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Properties

Label 1.6.N.12.4c
  
Name \(J(A(1,6)_2)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $12$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(D_6\)

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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:$D_6$
Order:$12$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{18}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{9}^{2} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{18}^{13} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{18}^{17} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{9}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{18}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $51$ $0$ $1310$ $0$ $39235$ $0$ $1250298$ $0$ $41033454$
$a_2$ $1$ $3$ $16$ $130$ $1438$ $18258$ $245373$ $3385581$ $47401794$ $669992062$ $9534545671$ $136396358211$ $1959439852867$
$a_3$ $1$ $0$ $18$ $0$ $4074$ $0$ $1294620$ $0$ $447174434$ $0$ $160017316488$ $0$ $58455958012892$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $16$ $6$ $23$ $51$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $18$ $130$ $65$ $255$ $141$ $573$ $1310$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $191$ $1438$ $770$ $428$ $3176$ $1749$ $7299$ $4005$ $16880$ $39235$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $2376$ $18258$ $1305$ $9924$ $5439$ $41981$ $22938$ $12556$ $97656$ $53281$ $227922$ $124173$ $533281$ $1250298$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $4074$ $31309$ $245373$ $17112$ $133414$ $72720$ $572547$ $39667$ $311427$ $169529$ $1341321$ $728844$ $396350$ $3148221$
$$ $1709162$ $7400253$ $4014360$ $17416602$ $41033454$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&7&0&1&0&4&0&0&13\\0&3&0&3&0&14&0&0&22&0&11&0&31&55&0\\2&0&11&0&7&0&28&42&0&24&0&65&0&0&167\\0&3&0&9&0&27&0&0&45&0&18&0&77&114&0\\0&0&7&0&21&0&41&42&0&50&0&103&0&0&238\\0&14&0&27&0&112&0&0&194&0&88&0&313&498&0\\1&0&28&0&41&0&130&144&0&132&0&303&0&0&761\\7&0&42&0&42&0&144&212&0&156&0&364&0&0&950\\0&22&0&45&0&194&0&0&357&0&155&0&571&911&0\\1&0&24&0&50&0&132&156&0&163&0&333&0&0&846\\0&11&0&18&0&88&0&0&155&0&78&0&252&411&0\\4&0&65&0&103&0&303&364&0&333&0&764&0&0&1916\\0&31&0&77&0&313&0&0&571&0&252&0&964&1495&0\\0&55&0&114&0&498&0&0&911&0&411&0&1495&2387&0\\13&0&167&0&238&0&761&950&0&846&0&1916&0&0&4938\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&11&9&21&112&130&212&357&163&78&764&964&2387&4938&2577&2680&6556&5846&1733\end{bmatrix}$

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$