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Properties

Label 1.6.N.9.2a
  
Name \(C(3,1)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $9$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3^2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$9$
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:$C_3^2$
Order:$9$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $54$ $0$ $1620$ $0$ $51030$ $0$ $1653372$ $0$ $54561276$
$a_2$ $1$ $1$ $11$ $135$ $1755$ $23571$ $323109$ $4491369$ $63069435$ $892525635$ $12707876241$ $181832006709$ $2612402220789$
$a_3$ $1$ $0$ $20$ $0$ $5244$ $0$ $1717220$ $0$ $595774060$ $0$ $213331967280$ $0$ $77939944022292$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $11$ $6$ $24$ $54$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $20$ $135$ $74$ $306$ $168$ $702$ $1620$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $230$ $1755$ $960$ $526$ $4050$ $2214$ $9396$ $5130$ $21870$ $51030$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $3028$ $23571$ $1656$ $12852$ $7014$ $54918$ $29916$ $16308$ $128304$ $69822$ $300348$ $163296$ $704214$ $1653372$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $5244$ $40938$ $323109$ $22308$ $175662$ $95562$ $757188$ $52020$ $411318$ $223560$ $1777302$ $964710$ $523908$ $4177170$
$$ $2265732$ $9828378$ $5327532$ $23147208$ $54561276$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&1&0&4&4&0&4&0&7&0&0&16\\0&2&0&4&0&16&0&0&28&0&12&0&44&70&0\\0&0&10&0&12&0&36&46&0&38&0&88&0&0&224\\0&4&0&10&0&32&0&0&60&0&28&0&96&154&0\\1&0&12&0&17&0&52&64&0&56&0&127&0&0&320\\0&16&0&32&0&144&0&0&256&0&112&0&416&656&0\\4&0&36&0&52&0&162&200&0&176&0&398&0&0&1008\\4&0&46&0&64&0&200&250&0&218&0&496&0&0&1264\\0&28&0&60&0&256&0&0&466&0&208&0&764&1210&0\\4&0&38&0&56&0&176&218&0&196&0&440&0&0&1120\\0&12&0&28&0&112&0&0&208&0&96&0&344&544&0\\7&0&88&0&127&0&398&496&0&440&0&1003&0&0&2560\\0&44&0&96&0&416&0&0&764&0&344&0&1264&2000&0\\0&70&0&154&0&656&0&0&1210&0&544&0&2000&3174&0\\16&0&224&0&320&0&1008&1264&0&1120&0&2560&0&0&6576\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&10&10&17&144&162&250&466&196&96&1003&1264&3174&6576&3406&3507&8704&7744&2270\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$8/9$$0$$8/9$$0$$0$$0$
$a_1=0$$8/9$$8/9$$0$$8/9$$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$