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Properties

Label 1.6.N.14.1a
  
Name \(J(A(1,7))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $14$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(D_7\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$14$
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:$D_7$
Order:$14$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{21}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{21}^{16} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{21}^{4} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{21}^{20} & 0& 0 \\0 & 0 & 0 & 0 & \zeta_{21}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{21}^{17} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(1,1))$, $A(1,7)$
Minimal supergroups:$J(C(1,7))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $45$ $0$ $1110$ $0$ $33285$ $0$ $1066338$ $0$ $35100450$
$a_2$ $1$ $3$ $15$ $114$ $1227$ $15528$ $209202$ $2893449$ $40570779$ $573880632$ $8169857220$ $116894299899$ $1679411908758$
$a_3$ $1$ $0$ $16$ $0$ $3468$ $0$ $1106710$ $0$ $383080124$ $0$ $137144494116$ $0$ $50104327877274$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $15$ $6$ $21$ $45$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $16$ $114$ $57$ $219$ $123$ $489$ $1110$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $165$ $1227$ $657$ $366$ $2697$ $1488$ $6195$ $3405$ $14325$ $33285$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $2022$ $15528$ $1116$ $8436$ $4629$ $35709$ $19512$ $10680$ $83142$ $45348$ $194190$ $105756$ $454629$ $1066338$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $3468$ $26652$ $209202$ $14571$ $113697$ $61965$ $488550$ $33810$ $265656$ $144579$ $1145445$ $622194$ $338223$ $2690112$
$$ $1459920$ $6326355$ $3430602$ $14894460$ $35100450$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&6&0&0&0&3&0&0&11\\0&3&0&3&0&12&0&0&18&0&9&0&27&45&0\\2&0&10&0&6&0&23&36&0&21&0&54&0&0&142\\0&3&0&7&0&23&0&0&39&0&16&0&63&97&0\\0&0&6&0&18&0&36&33&0&42&0&87&0&0&204\\0&12&0&23&0&94&0&0&165&0&74&0&267&422&0\\1&0&23&0&36&0&109&123&0&117&0&258&0&0&650\\6&0&36&0&33&0&123&180&0&129&0&309&0&0&813\\0&18&0&39&0&165&0&0&303&0&132&0&492&777&0\\0&0&21&0&42&0&117&129&0&134&0&288&0&0&724\\0&9&0&16&0&74&0&0&132&0&64&0&219&349&0\\3&0&54&0&87&0&258&309&0&288&0&651&0&0&1644\\0&27&0&63&0&267&0&0&492&0&219&0&816&1284&0\\0&45&0&97&0&422&0&0&777&0&349&0&1284&2041&0\\11&0&142&0&204&0&650&813&0&724&0&1644&0&0&4227\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&10&7&18&94&109&180&303&134&64&651&816&2041&4227&2193&2295&5596&5001&1457\end{bmatrix}$

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$