Properties

Label 1.6.N.6.2c
  
Name \(A(3,2)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $6$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$6$
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:$C_6$
Order:$6$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $6$ $0$ $114$ $0$ $2860$ $0$ $82530$ $0$ $2566116$ $0$ $83103636$
$a_2$ $1$ $3$ $25$ $257$ $3009$ $38053$ $504399$ $6883551$ $95702577$ $1347086789$ $19124879235$ $273229589251$ $3922295417807$
$a_3$ $1$ $0$ $40$ $0$ $8680$ $0$ $2642580$ $0$ $899450552$ $0$ $320516445720$ $0$ $116957293298692$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $25$ $14$ $52$ $114$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $40$ $257$ $146$ $566$ $320$ $1266$ $2860$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $422$ $3009$ $1680$ $946$ $6822$ $3802$ $15588$ $8650$ $35790$ $82530$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $5092$ $38053$ $2832$ $20996$ $11618$ $87754$ $48308$ $26668$ $203208$ $111606$ $472028$ $258664$ $1099322$ $2566116$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $8680$ $65402$ $504399$ $36012$ $276038$ $151334$ $1175208$ $83108$ $642066$ $351312$ $2744802$ $1497238$ $817852$ $6423450$
$$ $3498964$ $15057966$ $8191932$ $35351568$ $83103636$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&6&8&0&6&0&13&0&0&28\\0&6&0&8&0&32&0&0&48&0&24&0&72&114&0\\2&0&20&0&24&0&66&78&0&60&0&148&0&0&348\\0&8&0&18&0&60&0&0&96&0&44&0&160&238&0\\3&0&24&0&35&0&88&108&0&92&0&209&0&0&496\\0&32&0&60&0&244&0&0&408&0&188&0&656&1020&0\\6&0&66&0&88&0&268&316&0&266&0&630&0&0&1548\\8&0&78&0&108&0&316&398&0&342&0&772&0&0&1928\\0&48&0&96&0&408&0&0&726&0&328&0&1168&1850&0\\6&0&60&0&92&0&266&342&0&310&0&672&0&0&1704\\0&24&0&44&0&188&0&0&328&0&156&0&528&836&0\\13&0&148&0&209&0&630&772&0&672&0&1553&0&0&3888\\0&72&0&160&0&656&0&0&1168&0&528&0&1952&3032&0\\0&114&0&238&0&1020&0&0&1850&0&836&0&3032&4806&0\\28&0&348&0&496&0&1548&1928&0&1704&0&3888&0&0&9924\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&20&18&35&244&268&398&726&310&156&1553&1952&4806&9924&5162&5269&13116&11664&3462\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/3$$0$$1/3$$0$$0$$0$
$a_1=0$$1/3$$1/3$$0$$1/3$$0$$0$$0$
$a_3=0$$0$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$0$$0$$0$$0$$0$$0$$0$