Properties

Label 1.6.N.24.14b
  
Name \(J(A(6,2))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $24$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_2^2\times S_3\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$24$
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:$C_2^2\times S_3$
Order:$24$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0& 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $45$ $0$ $930$ $0$ $23625$ $0$ $684558$ $0$ $21406770$
$a_2$ $1$ $3$ $15$ $105$ $981$ $10983$ $136332$ $1795230$ $24478137$ $340930707$ $4812781770$ $68548275480$ $982420163484$
$a_3$ $1$ $0$ $16$ $0$ $2580$ $0$ $694030$ $0$ $227757404$ $0$ $80388358956$ $0$ $29263011957762$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $15$ $6$ $21$ $45$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $16$ $105$ $54$ $195$ $114$ $423$ $930$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $144$ $981$ $540$ $315$ $2079$ $1191$ $4644$ $2640$ $10440$ $23625$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $1548$ $10983$ $882$ $6108$ $3453$ $24627$ $13794$ $7770$ $56178$ $31338$ $128760$ $71526$ $296331$ $684558$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $2580$ $18348$ $136332$ $10278$ $75282$ $41829$ $313515$ $23316$ $173061$ $95814$ $725625$ $399396$ $220458$ $1684710$
$$ $924924$ $3922191$ $2148300$ $9153270$ $21406770$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&6&0&0&0&3&0&0&8\\0&3&0&3&0&12&0&0&15&0&9&0&21&33&0\\2&0&10&0&6&0&20&30&0&12&0&42&0&0&94\\0&3&0&7&0&20&0&0&27&0&13&0&48&64&0\\0&0&6&0&18&0&30&24&0&33&0&66&0&0&132\\0&12&0&20&0&76&0&0&114&0&56&0&180&272&0\\1&0&20&0&30&0&82&81&0&72&0&177&0&0&404\\6&0&30&0&24&0&81&129&0&81&0&198&0&0&498\\0&15&0&27&0&114&0&0&195&0&90&0&303&480&0\\0&0&12&0&33&0&72&81&0&95&0&180&0&0&436\\0&9&0&13&0&56&0&0&90&0&46&0&138&220&0\\3&0&42&0&66&0&177&198&0&180&0&417&0&0&996\\0&21&0&48&0&180&0&0&303&0&138&0&516&774&0\\0&33&0&64&0&272&0&0&480&0&220&0&774&1225&0\\8&0&94&0&132&0&404&498&0&436&0&996&0&0&2514\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&10&7&18&76&82&129&195&95&46&417&516&1225&2514&1317&1365&3310&2958&899\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$7/12$$0$$1/12$$0$$0$$1/2$
$a_1=0$$7/12$$7/12$$0$$1/12$$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$