Properties

Label 1.4.B.2.1a
  
Name \(N(\mathrm{SU}(2)\times\mathrm{SU}(2))\)
Weight $1$
Degree $4$
Real dimension $6$
Components $2$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)
Component group \(C_2\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$6$
Components:$2$
Contained in:$\mathrm{USp}(4)$
Rational:yes

Identity component

Name:$\mathrm{SU}(2)\times\mathrm{SU}(2)$
$\mathbb{R}$-dimension:$6$
Description:$\left\{\begin{bmatrix}A&0\\0&B\end{bmatrix}:A,B\in\mathrm{SU}(2)\right\}$ 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:$\begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $5$ $0$ $35$ $0$ $294$ $0$ $2772$ $0$ $28314$
$a_2$ $1$ $1$ $3$ $7$ $23$ $76$ $287$ $1135$ $4769$ $20788$ $93695$ $433148$ $2046266$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $3$ $3$ $5$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $7$ $10$ $18$ $35$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $23$ $36$ $69$ $140$ $294$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $76$ $138$ $278$ $584$ $1260$ $2772$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $287$ $557$ $1167$ $2522$ $5565$ $12474$ $28314$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&1&0&0&0&0\\0&1&0&0&1&0&0&0&1&0\\0&0&2&0&0&0&0&1&0&2\\0&0&0&2&0&1&0&1&0&0\\0&1&0&0&2&0&1&0&2&0\\1&0&0&1&0&3&0&0&0&0\\0&0&0&0&1&0&2&0&1&0\\0&0&1&1&0&0&0&4&0&2\\0&1&0&0&2&0&1&0&3&0\\0&0&2&0&0&0&0&2&0&4\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&2&2&3&2&4&3&4\end{bmatrix}$

Event probabilities

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