Properties

Label 1.4.F.24.12a
  
Name \(O_1\)
Weight $1$
Degree $4$
Real dimension $1$
Components $24$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)_2\)
Component group \(S_4\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$1$
Components:$24$
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:$S_4$
Order:$24$
Abelian:no
Generators:$\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}, \begin{bmatrix}\frac{i+1}{2}&\frac{i+1}{2}&0&0\\\frac{i-1}{2}&\frac{-i+1}{2}&0&0\\0&0&\frac{-i+1}{2}&\frac{-i+1}{2}\\0&0&\frac{-i-1}{2}&\frac{i+1}{2}\end{bmatrix}, \begin{bmatrix}0&0&0&\zeta_8\\0&0&-\zeta_8^7&0\\0&-\zeta_8^7&0&0\\\zeta_8&0&0&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$D_{3,2}$, $D_{4,1}$, $T$
Minimal supergroups:$J(O)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $6$ $0$ $60$ $0$ $770$ $0$ $10836$ $0$ $158004$
$a_2$ $1$ $1$ $3$ $8$ $30$ $126$ $617$ $3221$ $17586$ $98090$ $554673$ $3162501$ $18136681$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $3$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $8$ $12$ $26$ $60$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $30$ $55$ $128$ $310$ $770$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $126$ $276$ $672$ $1676$ $4242$ $10836$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $617$ $1462$ $3658$ $9282$ $23758$ $61152$ $158004$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&1&0&0&0&1\\0&1&0&0&1&0&1&0&2&0\\0&0&2&0&0&1&0&2&0&3\\0&0&0&3&0&1&0&4&0&2\\0&1&0&0&4&0&3&0&7&0\\1&0&1&1&0&5&0&4&0&4\\0&1&0&0&3&0&5&0&8&0\\0&0&2&4&0&4&0&14&0&10\\0&2&0&0&7&0&8&0&16&0\\1&0&3&2&0&4&0&10&0&12\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&3&4&5&5&14&16&12\end{bmatrix}$

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$$1/4$
$a_1=0$$5/8$$1/2$$0$$0$$1/4$$0$$1/4$