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Properties

Label 1.6.N.4.2a
  
Name \(A(2,2)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $4$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_2^2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$4$
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:$C_2^2$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(1,2)$${}^{\times 3}$
Minimal supergroups:$A(2,4)$, $J(A(2,2))$, $J_s(A(2,2))$, $B(3,2)$, $A(6,2)$, $B(1,4)_2$${}^{\times 2}$, $C(2,2)$, $A(2,6)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $6$ $0$ $126$ $0$ $3660$ $0$ $114870$ $0$ $3720276$ $0$ $122763564$
$a_2$ $1$ $3$ $27$ $309$ $3963$ $53073$ $727101$ $10105875$ $141907059$ $2008185033$ $28592728257$ $409122034335$ $5877905052117$
$a_3$ $1$ $0$ $44$ $0$ $11820$ $0$ $3864140$ $0$ $1340497564$ $0$ $479997013104$ $0$ $175364875326036$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $27$ $12$ $54$ $126$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $44$ $309$ $168$ $690$ $372$ $1578$ $3660$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $516$ $3963$ $2160$ $1194$ $9114$ $4986$ $21144$ $11520$ $49200$ $114870$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $6816$ $53073$ $3708$ $28920$ $15774$ $123570$ $67308$ $36732$ $288684$ $157116$ $675792$ $367332$ $1584450$ $3720276$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $11820$ $92112$ $727101$ $50208$ $395244$ $215022$ $1703682$ $116976$ $925470$ $502980$ $3998934$ $2170584$ $1178940$ $9398628$
$$ $5097960$ $22113882$ $11986632$ $52081092$ $122763564$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&4&12&0&10&0&15&0&0&40\\0&6&0&6&0&36&0&0&66&0&30&0&90&162&0\\2&0&22&0&24&0&86&102&0&80&0&198&0&0&500\\0&6&0&26&0&76&0&0&126&0&50&0&240&332&0\\3&0&24&0&45&0&108&150&0&144&0&285&0&0&720\\0&36&0&76&0&320&0&0&576&0&256&0&936&1480&0\\4&0&86&0&108&0&382&432&0&364&0&900&0&0&2272\\12&0&102&0&150&0&432&582&0&516&0&1110&0&0&2844\\0&66&0&126&0&576&0&0&1062&0&480&0&1686&2736&0\\10&0&80&0&144&0&364&516&0&492&0&984&0&0&2528\\0&30&0&50&0&256&0&0&480&0&224&0&744&1232&0\\15&0&198&0&285&0&900&1110&0&984&0&2259&0&0&5760\\0&90&0&240&0&936&0&0&1686&0&744&0&2928&4464&0\\0&162&0&332&0&1480&0&0&2736&0&1232&0&4464&7154&0\\40&0&500&0&720&0&2272&2844&0&2528&0&5760&0&0&14784\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&22&26&45&320&382&582&1062&492&224&2259&2928&7154&14784&7782&7911&19760&17520&5350\end{bmatrix}$

Event probabilities

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