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Properties

Label 1.6.N.1.1a
  
Name \(A(1,1)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $1$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_1\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$1$
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:$u\mapsto \mathrm{diag}(u,u, u, \bar u, \bar u, \bar u)$

Subgroups and supergroups

Maximal subgroups:
Minimal supergroups:$A(1,2)$, $J(A(1,1))$, $A(1,3)$, $A(3,1)$, $A(1,7)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $18$ $0$ $486$ $0$ $14580$ $0$ $459270$ $0$ $14880348$ $0$ $491051484$
$a_2$ $1$ $9$ $99$ $1215$ $15795$ $212139$ $2907981$ $40422321$ $567624915$ $8032730715$ $114370886169$ $1636488060381$ $23511619987101$
$a_3$ $1$ $0$ $164$ $0$ $47148$ $0$ $15454820$ $0$ $5361965980$ $0$ $1919987703504$ $0$ $701459496193236$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $9$ $18$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $99$ $54$ $216$ $486$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $164$ $1215$ $666$ $2754$ $1512$ $6318$ $14580$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $2070$ $15795$ $8640$ $4734$ $36450$ $19926$ $84564$ $46170$ $196830$ $459270$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $27252$ $212139$ $14904$ $115668$ $63126$ $494262$ $269244$ $146772$ $1154736$ $628398$ $2703132$ $1469664$ $6337926$ $14880348$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $47148$ $368442$ $2907981$ $200772$ $1580958$ $860058$ $6814692$ $468180$ $3701862$ $2012040$ $15995718$ $8682390$ $4715172$ $37594530$
$$ $20391588$ $88455402$ $47947788$ $208324872$ $491051484$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&8&0&9&0&28&36&0&28&0&63&0&0&160\\0&18&0&36&0&144&0&0&252&0&108&0&396&630&0\\8&0&82&0&108&0&332&414&0&350&0&792&0&0&2000\\0&36&0&74&0&304&0&0&540&0&236&0&864&1370&0\\9&0&108&0&153&0&468&576&0&504&0&1143&0&0&2880\\0&144&0&304&0&1280&0&0&2304&0&1024&0&3744&5920&0\\28&0&332&0&468&0&1450&1800&0&1576&0&3582&0&0&9088\\36&0&414&0&576&0&1800&2250&0&1962&0&4464&0&0&11376\\0&252&0&540&0&2304&0&0&4194&0&1872&0&6876&10890&0\\28&0&350&0&504&0&1576&1962&0&1740&0&3960&0&0&10112\\0&108&0&236&0&1024&0&0&1872&0&848&0&3096&4880&0\\63&0&792&0&1143&0&3582&4464&0&3960&0&9027&0&0&23040\\0&396&0&864&0&3744&0&0&6876&0&3096&0&11376&18000&0\\0&630&0&1370&0&5920&0&0&10890&0&4880&0&18000&28550&0\\160&0&2000&0&2880&0&9088&11376&0&10112&0&23040&0&0&59136\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&18&82&74&153&1280&1450&2250&4194&1740&848&9027&11376&28550&59136&30654&31515&78320&69696&20350\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.