Properties

Label 1.4.C.1.1a
  
Name \(\mathrm{U}(1)\times\mathrm{SU}(2)\)
Weight $1$
Degree $4$
Real dimension $4$
Components $1$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)\)
Component group \(C_1\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$4$
Components:$1$
Contained in:$\mathrm{USp}(4)$
Rational:yes

Identity component

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

Subgroups and supergroups

Maximal subgroups:
Minimal supergroups:$N(\mathrm{U}(1)\times\mathrm{SU}(2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $20$ $0$ $175$ $0$ $1764$ $0$ $19404$ $0$ $226512$
$a_2$ $1$ $2$ $6$ $20$ $76$ $312$ $1364$ $6232$ $29460$ $142952$ $708328$ $3570096$ $18251248$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $6$ $10$ $20$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $20$ $38$ $80$ $175$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $76$ $156$ $342$ $770$ $1764$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $312$ $678$ $1532$ $3528$ $8232$ $19404$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1364$ $3076$ $7112$ $16672$ $39480$ $94248$ $226512$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&1&0&1&0&1&0&1\\0&3&0&0&4&0&3&0&4&0\\1&0&3&3&0&3&0&5&0&3\\1&0&3&6&0&4&0&9&0&4\\0&4&0&0&10&0&8&0&12&0\\1&0&3&4&0&6&0&9&0&6\\0&3&0&0&8&0&10&0&11&0\\1&0&5&9&0&9&0&21&0&11\\0&4&0&0&12&0&11&0&21&0\\1&0&3&4&0&6&0&11&0&10\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&3&6&10&6&10&21&21&10\end{bmatrix}$

Event probabilities

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