Properties

Label 1.6.N.72.49a
  
Name \(J(A(6,6))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $72$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_2^2\times C_3:S_3\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$72$
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:$C_2^2\times C_3:S_3$
Order:$72$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{18}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{18}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{8} & 0 & 0 & 0 \\0 & 0 & 0 &\zeta_{18}^{17} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{18}^{17} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0& 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(2,6))$${}^{\times 3}$, $J(A(3,6))$${}^{\times 6}$, $A(6,6)$, $J(A(6,2))$
Minimal supergroups:$J(B(6,6))$, $J(C(6,6))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $45$ $0$ $930$ $0$ $22365$ $0$ $586278$ $0$ $16250850$
$a_2$ $1$ $3$ $15$ $105$ $963$ $10233$ $117432$ $1409670$ $17449479$ $221132301$ $2857056750$ $37531830240$ $500300094966$
$a_3$ $1$ $0$ $16$ $0$ $2460$ $0$ $554620$ $0$ $147080780$ $0$ $43065258516$ $0$ $13554887266140$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $15$ $6$ $21$ $45$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $16$ $105$ $54$ $195$ $114$ $423$ $930$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $144$ $963$ $534$ $312$ $2025$ $1170$ $4485$ $2580$ $9990$ $22365$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $1500$ $10233$ $864$ $5778$ $3312$ $22599$ $12882$ $7368$ $50751$ $28860$ $114390$ $64890$ $258615$ $586278$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $2460$ $16740$ $117432$ $9540$ $66222$ $37548$ $264735$ $21336$ $149454$ $84552$ $600471$ $338340$ $191040$ $1365390$
$$ $767970$ $3111507$ $1747116$ $7104510$ $16250850$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&6&0&0&0&3&0&0&8\\0&3&0&3&0&12&0&0&15&0&9&0&21&33&0\\2&0&10&0&6&0&20&30&0&12&0&42&0&0&88\\0&3&0&7&0&20&0&0&27&0&13&0&45&61&0\\0&0&6&0&18&0&30&24&0&30&0&66&0&0&120\\0&12&0&20&0&76&0&0&108&0&56&0&168&248&0\\1&0&20&0&30&0&79&78&0&66&0&165&0&0&344\\6&0&30&0&24&0&78&120&0&66&0&180&0&0&408\\0&15&0&27&0&108&0&0&171&0&81&0&261&393&0\\0&0&12&0&30&0&66&66&0&74&0&150&0&0&328\\0&9&0&13&0&56&0&0&81&0&46&0&126&193&0\\3&0&42&0&66&0&165&180&0&150&0&375&0&0&792\\0&21&0&45&0&168&0&0&261&0&126&0&426&621&0\\0&33&0&61&0&248&0&0&393&0&193&0&621&943&0\\8&0&88&0&120&0&344&408&0&328&0&792&0&0&1800\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&10&7&18&76&79&120&171&74&46&375&426&943&1800&843&891&1984&1632&443\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$19/36$$0$$1/36$$0$$0$$1/2$
$a_1=0$$19/36$$19/36$$0$$1/36$$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$