Properties

Label 1.6.N.8.5a
  
Name \(J(A(2,2))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $8$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_2^3\)

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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:$C_2^3$
Order:$8$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0& 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(2,2)$, $J(A(1,2))$${}^{\times 6}$
Minimal supergroups:$J(B(3,2))$, $J(A(2,4))$${}^{\times 2}$, $J(A(2,6))$, $J(B(1,4)_2)$${}^{\times 2}$, $J(A(6,2))$, $J(C(2,2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $63$ $0$ $1830$ $0$ $57435$ $0$ $1860138$ $0$ $61381782$
$a_2$ $1$ $3$ $18$ $168$ $2022$ $26658$ $363915$ $5054031$ $70956810$ $1004102358$ $14296393653$ $204561105741$ $2938952791779$
$a_3$ $1$ $0$ $22$ $0$ $5910$ $0$ $1932070$ $0$ $670248782$ $0$ $239998506552$ $0$ $87682437663018$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $18$ $6$ $27$ $63$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $22$ $168$ $84$ $345$ $186$ $789$ $1830$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $258$ $2022$ $1080$ $597$ $4557$ $2493$ $10572$ $5760$ $24600$ $57435$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $3408$ $26658$ $1854$ $14460$ $7887$ $61785$ $33654$ $18366$ $144342$ $78558$ $337896$ $183666$ $792225$ $1860138$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $5910$ $46056$ $363915$ $25104$ $197622$ $107511$ $851841$ $58488$ $462735$ $251490$ $1999467$ $1085292$ $589470$ $4699314$
$$ $2548980$ $11056941$ $5993316$ $26040546$ $61381782$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&9&0&3&0&6&0&0&20\\0&3&0&3&0&18&0&0&33&0&15&0&45&81&0\\2&0&13&0&9&0&41&57&0&36&0&96&0&0&250\\0&3&0&13&0&38&0&0&63&0&25&0&120&166&0\\0&0&9&0&27&0&57&66&0&78&0&147&0&0&360\\0&18&0&38&0&160&0&0&288&0&128&0&468&740&0\\1&0&41&0&57&0&193&210&0&186&0&453&0&0&1136\\9&0&57&0&66&0&210&309&0&246&0&546&0&0&1422\\0&33&0&63&0&288&0&0&531&0&240&0&843&1368&0\\3&0&36&0&78&0&186&246&0&254&0&498&0&0&1264\\0&15&0&25&0&128&0&0&240&0&112&0&372&616&0\\6&0&96&0&147&0&453&546&0&498&0&1134&0&0&2880\\0&45&0&120&0&468&0&0&843&0&372&0&1464&2232&0\\0&81&0&166&0&740&0&0&1368&0&616&0&2232&3577&0\\20&0&250&0&360&0&1136&1422&0&1264&0&2880&0&0&7392\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&13&13&27&160&193&309&531&254&112&1134&1464&3577&7392&3891&3996&9880&8778&2675\end{bmatrix}$

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$