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Properties

Label 1.6.N.2.1b
  
Name \(J(A(1,1))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $2$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$2$
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:$C_2$
Order:$2$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 &0 & 0 \\0 & 0 & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(1,1)$
Minimal supergroups:$J(A(1,3))$, $J(A(1,2))$${}^{\times 2}$, $J(A(3,1))$, $J(A(1,7))$, $J_s(A(3,1))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $9$ $0$ $243$ $0$ $7290$ $0$ $229635$ $0$ $7440174$ $0$ $245525742$
$a_2$ $1$ $6$ $54$ $621$ $7938$ $106191$ $1454355$ $20212254$ $283815738$ $4016375199$ $57185472609$ $818244118764$ $11755810259271$
$a_3$ $1$ $0$ $82$ $0$ $23574$ $0$ $7727410$ $0$ $2680982990$ $0$ $959993851752$ $0$ $350729748096618$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $6$ $9$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $54$ $27$ $108$ $243$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $82$ $621$ $333$ $1377$ $756$ $3159$ $7290$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $1035$ $7938$ $4320$ $2367$ $18225$ $9963$ $42282$ $23085$ $98415$ $229635$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $13626$ $106191$ $7452$ $57834$ $31563$ $247131$ $134622$ $73386$ $577368$ $314199$ $1351566$ $734832$ $3168963$ $7440174$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $23574$ $184221$ $1454355$ $100386$ $790479$ $430029$ $3407346$ $234090$ $1850931$ $1006020$ $7997859$ $4341195$ $2357586$ $18797265$
$$ $10195794$ $44227701$ $23973894$ $104162436$ $245525742$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&5&0&3&0&13&21&0&12&0&30&0&0&80\\0&9&0&18&0&72&0&0&126&0&54&0&198&315&0\\5&0&43&0&51&0&164&213&0&171&0&393&0&0&1000\\0&18&0&37&0&152&0&0&270&0&118&0&432&685&0\\3&0&51&0&81&0&237&279&0&258&0&576&0&0&1440\\0&72&0&152&0&640&0&0&1152&0&512&0&1872&2960&0\\13&0&164&0&237&0&727&894&0&792&0&1794&0&0&4544\\21&0&213&0&279&0&894&1143&0&969&0&2223&0&0&5688\\0&126&0&270&0&1152&0&0&2097&0&936&0&3438&5445&0\\12&0&171&0&258&0&792&969&0&878&0&1986&0&0&5056\\0&54&0&118&0&512&0&0&936&0&424&0&1548&2440&0\\30&0&393&0&576&0&1794&2223&0&1986&0&4518&0&0&11520\\0&198&0&432&0&1872&0&0&3438&0&1548&0&5688&9000&0\\0&315&0&685&0&2960&0&0&5445&0&2440&0&9000&14275&0\\80&0&1000&0&1440&0&4544&5688&0&5056&0&11520&0&0&29568\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&9&43&37&81&640&727&1143&2097&878&424&4518&5688&14275&29568&15327&15798&39160&34866&10175\end{bmatrix}$

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$