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $67$ $139$ $348$ $880$ $2240$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $311$ $758$ $1920$ $4896$ $12544$ $32256$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1694$ $4194$ $10712$ $27488$ $70784$ $182784$ $473088$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&3&0&0&0&0\\0&1&0&0&3&0&3&0&6&0\\0&0&4&0&0&0&0&9&0&12\\0&0&0&7&0&6&0&11&0&4\\0&3&0&0&10&0&11&0&21&0\\3&0&0&6&0&16&0&9&0&3\\0&3&0&0&11&0&13&0&24&0\\0&0&9&11&0&9&0&43&0&38\\0&6&0&0&21&0&24&0&46&0\\0&0&12&4&0&3&0&38&0&44\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&4&7&10&16&13&43&46&44\end{bmatrix}$

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$$1/8$$0$$0$$0$$3/8$
$a_1=0$$7/8$$1/2$$1/8$$0$$0$$0$$3/8$