↑

Properties

Label 1.6.N.8.3d
  
Name \(J(A(1,4)_1)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $8$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(D_4\)

Downloads

Learn more

Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$8$
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:$D_4$
Order:$8$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0\\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{7} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(1,4)_1$, $J(A(1,2))$${}^{\times 2}$
Minimal supergroups:$J(A(2,4))$${}^{\times 2}$, $J_s(A(2,4))$, $J(A(3,4))$, $J_s(A(3,4))$, $J(A(1,8)_2)$${}^{\times 2}$, $J_s(A(1,8)_2)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $5$ $0$ $99$ $0$ $2450$ $0$ $68355$ $0$ $2056950$ $0$ $64990926$
$a_2$ $1$ $4$ $26$ $235$ $2586$ $31489$ $405911$ $5423464$ $74237002$ $1033441417$ $14560357821$ $206947415734$ $2960611792863$
$a_3$ $1$ $0$ $34$ $0$ $7222$ $0$ $2086690$ $0$ $689426766$ $0$ $242452083624$ $0$ $88002695271370$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $4$ $5$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $26$ $12$ $45$ $99$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $34$ $235$ $126$ $487$ $276$ $1087$ $2450$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $362$ $2586$ $1428$ $807$ $5739$ $3215$ $13042$ $7280$ $29790$ $68355$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $4272$ $31489$ $2388$ $17404$ $9683$ $71871$ $39790$ $22090$ $165432$ $91386$ $382148$ $210644$ $885423$ $2056950$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $7222$ $53476$ $405911$ $29602$ $223068$ $122933$ $939463$ $67872$ $515807$ $283682$ $2182723$ $1196356$ $656802$ $5083740$
$$ $2782094$ $11865693$ $6484296$ $27746838$ $64990926$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&3&0&1&0&4&10&0&3&0&10&0&0&24\\0&5&0&7&0&28&0&0&41&0&21&0&61&97&0\\3&0&19&0&18&0&55&73&0&46&0&123&0&0&284\\0&7&0&15&0&52&0&0&81&0&37&0&132&196&0\\1&0&18&0&35&0&78&83&0&80&0&182&0&0&400\\0&28&0&52&0&208&0&0&336&0&160&0&536&832&0\\4&0&55&0&78&0&225&256&0&218&0&518&0&0&1232\\10&0&73&0&83&0&256&345&0&260&0&619&0&0&1524\\0&41&0&81&0&336&0&0&585&0&264&0&929&1464&0\\3&0&46&0&80&0&218&260&0&252&0&532&0&0&1328\\0&21&0&37&0&160&0&0&264&0&132&0&426&672&0\\10&0&123&0&182&0&518&619&0&532&0&1262&0&0&3040\\0&61&0&132&0&536&0&0&929&0&426&0&1540&2384&0\\0&97&0&196&0&832&0&0&1464&0&672&0&2384&3763&0\\24&0&284&0&400&0&1232&1524&0&1328&0&3040&0&0&7664\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&19&15&35&208&225&345&585&252&132&1262&1540&3763&7664&3949&4080&9988&8860&2613\end{bmatrix}$

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$