Properties

Label 1.6.N.8.1a
  
Name \(J_n(A(1,4)_2)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $8$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_8\)

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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:$C_8$
Order:$8$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{7} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0\\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -i\\0 & 0 & 0 & 0 & 1 & 0 \\-i & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -i & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(1,4)_2$
Minimal supergroups:$J_n(A(2,4))$${}^{\times 2}$, $J_s(A(1,8)_1)$, $J_n(E(36))$, $J_n(A(1,12))$, $J_s(B(1,4)_2)$, $J_s(B(1,4;2)_2)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $63$ $0$ $1830$ $0$ $57435$ $0$ $1860138$ $0$ $61381782$
$a_2$ $1$ $2$ $14$ $155$ $1982$ $26537$ $363551$ $5052938$ $70953530$ $1004092517$ $14296364129$ $204561017168$ $2938952526059$
$a_3$ $1$ $0$ $22$ $0$ $5910$ $0$ $1932070$ $0$ $670248782$ $0$ $239998506552$ $0$ $87682437663018$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $14$ $7$ $28$ $63$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $22$ $155$ $85$ $347$ $190$ $793$ $1830$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $261$ $1982$ $1084$ $597$ $4563$ $2497$ $10580$ $5775$ $24615$ $57435$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $3414$ $26537$ $1866$ $14470$ $7899$ $61801$ $33670$ $18366$ $144366$ $78573$ $337926$ $183722$ $792281$ $1860138$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $5910$ $46075$ $363551$ $25116$ $197651$ $107535$ $851886$ $58534$ $462775$ $251536$ $1999531$ $1085353$ $589470$ $4699405$
$$ $2549036$ $11057053$ $5993526$ $26040756$ $61381782$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&3&5&0&4&0&8&0&0&20\\0&3&0&4&0&18&0&0&32&0&14&0&48&79&0\\1&0&11&0&13&0&42&51&0&43&0&99&0&0&250\\0&4&0&11&0&38&0&0&66&0&28&0&112&169&0\\1&0&13&0&21&0&57&73&0&66&0&142&0&0&360\\0&18&0&38&0&160&0&0&288&0&128&0&468&740&0\\3&0&42&0&57&0&185&222&0&192&0&448&0&0&1136\\5&0&51&0&73&0&222&285&0&249&0&557&0&0&1422\\0&32&0&66&0&288&0&0&527&0&236&0&854&1363&0\\4&0&43&0&66&0&192&249&0&228&0&494&0&0&1264\\0&14&0&28&0&128&0&0&236&0&108&0&382&612&0\\8&0&99&0&142&0&448&557&0&494&0&1130&0&0&2880\\0&48&0&112&0&468&0&0&854&0&382&0&1436&2244&0\\0&79&0&169&0&740&0&0&1363&0&612&0&2244&3573&0\\20&0&250&0&360&0&1136&1422&0&1264&0&2880&0&0&7392\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&11&11&21&160&185&285&527&228&108&1130&1436&3573&7392&3853&3946&9820&8730&2589\end{bmatrix}$

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