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Properties

Label 1.6.N.32.34a
  
Name \(J(A(4,4))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $32$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_4:D_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$32$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)_3$
$\mathbb{R}$-dimension:$1$
Description:$\left\{\begin{bmatrix}\alpha I_3&0,\\ 0&\bar \alpha I_3\end{bmatrix}: \alpha\bar\alpha=1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form:$\begin{bmatrix} 0 & I_3\\ -I_3 & 0\end{bmatrix}$
Hodge circle:$u\mapsto \mathrm{diag}(u,u, u, \bar u, \bar u, \bar u)$

Component group

Name:$C_4:D_4$
Order:$32$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & i & 0 & 0 & 0 & 0 \\0 & 0 & -i & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i\\\end{bmatrix}, \begin{bmatrix}\zeta_{12}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 &\zeta_{12}^{11} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0& 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(4,4)$, $J(A(2,4))$${}^{\times 6}$
Minimal supergroups:$J(C(4,4))$, $J(B(4,4))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $45$ $0$ $930$ $0$ $22575$ $0$ $612738$ $0$ $18052650$
$a_2$ $1$ $3$ $15$ $105$ $966$ $10398$ $122802$ $1541550$ $20202222$ $273038442$ $3772097040$ $52930084980$ $750982675719$
$a_3$ $1$ $0$ $16$ $0$ $2478$ $0$ $599200$ $0$ $181948718$ $0$ $61839853056$ $0$ $22160803260882$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $15$ $6$ $21$ $45$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $16$ $105$ $54$ $195$ $114$ $423$ $930$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $144$ $966$ $534$ $312$ $2031$ $1170$ $4503$ $2580$ $10050$ $22575$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1506$ $10398$ $864$ $5832$ $3324$ $23025$ $13026$ $7404$ $51921$ $29280$ $117690$ $66150$ $267981$ $612738$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2478$ $17094$ $122802$ $9666$ $68514$ $38466$ $278715$ $21660$ $155520$ $87036$ $637359$ $354660$ $197940$ $1463214$
$$ $812154$ $3370899$ $1866564$ $7789950$ $18052650$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&6&0&0&0&3&0&0&8\\0&3&0&3&0&12&0&0&15&0&9&0&21&33&0\\2&0&10&0&6&0&20&30&0&12&0&42&0&0&88\\0&3&0&7&0&20&0&0&27&0&13&0&45&61&0\\0&0&6&0&18&0&30&24&0&30&0&66&0&0&120\\0&12&0&20&0&76&0&0&108&0&56&0&168&254&0\\1&0&20&0&30&0&79&78&0&66&0&165&0&0&356\\6&0&30&0&24&0&78&123&0&69&0&183&0&0&432\\0&15&0&27&0&108&0&0&177&0&81&0&267&417&0\\0&0&12&0&30&0&66&69&0&83&0&153&0&0&364\\0&9&0&13&0&56&0&0&81&0&46&0&126&199&0\\3&0&42&0&66&0&165&183&0&153&0&384&0&0&840\\0&21&0&45&0&168&0&0&267&0&126&0&444&663&0\\0&33&0&61&0&254&0&0&417&0&199&0&663&1039&0\\8&0&88&0&120&0&356&432&0&364&0&840&0&0&2052\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&10&7&18&76&79&123&177&83&46&384&444&1039&2052&1047&1101&2596&2295&689\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/2$$0$$0$$0$$0$$1/2$
$a_1=0$$1/2$$1/2$$0$$0$$0$$0$$1/2$
$a_3=0$$1/2$$1/2$$0$$0$$0$$0$$1/2$
$a_1=a_3=0$$1/2$$1/2$$0$$0$$0$$0$$1/2$