Properties

Label 1.4.B.1.1a
  
Name \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)
Weight $1$
Degree $4$
Real dimension $6$
Components $1$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)
Component group \(C_1\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$6$
Components:$1$
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:$u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $10$ $0$ $70$ $0$ $588$ $0$ $5544$ $0$ $56628$
$a_2$ $1$ $2$ $5$ $14$ $44$ $152$ $569$ $2270$ $9524$ $41576$ $187348$ $866296$ $4092400$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $5$ $6$ $10$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $14$ $20$ $36$ $70$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $44$ $72$ $138$ $280$ $588$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $152$ $276$ $556$ $1168$ $2520$ $5544$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $569$ $1114$ $2334$ $5044$ $11130$ $24948$ $56628$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&1&0&0&0&1\\0&2&0&0&2&0&0&0&2&0\\1&0&2&1&0&2&0&1&0&2\\0&0&1&3&0&1&0&3&0&1\\0&2&0&0&4&0&2&0&4&0\\1&0&2&1&0&3&0&2&0&3\\0&0&0&0&2&0&4&0&2&0\\0&0&1&3&0&2&0&6&0&2\\0&2&0&0&4&0&2&0&6&0\\1&0&2&1&0&3&0&2&0&4\end{bmatrix}$

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

Event probabilities

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

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