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Properties

Label 1.6.N.12.1b
  
Name \(J_n(A(1,6)_1)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $12$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3:C_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$12$
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:$C_3:C_4$
Order:$12$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{9}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{18}^{17} & 0 & 0& 0 & 0 \\0 & 0 & \zeta_{18}^{17} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{8} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{18}^{1} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{18}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 &1 & 0 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(1,6)_1$, $J_n(A(1,2))$
Minimal supergroups:$J_s(A(2,6))$, $J_n(A(3,6))$, $J_s(A(1,12))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $5$ $0$ $99$ $0$ $2450$ $0$ $67235$ $0$ $1956150$ $0$ $59151246$
$a_2$ $1$ $2$ $22$ $221$ $2530$ $30647$ $385787$ $4989490$ $65877626$ $884357783$ $12036457897$ $165738690680$ $2304954707127$
$a_3$ $1$ $0$ $34$ $0$ $7122$ $0$ $1934400$ $0$ $589153362$ $0$ $192537250344$ $0$ $66056057908676$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $5$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $22$ $12$ $45$ $99$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $34$ $221$ $126$ $487$ $276$ $1087$ $2450$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $362$ $2530$ $1424$ $806$ $5695$ $3202$ $12913$ $7240$ $29410$ $67235$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $4232$ $30647$ $2376$ $17112$ $9574$ $69951$ $38992$ $21780$ $160257$ $89180$ $368174$ $204540$ $847791$ $1956150$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $7122$ $51932$ $385787$ $28946$ $213972$ $118860$ $888303$ $66121$ $491988$ $272869$ $2050113$ $1133905$ $628018$ $4740447$
$$ $2618574$ $10979409$ $6057576$ $25467078$ $59151246$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&3&0&6&8&0&3&0&10&0&0&24\\0&5&0&7&0&28&0&0&41&0&21&0&61&97&0\\1&0&19&0&20&0&55&65&0&54&0&129&0&0&280\\0&7&0&15&0&52&0&0&81&0&37&0&131&193&0\\3&0&20&0&31&0&76&93&0&71&0&176&0&0&392\\0&28&0&52&0&208&0&0&332&0&160&0&528&808&0\\6&0&55&0&76&0&224&261&0&204&0&504&0&0&1176\\8&0&65&0&93&0&261&320&0&255&0&609&0&0&1432\\0&41&0&81&0&332&0&0&561&0&259&0&891&1375&0\\3&0&54&0&71&0&204&255&0&231&0&508&0&0&1208\\0&21&0&37&0&160&0&0&259&0&132&0&416&647&0\\10&0&129&0&176&0&504&609&0&508&0&1222&0&0&2840\\0&61&0&131&0&528&0&0&891&0&416&0&1462&2223&0\\0&97&0&193&0&808&0&0&1375&0&647&0&2223&3449&0\\24&0&280&0&392&0&1176&1432&0&1208&0&2840&0&0&6852\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&19&15&31&208&224&320&561&231&132&1222&1462&3449&6852&3373&3450&8304&7102&2009\end{bmatrix}$

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$$1/2$$0$$0$$0$$0$
$a_1=0$$1/2$$1/2$$1/2$$0$$0$$0$$0$
$a_3=0$$1/2$$1/2$$1/2$$0$$0$$0$$0$
$a_1=a_3=0$$1/2$$1/2$$1/2$$0$$0$$0$$0$