Properties

Label 1.4.F.3.1a
  
Name \(C_3\)
Weight $1$
Degree $4$
Real dimension $1$
Components $3$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)_2\)
Component group \(C_3\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$1$
Components:$3$
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:$C_3$
Order:$3$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_6&0&0&0\\0&\zeta_6^5&0&0\\0&0&\zeta_6^5&0\\0&0&0&\zeta_6\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$C_1$
Minimal supergroups:$D_3$, $D_{3,2}$, $C_6$, $J(C_3)$, $C_{6,1}$, $T$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $36$ $0$ $440$ $0$ $6020$ $0$ $86184$ $0$ $1262184$
$a_2$ $1$ $2$ $8$ $34$ $164$ $842$ $4506$ $24726$ $137892$ $777418$ $4417178$ $25244606$ $144936754$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $8$ $16$ $36$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $34$ $76$ $180$ $440$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $164$ $388$ $952$ $2380$ $6020$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $842$ $2068$ $5184$ $13144$ $33572$ $86184$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $4506$ $11312$ $28740$ $73540$ $189084$ $487872$ $1262184$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&2&0&2&0&3&0&4\\0&4&0&0&8&0&8&0&16&0\\1&0&5&6&0&8&0&17&0&14\\2&0&6&12&0&14&0&32&0&26\\0&8&0&0&28&0&28&0&56&0\\2&0&8&14&0&20&0&42&0&34\\0&8&0&0&28&0&36&0&64&0\\3&0&17&32&0&42&0&101&0&78\\0&16&0&0&56&0&64&0&124&0\\4&0&14&26&0&34&0&78&0&66\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&5&12&28&20&36&101&124&66\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2$ and $n\in\mathbb{Z}$.