Properties

Label 1.6.D.2.1a
  
Name \(N(\mathrm{U}(1)\times \mathrm{USp}(4))\)
Weight $1$
Degree $6$
Real dimension $11$
Components $2$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{USp}(4)\)
Component group \(C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$11$
Components:$2$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)\times\mathrm{USp}(4)$
$\mathbb{R}$-dimension:$11$
Description:$\left\{\begin{bmatrix}A&0\\0&B\end{bmatrix}: A\in \mathrm{U}(1)\subseteq\mathrm{SU}(2),\ B\in\mathrm{USp}(4)\right\}$ Symplectic form:$\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
Hodge circle:$\mathrm{diag}(u,\bar u, u,u,\bar u,\bar u)$

Component group

Name:$C_2$
Order:$2$
Abelian:yes
Generators:$\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$\mathrm{U}(1)\times \mathrm{USp}(4)$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $12$ $0$ $114$ $0$ $1421$ $0$ $21195$ $0$ $358776$
$a_2$ $1$ $2$ $6$ $23$ $112$ $668$ $4656$ $36312$ $307075$ $2758452$ $25973562$ $254074659$ $2565728028$
$a_3$ $1$ $0$ $6$ $0$ $198$ $0$ $15216$ $0$ $1786372$ $0$ $265868064$ $0$ $46182504984$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $6$ $3$ $7$ $12$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $6$ $23$ $13$ $33$ $21$ $59$ $114$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $26$ $112$ $67$ $43$ $188$ $120$ $356$ $228$ $702$ $1421$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $143$ $668$ $90$ $410$ $259$ $1243$ $783$ $498$ $2457$ $1550$ $4971$ $3129$ $10206$ $21195$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $198$ $926$ $4656$ $580$ $2868$ $1790$ $9211$ $1124$ $5721$ $3572$ $18769$ $11646$ $7261$ $38771$
$$ $24010$ $80841$ $49950$ $169785$ $358776$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&1&0&0&0&0&0&0&0\\0&2&0&1&0&2&0&0&1&0&1&0&0&2&0\\1&0&3&0&1&0&2&3&0&0&0&2&0&0&2\\0&1&0&2&0&2&0&0&2&0&0&0&2&2&0\\0&0&1&0&4&0&2&1&0&1&0&5&0&0&2\\0&2&0&2&0&7&0&0&4&0&4&0&6&8&0\\0&0&2&0&2&0&6&3&0&2&0&6&0&0&8\\1&0&3&0&1&0&3&7&0&1&0&5&0&0&8\\0&1&0&2&0&4&0&0&7&0&1&0&6&8&0\\0&0&0&0&1&0&2&1&0&4&0&2&0&0&6\\0&1&0&0&0&4&0&0&1&0&6&0&4&5&0\\0&0&2&0&5&0&6&5&0&2&0&15&0&0&12\\0&0&0&2&0&6&0&0&6&0&4&0&13&10&0\\0&2&0&2&0&8&0&0&8&0&5&0&10&18&0\\0&0&2&0&2&0&8&8&0&6&0&12&0&0&25\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&2&4&7&6&7&7&4&6&15&13&18&25&13&13&18&15&6\end{bmatrix}$

Event probabilities

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