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Properties

Label 1.4.F.6.2a
  
Name \(J(C_3)\)
Weight $1$
Degree $4$
Real dimension $1$
Components $6$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)_2\)
Component group \(C_6\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$1$
Components:$6$
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:$C_6$
Order:$6$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_6&0&0&0\\0&\zeta_6^5&0&0\\0&0&\zeta_6^5&0\\0&0&0&\zeta_6\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(C_1)$, $C_3$
Minimal supergroups:$J(D_3)$, $J(C_6)$, $J(T)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $18$ $0$ $220$ $0$ $3010$ $0$ $43092$ $0$ $631092$
$a_2$ $1$ $1$ $5$ $16$ $85$ $416$ $2264$ $12342$ $68989$ $388624$ $2208760$ $12621962$ $72469060$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $5$ $8$ $18$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $16$ $38$ $90$ $220$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $85$ $194$ $476$ $1190$ $3010$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $416$ $1034$ $2592$ $6572$ $16786$ $43092$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $2264$ $5656$ $14370$ $36770$ $94542$ $243936$ $631092$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&2&0&1&0&0\\0&2&0&0&4&0&4&0&8&0\\0&0&4&2&0&1&0&11&0&12\\1&0&2&7&0&9&0&14&0&10\\0&4&0&0&14&0&14&0&28&0\\2&0&1&9&0&16&0&16&0&7\\0&4&0&0&14&0&18&0&32&0\\1&0&11&14&0&16&0&55&0&47\\0&8&0&0&28&0&32&0&62&0\\0&0&12&10&0&7&0&47&0&50\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&7&14&16&18&55&62&50\end{bmatrix}$

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$$1/6$$0$$0$$1/3$$0$
$a_1=0$$1/2$$1/2$$1/6$$0$$0$$1/3$$0$