Properties

Label 1.4.C.2.1a
  
Name \(N(\mathrm{U}(1)\times\mathrm{SU}(2))\)
Weight $1$
Degree $4$
Real dimension $4$
Components $2$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)\)
Component group \(C_2\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$4$
Components:$2$
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:$u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$

Component group

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

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $11$ $0$ $90$ $0$ $889$ $0$ $9723$ $0$ $113322$
$a_2$ $1$ $2$ $5$ $14$ $46$ $172$ $714$ $3180$ $14858$ $71732$ $354676$ $1786072$ $9127672$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $5$ $6$ $11$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $14$ $21$ $42$ $90$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $46$ $82$ $175$ $390$ $889$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $172$ $347$ $774$ $1774$ $4130$ $9723$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $714$ $1554$ $3572$ $8356$ $19768$ $47166$ $113322$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&1&0&0&0&1\\0&2&0&0&2&0&1&0&2&0\\1&0&2&1&0&2&0&2&0&2\\0&0&1&4&0&1&0&5&0&1\\0&2&0&0&5&0&4&0&6&0\\1&0&2&1&0&4&0&4&0&4\\0&1&0&0&4&0&6&0&5&0\\0&0&2&5&0&4&0&11&0&5\\0&2&0&0&6&0&5&0&11&0\\1&0&2&1&0&4&0&5&0&6\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&2&4&5&4&6&11&11&6\end{bmatrix}$

Event probabilities

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