Properties

Label 1.4.F.4.1b
  
Name \(C_4\)
Weight $1$
Degree $4$
Real dimension $1$
Components $4$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)_2\)
Component group \(C_4\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$1$
Components:$4$
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:$C_4$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_8&0&0&0\\0&\zeta_8^7&0&0\\0&0&\zeta_8^7&0\\0&0&0&\zeta_8\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$C_2$
Minimal supergroups:$J(C_4)$, $D_4$, $D_{4,2}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $36$ $0$ $400$ $0$ $5040$ $0$ $68544$ $0$ $975744$
$a_2$ $1$ $2$ $8$ $32$ $150$ $732$ $3776$ $20064$ $109318$ $605804$ $3400848$ $19273344$ $110017980$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $8$ $16$ $36$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $32$ $72$ $168$ $400$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $150$ $348$ $836$ $2040$ $5040$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $732$ $1768$ $4344$ $10800$ $27104$ $68544$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $3776$ $9312$ $23272$ $58672$ $148960$ $380352$ $975744$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&2&0&2&0&3&0&2\\0&4&0&0&8&0&8&0&12&0\\1&0&5&6&0&6&0&15&0&12\\2&0&6&12&0&12&0&26&0&18\\0&8&0&0&24&0&24&0&44&0\\2&0&6&12&0&18&0&32&0&24\\0&8&0&0&24&0&28&0&48&0\\3&0&15&26&0&32&0&79&0&62\\0&12&0&0&44&0&48&0&96&0\\2&0&12&18&0&24&0&62&0&54\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&5&12&24&18&28&79&96&54\end{bmatrix}$

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$$0$
$a_1=0$$1/4$$0$$0$$0$$0$$0$$0$