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Properties

Label 1.6.N.24.10b
  
Name \(J_s(A(3,4))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $24$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3\times D_4\)

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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:$C_3\times D_4$
Order:$24$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0& \zeta_{12}^{7} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & -1 & 0 & 0 \\0 & 0& 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(1,4)_1)$, $J_s(A(3,2))$${}^{\times 2}$, $A(3,4)$
Minimal supergroups:$J(B(3,4;4))$, $J(B(3,4))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $45$ $0$ $950$ $0$ $24465$ $0$ $708498$ $0$ $21988890$
$a_2$ $1$ $2$ $12$ $97$ $978$ $11267$ $140669$ $1846392$ $25029118$ $346595491$ $4869404607$ $69103723922$ $987797588853$
$a_3$ $1$ $0$ $16$ $0$ $2644$ $0$ $713050$ $0$ $231277788$ $0$ $80947583556$ $0$ $29346093039010$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $12$ $6$ $21$ $45$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $16$ $97$ $54$ $197$ $114$ $429$ $950$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $146$ $978$ $550$ $317$ $2133$ $1213$ $4774$ $2700$ $10770$ $24465$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1588$ $11267$ $900$ $6292$ $3541$ $25445$ $14214$ $7974$ $58056$ $32326$ $133132$ $73892$ $306565$ $708498$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2644$ $18942$ $140669$ $10578$ $77764$ $43147$ $323645$ $24006$ $178613$ $98818$ $748257$ $411964$ $227350$ $1735572$
$$ $953498$ $4036767$ $2213400$ $9411570$ $21988890$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&2&4&0&1&0&4&0&0&8\\0&3&0&3&0&12&0&0&15&0&9&0&21&35&0\\1&0&9&0&8&0&21&27&0&16&0&45&0&0&98\\0&3&0&7&0&20&0&0&29&0&13&0&48&68&0\\1&0&8&0&15&0&28&31&0&28&0&66&0&0&136\\0&12&0&20&0&78&0&0&118&0&58&0&186&286&0\\2&0&21&0&28&0&83&90&0&74&0&180&0&0&420\\4&0&27&0&31&0&90&121&0&90&0&213&0&0&516\\0&15&0&29&0&118&0&0&203&0&90&0&317&496&0\\1&0&16&0&28&0&74&90&0&88&0&180&0&0&448\\0&9&0&13&0&58&0&0&90&0&50&0&144&230&0\\4&0&45&0&66&0&180&213&0&180&0&434&0&0&1024\\0&21&0&48&0&186&0&0&317&0&144&0&528&802&0\\0&35&0&68&0&286&0&0&496&0&230&0&802&1267&0\\8&0&98&0&136&0&420&516&0&448&0&1024&0&0&2574\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&9&7&15&78&83&121&203&88&50&434&528&1267&2574&1327&1366&3338&2964&877\end{bmatrix}$

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$$0$$5/12$$0$$0$$1/6$
$a_1=0$$7/12$$7/12$$0$$5/12$$0$$0$$1/6$
$a_3=0$$1/2$$1/2$$0$$1/3$$0$$0$$1/6$
$a_1=a_3=0$$1/2$$1/2$$0$$1/3$$0$$0$$1/6$