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Properties

Label 1.6.N.16.7b
  
Name \(J(A(1,8)_1)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $16$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(D_8\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$16$
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:$D_8$
Order:$16$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{24}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{24}^{19} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{24}^{23} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{24}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(1,4)_2)$${}^{\times 2}$, $A(1,8)_1$
Minimal supergroups:$J(B(1,8)_1)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $45$ $0$ $1050$ $0$ $30135$ $0$ $945378$ $0$ $30859290$
$a_2$ $1$ $3$ $15$ $111$ $1146$ $14058$ $186102$ $2551356$ $35624286$ $502936914$ $7153621680$ $102313984446$ $1469683383411$
$a_3$ $1$ $0$ $16$ $0$ $3178$ $0$ $977530$ $0$ $335751374$ $0$ $120034532016$ $0$ $43843243583854$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $15$ $6$ $21$ $45$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $16$ $111$ $56$ $211$ $120$ $467$ $1050$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $158$ $1146$ $618$ $349$ $2493$ $1389$ $5684$ $3150$ $13050$ $30135$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1866$ $14058$ $1038$ $7676$ $4241$ $32129$ $17648$ $9722$ $74460$ $40798$ $173220$ $94696$ $404173$ $945378$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $3178$ $23976$ $186102$ $13176$ $101434$ $55487$ $433215$ $30402$ $236187$ $128940$ $1013201$ $551568$ $300628$ $2374698$
$$ $1291108$ $5575367$ $3027948$ $13108830$ $30859290$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&6&0&0&0&3&0&0&10\\0&3&0&3&0&12&0&0&17&0&9&0&25&41&0\\2&0&10&0&6&0&22&34&0&18&0&50&0&0&126\\0&3&0&7&0&22&0&0&35&0&15&0&58&86&0\\0&0&6&0&18&0&34&30&0&39&0&80&0&0&180\\0&12&0&22&0&88&0&0&148&0&68&0&238&374&0\\1&0&22&0&34&0&100&109&0&102&0&231&0&0&572\\6&0&34&0&30&0&109&164&0&114&0&273&0&0&714\\0&17&0&35&0&148&0&0&269&0&118&0&431&684&0\\0&0&18&0&39&0&102&114&0&124&0&253&0&0&636\\0&9&0&15&0&68&0&0&118&0&58&0&192&308&0\\3&0&50&0&80&0&231&273&0&253&0&576&0&0&1440\\0&25&0&58&0&238&0&0&431&0&192&0&722&1124&0\\0&41&0&86&0&374&0&0&684&0&308&0&1124&1789&0\\10&0&126&0&180&0&572&714&0&636&0&1440&0&0&3704\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&10&7&18&88&100&164&269&124&58&576&722&1789&3704&1933&2015&4914&4389&1297\end{bmatrix}$

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$